Geometry

New Formula for the Area of a General Quadrilateral Using Opposite Sides and Four Angles

Authors: Ryo Takayama
Comments: 5 Pages. In Japanese

In this paper, we derive an explicit formula for the area of a particular quadrilateral using only the lengths of a pair of opposite sides and the four interior angles. The formula is based on known geometric relationships and is presented after algebraic simplification and manipulation. Furthermore, we clarify the conditions under which the formula is valid and elucidate its geometric background. As a result, we demonstrate its applicability not only to convex and concave quadrilaterals but also to certain self-intersecting quadrilaterals.
Category: Geometry

Solid Angle Subtended by a Circular Plane at an Arbitrary Point in Space

Authors: Harish Chandra Rajpoot
Comments: 8 Pages. 2 Figures

This paper presents the derivation of analytical formulae for estimating the solid angle subtended by a circular plane at an arbitrary point in space. The proposed relations are also employed to determine the geometric parameters of the elliptical projection of a circular plane when observed from an off-center position, including the major axis, minor axis, and eccentricity. All mathematical expressions are derived using the Approximation Formula for the Solid Angle of Symmetrical Planes, as developed by the author and presented in his book Advanced Geometry. The resulting formulae provide a simple and effective approach for the approximate evaluation of solid angles and associated projection parameters in related geometric problems.
Category: Geometry

Analytical Computation of the Area of a Spherical Triangle

Authors: Harish Chandra Rajpoot
Comments: 4 Pages. 1 Figure

This paper presents the derivation of the general formula to compute the area of the spherical triangle having each side as a great circle arc on the spherical surface when (1) the aperture angle subtended by each of the three sides at the center of the sphere is known, and (2) the arc length of each of the three sides is known. These formulas are applicable to any spherical triangle to compute the area on the sphere.
Category: Geometry

Mathematical Analysis of Regular Spherical Polygons

Authors: Harish Chandra Rajpoot
Comments: 16 Pages. 4 Figures

In this paper, a new analytic formula governing all regular spherical polygons (having sides in the form of great circle arcs) has been derived to compute the important parameters such as solid angle, surface area & check the feasibility of the existence of a regular spherical polygon. A regular spherical polygon will exist only if it duly satisfies HCR's formula for a regular spherical polygon.
Category: Geometry

Mathematical Analysis of an Icosidodecahedron

Authors: Harish Chandra Rajpoot
Comments: 10 Pages. 2 Figures

In this work, the principal geometric parameters of an icosidodecahedron comprising 20 congruent equilateral triangular faces and 12 congruent regular pentagonal faces of equal edge length are analytically determined. These parameters include the normal distances and solid angles subtended by the faces, as well as the inner radius, outer radius, mean radius, surface area, and volume. The calculations are carried out using HCR’s formula for regular polyhedra, a generalized dimensional formulation applicable to all five Platonic solids, namely the regular tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron. The versatility of the proposed formula further enables its application to the analysis, design, and geometric modeling of truncated polyhedra.
Category: Geometry

Derivation of Great-Circle Distance Formula (Minimum Distance Between Any Two Points on Globe Given Latitude and Longitude)

Authors: Harish Chandra Rajpoot
Comments: 5 Pages, 3 Figures, Original Research

In this work, the great-circle distance formula is derived using hcr's inverse cosine formula. An analytic and the most generalized formula has been derived to accurately compute the minimum distance or great circle distance between any two arbitrary points on a sphere of finite radius which is equally applicable in the geometry of sphere. This formula is extremely useful to calculate the geographical distance between any two points on the globe for the given latitudes & longitudes. This formula is the most power tool which is applicable for all the distances on the tiny sphere as well as the large sphere like giant planet assuming them the perfect spheres.
Category: Geometry

Arrangement of Identical Touching Circles on a Spherical Surface Associated with Platonic Solids

Authors: Harish Chandra Rajpoot
Comments: 13 Pages, 10 Figures, Original Research

All the articles discussed and analyzed in this work are related to the five Platonic solids. A geometrical problem involving a finite number of identical circles mutually touching one another on the entire surface of a sphere of given radius is considered. Using elementary geometric relations together with tabulated parameters corresponding to the five Platonic solids, all important quantities, including the flat radius and arc radius of each circle, the total surface area covered by the circles, and the percentage of spherical surface coverage, are systematically evaluated. The derived parameters are useful for accurately drawing identical circles on a spherical surface and for the design and modeling of the five Platonic solids with identical flat circular faces.
Category: Geometry

Solid Angle Subtended by a Beam with Rectangular Profile Given Horizontal and Vertical Beam Angles

Authors: Harish Chandra Rajpoot
Comments: 6 Pages, 3 Figures, Original Research

In this work, previously reported analytical formulations are systematically discussed and analyzed to evaluate key beam parameters, including the solid angle subtended by a beam at a point source, the total area intercepted by the beam on a spherical surface, and the cone angle of an equivalent beam with a circular cross section. These formulations enable the transformation of a beam with a rectangular profile into an equivalent circular profile, and vice versa, while preserving the total radiation energy or luminous flux associated with the original beam emitted by a uniformly radiating point source. The presented analysis is particularly useful for radiometric applications involving the evaluation of radiation energy and directional intensity of uniform point sources, as well as for photometric applications concerned with luminous flux and luminous intensity in specified directions. Furthermore, the results facilitate the replacement of rectangular apertures with circular apertures, and conversely, without altering the total transmitted radiation energy or luminous flux. Consequently, the discussed formulations provide a valuable theoretical framework for the analysis and design of optical and beam-emitting devices, such as laser systems, based on uniformly radiating point sources.
Category: Geometry

Solid Angles Subtended by Archimedean Solids at Their Vertices

Authors: Harish Chandra Rajpoot
Comments: 3 Pages,

This paper presents a comprehensive tabulation of the solid angles subtended at the vertices of all thirteen Archimedean solids (convex uniform polyhedra). The solid angles are analytically evaluated using the standard solid angle formula in conjunction with tetrahedral decomposition. The resulting values constitute a consistent and complete set of reference data for the vertex geometry of the Archimedean solids, including the truncated tetrahedron, truncated hexahedron (cube), truncated octahedron, truncated dodecahedron, truncated icosahedron, cuboctahedron, icosidodecahedron, small rhombicuboctahedron, small rhombicosidodecahedron, snub cube, snub dodecahedron, great rhombicuboctahedron, and great rhombicosidodecahedron. These results provide useful quantitative tools for the geometric analysis and comparative study of uniform polyhedra.
Category: Geometry

Analytical Solutions for a Non-Uniform Tetradecahedron with Two Regular Hexagonal Faces and 12 Trapezoidal Faces

Authors: Harish Chandra Rajpoot
Comments: 11 Pages, 4 Figures

This paper presents an analytical derivation of the fundamental geometric parameters of a non-uniform tetradecahedron composed of two congruent regular hexagonal faces, twelve congruent trapezoidal faces, and eighteen vertices lying on a common circumscribed sphere. Using HCR’s Theory of Polygon, explicit closed-form expressions are obtained for the solid angles subtended at the center by the hexagonal and trapezoidal faces, as well as for the corresponding normal distances of these faces from the center. From these results, exact formulas for the inradius, circumradius, mean radius, total surface area, and enclosed volume of the polyhedron are systematically derived. The analytical framework developed herein provides a general and rigorous method for the geometric characterization of non-uniform polyhedra.
Category: Geometry

Mathematical Analysis of a Tetrahedron: Dihedral Angles Between Consecutive Faces and Solid Angle at Its Vertex Given the Apex Angles

Authors: Harish Chandra Rajpoot
Comments: 12 Pages, 3 Figures, Original Research

In this paper, analytical formulas are derived using HCR’s Inverse Cosine Formula in conjunction with HCR’s Theory of Polygon. These formulas provide a simple and practical method for computing the internal (dihedral) angles between consecutive lateral faces of an arbitrary tetrahedron at any of its four vertices, as well as the solid angle subtended by the tetrahedron at a vertex when the apex angles between consecutive lateral edges meeting at that vertex are known. The resulting expressions are fully generalized and can also be applied to configurations in which three faces meet at a vertex of various regular and uniform polyhedra, enabling the calculation of the solid angle subtended by the polyhedron at that vertex.
Category: Geometry

Solid Angles Subtended by the Platonic Solids (Regular Polyhedrons) at Their Vertices

Authors: Harish Chandra Rajpoot
Comments: 5 Pages, 5 Figures

In this paper, the solid angles subtended at the vertices by all five platonic solids (regular polyhedrons) have been calculated by the author using the standard formula of solid angle. These are the standard values of solid angles for all five platonic solids i.e. regular tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron useful for the analysis of platonic solids.
Category: Geometry

Mathematical Analysis of Spherical Rectangle

Authors: Harish Chandra Rajpoot
Comments: 6 Pages, 2 Figures

In this work, all the articles have been derived using simple geometry & trigonometry. All the formulas are very practical & simple to apply in the case of any spherical rectangle to calculate all its important parameters, such as solid angle, surface area covered, interior angles, etc. & also useful for calculating all the parameters of the corresponding plane rectangle obtained by joining all the vertices of a spherical rectangle by the straight lines. These formulas can also be used to calculate all the parameters of the right pyramid obtained by joining all the vertices of a spherical rectangle to the center of the sphere, such as normal height, angle between the consecutive lateral edges, area of the rectangular base, etc.
Category: Geometry

Geometric Analysis of a Spherical Triangle

Authors: Harish Chandra Rajpoot
Comments: 11 Pages, 3 Figures

In this paper, the principal geometric parameters of a spherical triangle are derived using elementary geometry and trigonometry. The resulting formulae are practical and straightforward to apply for computing key quantities such as the solid angle subtended at the center, the covered spherical surface area, and the interior angles. The analysis is further extended to the corresponding plane triangle obtained by joining the vertices of the spherical triangle with straight line segments, allowing for the evaluation of its geometric parameters. In addition, the derived relations are applied to the right pyramid formed by connecting the vertices of a spherical triangle to the center of the sphere, enabling analytical computation of parameters such as the normal height, angles between consecutive lateral edges, and the area of the base.
Category: Geometry

Mathematical Analysis of a Great Rhombicuboctahedron

Authors: Harish Chandra Rajpoot
Comments: 15 Pages. 5 Figures (Note by viXra Admin: An abstract in the article is required!)

In this work, the principal geometric parameters of the great rhombicuboctahedron, an Archimedean solid, are analytically derived. This polyhedron consists of 12 congruent square faces, 8 regular hexagonal faces, and 6 congruent regular octagonal faces of equal edge length, with 72 edges and 48 vertices lying on a circumscribed spherical surface. By applying HCR’s Theory of Polygon, explicit expressions are obtained for the solid angles subtended by each square, hexagonal, and octagonal face, as well as their corresponding normal distances from the center of the great rhombicuboctahedron. The formulation further yields the dihedral angles between adjacent faces, the inscribed radius, circumscribed radius, mean radius, surface area, and volume. The derived formulas are useful for the geometric analysis, design, and modeling of uniform (convex or non-convex) polyhedra.
Category: Geometry

Mathematical Analysis of a Great Rhombicosidodecahedron

Authors: Harish Chandra Rajpoot
Comments: 16 Pages. 5 Figures (Note by viXra Admin: An abstract in the article is required!)

In this paper, the principal geometric parameters of the great rhombicosidodecahedron, the largest Archimedean solid, are analytically derived. This polyhedron consists of 30 congruent square faces, 20 regular hexagonal faces, and 12 congruent regular decagonal faces, all of equal edge length, with 180 edges and 120 vertices lying on a circumscribed sphere. By applying HCR’s Theory of Polygon, explicit expressions are obtained for the solid angles subtended by each square, hexagonal, and decagonal face, along with their corresponding normal distances from the center of the great rhombicosidodecahedron. The derived formulation further yields the dihedral angles between adjacent faces, the inscribed radius, circumscribed radius, mean radius, surface area, and volume. The resulting formulas are useful for the geometric analysis, design, and modeling of uniform polyhedra.
Category: Geometry

A New Polyhedron Obtained by Truncation of Rhombic Dodecahedron

Authors: Harish Chandra Rajpoot
Comments: 19 Pages, 15 Figures

In this paper, a new convex polyhedron is introduced, obtained by systematically truncating all 24 edges of a rhombic dodecahedron such that the newly generated 24 congruent vertices lie exactly on a common spherical surface. The resulting truncated rhombic dodecahedron is a non-uniform convex polyhedron composed of 12 congruent rectangular faces, 6 congruent square faces, and 8 congruent equilateral triangular faces, with a total of 48 edges and 24 identical vertices. Using HCR’s Theory of Polygon, closed-form analytical expressions are derived for the radius of the circumscribed sphere passing through all vertices, the normal distances of the rectangular, square, and equilateral triangular faces from the center of the polyhedron, as well as its total surface area and enclosed volume. In addition, analytical formulae are obtained for the solid angles subtended at the center by each type of face, the dihedral angles between any two faces meeting at each of the 24 vertices, and the solid angle subtended by the truncated rhombic dodecahedron at each of its vertices.
Category: Geometry

Mathematical Analysis of Rhombic Dodecahedron Using Theory of Polygon

Authors: Harish Chandra Rajpoot
Comments: 11 Pages, 9 Figures

In this paper, a comprehensive mathematical analysis of the rhombic dodecahedron is presented, and closed-form analytical expressions are derived for a polyhedron consisting of 12 congruent rhombic faces, 24 edges, and 14 vertices lying on a common circumscribed sphere. Using HCR’s Theory of Polygon, generalized formulae are obtained for the face angles and diagonals of the rhombic faces, as well as for the radii of the circumscribed sphere, inscribed sphere, and midsphere. Analytical expressions are further derived for the total surface area and enclosed volume in terms of the edge length. In addition, the solid angles subtended at the vertices and the dihedral angles between adjacent faces are evaluated. It is also shown that this convex polyhedron can be constructed by assembling twelve congruent right pyramids with rhombic bases and a specific normal height.
Category: Geometry

Analytical Study of the Rhombicuboctahedron Using the Theory of Polygon

Authors: Harish Chandra Rajpoot
Comments: 18 Pages, 11 Figures

In this paper, the circumscribed radius of a rhombicuboctahedron is derived using an alternative geometrical approach based on HCR’s Theory of Polygon. An explicit analytical expression is obtained for the radius of the circumscribed sphere passing through all 24 congruent vertices of a rhombicuboctahedron with a given edge length. Using the same theoretical framework, closed-form formulae are subsequently derived for the normal distances of the equilateral triangular and square faces from the centre, the total surface area, and the enclosed volume. In addition, analytical expressions are presented for the solid angles subtended at the centre by each equilateral triangular face and each square face, the dihedral angles between any two faces meeting at a vertex, and the solid angle subtended by the rhombicuboctahedron at any of its 24 identical vertices.
Category: Geometry

Mathematical Analysis and Modeling of Pyramidal Containers, Right Pyramids, and Related Polyhedra

Authors: Harish Chandra Rajpoot
Comments: 32 Pages, 22 Figures

This paper presents generalized analytical formulas for computing key geometric parameters of pyramidal flat containers with regular polygonal bases, right pyramids, and related polyhedra. The derived expressions include the V-cut angle (obtained using HCR’s Theorem), edge length of the open end, lateral edge length, dihedral angle (derived using HCR’s Corollary), surface area, and volume. These formulas provide a unified framework for the systematic analysis and construction of such structures for arbitrary regular polygonal bases. The results are applied to pyramidal flat containers with square, regular pentagonal, hexagonal, heptagonal, and octagonal bases, as well as to right pyramids and bipyramidal polyhedra. The underlying geometric construction is based on the rotation or folding of two coplanar planes about their intersecting straight edges, as formulated in HCR’s Theorem, and is illustrated through practical paper models.
Category: Geometry

Two Mathematical Proofs of Bond Angle in a Regular Tetrahedral Structure

Authors: Harish Chandra Rajpoot
Comments: 9 Pages, 5 Figures

This paper presents a proof of the angle between any two bonds in a molecule possessing a tetrahedral structure, such as the methane molecule, in which all four σ-bonds (corresponding to four hydrogen atoms bonded to a central carbon atom) are equally inclined in three-dimensional space. The bond angle is derived using two independent approaches. The first method involves formulating the geometry of a right pyramid with a regular n-gonal base, while the second employs HCR’s formula for regular polyhedra. Both approaches lead to the same result, thereby providing a simple and rigorous geometric justification of the tetrahedral bond angle.
Category: Geometry

Mathematical Analysis of Non-Uniform Polyhedra with Two Regular N-Gonal Faces and 2n Trapezoidal Faces Inscribed in a Sphere

Authors: Harish Chandra Rajpoot
Comments: 12 Pages, 4 Figures

In this paper, all the important formulas have been generalized which are applicable to calculate the important parameters, of any non-uniform polyhedron having 2 congruent regular n-gonal faces, 2n congruent trapezoidal faces each with three equal sides, 5n edges & 3n vertices lying on a spherical surface, such as solid angle subtended by each face at the center, normal distance of each face from the center, inner radius, outer radius, mean radius, surface area & volume. These are useful for the analysis, designing & modeling of different non-uniform polyhedrons.
Category: Geometry

Mathematical Derivations of Inscribed & Circumscribed Radii for Three Externally Tangent Circles

Authors: Harish Chandra Rajpoot
Comments: 14 Pages, 6 Figures

In this paper, the generalized formula have been derived to analytically compute the radii of the circles inscribed and circumscribed by three mutually tangent circles of known radii. The formulation and analysis of three external tangent circles packed in the smallest rectangle and the intersection circles have also been done. . The analytic formula derived here can also be used in case of three tangent spheres in three dimensions. These formula are also used for calculating any of three radii if rest two are known & computing the length of common chord, angle of intersection & area of intersection of two intersecting circles. The generalized formula derived here can also be used to derive the recurrence relations for circle packing over a plane which can further be extended into 3 dimensions for sphere packings.
Category: Geometry

Mathematical Analysis of Trapezohedron/deltohedron Having 2n Congruent Right Kite Faces

Authors: Harish Chandra Rajpoot
Comments: 19 Pages, 8 Figures

The generalized formula, derived here, are equally applicable on any n-gonal trapezohedron having 2n congruent right kite faces, 4n edges & 2n+2 vertices lying on a spherical surface with a certain radius. These formula have been derived by the author Mr H.C. Rajpoot to analyse infinite no. of the trapezohedrons having congruent right kite faces simply by setting n=3,4,5,6,7,u2026u2026u2026u2026u2026u2026upto infinity, to analytically compute all the important parameters such as ratio of unequal edges, outer radius, inner radius, mean radius, surface area, volume, solid angles subtended by the polyhedron at its vertices, dihedral angles between the adjacent right kite faces etc. These formula are very useful for the analysis, modeling & designing of various n-gonal trapezohedrons/deltohedrons.
Category: Geometry

Mathematical Analysis of Spheres at Polyhedral Vertices, Face Filleting, and Sphere Packing in Right Pyramids and Platonic Solids

Authors: Harish Chandra Rajpoot
Comments: 28 Pages, 6 Figures

In this paper, the generalized formula has been derived that is applicable to locate any sphere, with a certain radius, resting in a vertex (corner) at which n no. of edges meet together at angle α between any two consecutive of them such as the vertex of platonic solids, any of two identical & diagonally opposite vertices of trapezohedron (uniform polyhedron with congruent right kite faces) & the vertex of right pyramid with regular n-gonal base. These are also useful for filleting the faces meeting at the vertex of the polyhedron to best fit the sphere in that vertex. These are used to determine the distance of the sphere from the vertex, the distance of the sphere from the edges, the fillet radius of the faces, etc. The formula has been generalized for packing the spheres in the vertices of right pyramids & all five platonic solids.
Category: Geometry

Mathematical Analysis of Disphenoid (Isosceles Tetrahedron)

Authors: Harish Chandra Rajpoot
Comments: 12 Pages. 4 Figures

In the present work, general analytical expressions are derived for the fundamental geometric properties of a disphenoid using three-dimensional coordinate geometry. Specifically, closed-form formulae are obtained for the volume and total surface area, as well as for the radii of the inscribed and circumscribed spheres, and solid angle subtended by disphenoid at its vertex. In addition, explicit coordinates of the four vertices are determined, together with the coordinates of the in-center, circum-center, and centroid of the disphenoid. The proof for in-center, circum-center and centroid to be coincident is presented and mathematical equation governing all disphenoids is also derived in a closed form.
Category: Geometry

Dihedral Angles Between the Polygonal Faces in Various Regular & Uniform Polyhedra: Platonic and Archimedean Solids

Authors: Harish Chandra Rajpoot
Comments: 29 Pages, 8 Figures, Original Research

This paper presents a set of practically oriented analytical formulas and tables compiled from geometric data of various convex uniform polyhedra, listing the dihedral angles between two faces with or without sharing a common edge. These formulas and tables are specifically intended to support the physical and computational construction of convex uniform polyhedral shells composed of different regular polygonal faces. By using the tabulated dihedral angles, wire-frame and shell models of polyhedra can be constructed efficiently by successively joining adjacent planar faces at their common edges with the correct angular orientation. The presented data are particularly useful for applications in geometric modeling, structural design, educational model fabrication, and the development of computational algorithms for polyhedral assembly. Overall, the tables provide a convenient and reliable reference for the practical realization and analysis of convex uniform polyhedral structures.
Category: Geometry

Solid Angle of Torus on Normal Geometrical Axis Using [a] Method of Concentric Cones

Authors: Harish Chandra Rajpoot
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please don't name title, equation/formula etc after the author's name) 3 Figures.

An analytical method is developed for the exact evaluation of the solid angle subtended by a torus at a point lying on its axis of symmetry, i.e., on the line perpendicular to the mid-plane of the torus and passing through its centre. The method is based on a geometric enclosure of the torus between two coaxial right circular cones with a common apex at the observation point and with axes coincident with the torus axis. It is shown that the solid angle associated with the torus equals the algebraic sum i.e. the difference between the solid angles subtended by the outer and inner bounding cones at the apex [1,2,3]. This construction leads to closed-form expressions for the solid angle as a function of the torus radii and the axial distance of the observation point, without recourse to surface integration. The resulting formulation provides a concise geometric characterization of toroidal visibility and is well suited for applications in geometric modeling, where exact angular measures are required, and in photometry and radiative transfer, particularly for the evaluation of irradiance, flux, and angular response of axially symmetric toroidal sources and apertures.
Category: Geometry

Trivolutions From Cubic Pencils: A Synthetic Viewpoint

Authors: Yerkebulan Bolat
Comments: 7 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

A pencil of plane conics induces an involution on any transversal line (Desargues’ Involution Theorem). For cubics, the analogous construction yields a natural degree-3 correspondence on a line, which we call a trivolution. Although the underlying mechanism is classical (Cayley—Bacharach and genus—one geometry), we give a synthetic treatment focused on the projective picture and (when relevant) circular cubics. We describe the induced degree—3 covering of a line, its monodromy via discriminants, and how this explains the rigidity of involution methods in degree 2.As concrete consequences we include a projective butterfly theorem for 2—torsion points and a cubic—method collinearity theorem of Sakhipov.
Category: Geometry

[ a New] Theorem for Dihedral Angles in a Regular N-Gonal Right Pyramid

Authors: Harish Chandra Rajpoot
Comments: 14 Pages. (Note by viXra Admin: Please don't name a theorem/formula/equation etc after the author's name)

In this paper, a theorem is formulated and proved that yields generalized closed-form expressions for the dihedral angle between any two arbitrary lateral faces of a regular n-gonal right pyramid. The dihedral angles are expressed in terms of the apex angle, defined as the angle between two adjacent lateral edges meeting at the apex. The proposed formulation establishes a direct analytical relationship between the edge geometry at the apex and the corresponding dihedral angles of the pyramid. Due to its generality, the theorem applies to all regular and uniform polyhedra whose vertex configuration coincides with that of a right pyramid, as well as to regular n-gonal right prisms with an arbitrary number of sides. The resulting formulas are useful for geometric modeling, construction of physical models, and the development of computational algorithms for the analysis of polyhedral structures and equally inclined sets of concurrent vectors in three-dimensional space.
Category: Geometry

Infinitely Algebraic Classes

Authors: Bin Wang
Comments: 12 Pages.

We show that on a complex projective manifold $X$, for $mathbb G=mathbb R$ or $mathbb Q$, a class in $H^{p, p}(X;mathbb Z)otimes mathbb G$ is represented by a convergent infinite series of integration currents over algebraic cycles with real coefficients. It implies that a Hodge class is represented by an algebraic cycle with rational coefficients.
Category: Geometry

The Ellipse's Dynamic Formation: the Four-Quarter Model and Intrinsic Balance the Four-Quarter Model and Intrinsic Balance

Authors: Erkan Gürkan
Comments: 6 Pages.

This study positions the ellipse not merely as a static definition, but as the embodiment of a dialogue between perception and mathematics, a living expression of balance. Through a poetic-philosophical narrative, the (F1 and F2) foci are revealed as symbols of complementary opposites, approaching and separating in a continuous act of creation. The research demonstrates that the ellipse functions not only as a static locus of points but as a self-regulating, dynamic structure governed by an "Internal Law of Balance" and a "Four-Quarter Mathematical Repetition Program." This structure manifests as a continuous unit value exchange between the axes, analytically detailing how the ellipse is cyclically regenerated across four symmetrical quarters. This approach expands the current understanding of the ellipse, positioning it not merely as a defined curve, but as a structure that reveals an intrinsic and continuous mathematical process that necessitates radical revisions in the field of geometry.
Category: Geometry

Fast Method for Solving the Minimal Overlapping Circle Expansion Problems

Authors: Andy Zhuang
Comments: 18 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

In this paper, we first introduce the Minimal Overlapping Circle Expansion (MOCE) problem. Solution to such a problem has real-world applications, such as finding the location to best communicate with a number of wireless devices, finding the quickest way for a number of vehicles to get to a rendezvous location etc. We present several algorithms to compute the solution with different running time and accuracy. The first uses enclosing square to get an approximate solution; the second only considers pair-wise overlap to approximate; the third uses the the results of the first two and a few other methods to speed up the computation. Our results show that (1) the approximate algorithm can be 1000 times faster than the accurate algorithm, and get to 99.9% of the correct value. (2) improvements can cut down the compute time by 50% for the accurate algorithm.
Category: Geometry

A Recursive Discrete-Rotation Framework for Waveform Reconstruction and Computational Geometry

Authors: Himanshu M. Chavda
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific references, list scientific references in a complete manner, and submit article written with AI assistance to ai.viXra.org)

"This paper introduces a novel recursive framework for approximating circular geometry and waveforms using discrete segment rotations. Traditional analytic methods, such as the classical circumference formula $C=2pi r$, rely on continuous functions that abstract away the geometric essence of rotation and introduce computational inefficiencies in discrete digital environments. By re-evaluating the 'Method of Exhaustion,' this work derives an original Discrete Radius Formula ($r = frac{C}{2n sin(Deltatheta/2)}$) that eliminates the inherent path-drift found in standard step-based systems. A recursive update algorithm is developed to reconstruct complex signals with $O(1)$ computational complexity, transforming global trigonometric evaluations into local iterative additions. Numerical validation demonstrates high-precision convergence to continuous limits, achieving an absolute error of approximately $7.97 times 10^{-9}$ at high resolution. The results establish a robust bridge between classical geometry and modern digital implementation, offering significant improvements in speed and accuracy for robotics, AI graphics, and signal processing."
Category: Geometry

Metamathematics: Spatial Dimensionality, Infinity, Zero, Pure Geometry

Authors: Dan Howitt
Comments: 9 Pages. © 2025 Dan Howitt

Via metamathematical analyses, proven is that mathematics are inaccurate that include discussion of "zero dimensional", "one dimensional", "two dimensional", "four dimensional" et seq, "infinity", and "zero": The concepts of each, except the concept "infinity", are arrived at via conceptual dissociation, and as such cannot represent facets of the universe, and the language of each are of linguistic alteration ("infinity" is an antonymic linguistic alteration), and as such are absent of mathematic meaning. Of the three dimensional, the three dimensions that comprise it are inextricable, such that none exist without the others: It cannot be constructed by, nor deconstructed into, "lower dimensions". Regarding the concept "pure geometry": It is a conceptual dissociation of the geometry of existents from the existents; and anything arrived at via conceptual dissociation cannot represent facets of the universe.
Category: Geometry

Simplified Master Formula for a Right Triangular Plane and Solid-Angle Corollaries in the Theory of Polygon

Authors: Harish Chandra Rajpoot
Comments: 15 Pages. Original Research Work

A generalized framework from HCR's Theory of Polygon is presented for computing the solid angle subtended by an arbitrary polygonal plane, regular or irregular, at any point in three-dimensional space. The approach is unified and systematic, relying on a single master formula derived for a right triangular plane. This formula is simplified and equivalently expressed in terms of inverse trigonometric functions, including arcsine, arccosine, and arctangent. The variation of the solid angle with respect to the orthogonal sides of the triangle and the distance of the observation point is illustrated graphically. In addition, several corollaries are established for the solid angle subtended by planar surfaces, both polygonal and non-polygonal, at different coplanar locations of the observation point. The results are derived using the standard formula for right-triangle geometry and the concept of the angle of vision for observation of two-dimensional figures.
Category: Geometry

Complete Mathematical Framework of the Hopf-Fibered 3-Sphere

Authors: Peter Kugelmann
Comments: 14 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This document presents a comprehensive mathematical framework for the Hopf-fibered 3-sphere $S^3$. We systematically derive the full geometric, topological, and analytic structure of $S^3$ equipped with its canonical round metric and Hopf fibration $S^1 hookrightarrow S^3 to S^2$. The framework establishes $S^3$'s uniqueness properties, rigidity theorems, and advanced geometric consequences emerging from combinations of its basic structures. All results are presented with complete proofs or references to standard mathematical literature. This article should be viewed as a comprehensive synthesis of canonical structures and standard results associated with the Hopf-fibered round 3-sphere, rather than a source of new classification theorems.
Category: Geometry

A Universal Notation for Plotting Lines

Authors: Faiz Qamar
Comments: 64 Pages.

When faced with curves that defy standard expressions, those that one may regard as, in simpler terms, chaotic or random, this may prompt a question: Is there a way to describe them uniformly, without patching new rules each time? Herein, we will explore one such attempt.
Category: Geometry

Higher-Degree Generalizations of the Method of Moving Points

Authors: Alex Wang
Comments: 14 Pages.

This report will be presenting a generalization to a previous method to solveEuclidean geometry problems which are parameterizable in one variable, known asthe Method of Moving Points". This method sometimes faces limitations, oftenunable to directly intersect or parameterize curves with degrees greater than onewithout tailored geometric analysis. We generalize this method through applying theVeronese map to be able to parameterize higher-degree moving curves, and extendthe notion of multiplicity of point-point concurrence to the degree of vanishing of adeterminant, to find effective bounds on the degree of higher-degree moving curves.Additionally, through an application of polynomial resultants, we bound the degreeof the locus of intersections of higher-degree moving curves. Finally, we present acollection of examples and applications of this theory to solving olympiad geometry problems involving moving circles and factoring their resultant bounds.
Category: Geometry

Structures of Finite and Infinite Types of Non-Compact Riemann Surfaces Via Fundamental Groups

Authors: Absos Ali Shaikh, Uddhab Roy
Comments: 32 Pages.

The purpose of this article is to introduce the notion of constructing any arbitrary finite and infinite types of non-compact hyperbolic Riemann surfaces via non-abelian fundamental groups equipped with various types of classical Schottky structures, with limit sets as uncountable sets (but not necessarily Cantor sets), emphasising the cases in which the surfaces are of infinite hyperbolic areas. At first, in this paper, we have fabricated three types of caconical non-compact infinite area Fuchsian polygons in the hyperbolic plane with different kinds of classical Schottky structures. After that, we have initiated a structure of an arbitrary finite type non-compact hyperbolic Riemann surface with genus, conformal holes, cusps, and funnel ends by using the canonical Fuchsian Schottky polygons. Furthermore, in this manuscript, we have proposed the notions of infinite types conformally compact and semi-conformally compact hyperbolic Riemann surfaces. In particular, we have introduced eight new and interesting types of infinite type hyperbolic Riemann surfaces (we call generalized flute surfaces) that are constructed from infinite sequences of infinite area hyperbolic pair of pants, each glued to the next along a common geodesic boundary with certain strategies.
Category: Geometry

Geometry and Trigonometry

Authors: Teo Banica
Comments: 400 Pages.

This is an introduction to plane geometry, angles and trigonometry, starting from zero or almost. We first discuss basic plane geometry, with the main results regarding the triangles explained. Then we get into trigonometry, with the basic properties of the sine, cosine and tangent discussed. We then go on a more advanced discussion, using affine and polar coordinates, then complex numbers, and with a look into trilinear coordinates too. Finally, we get into calculus methods, with an even more advanced study of the trigonometric functions, and with some applications discussed too.
Category: Geometry

Different Perspectives on Power of a Point

Authors: Yerkebulan Bolat
Comments: 25 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This paper is devoted to an in-depth study of the concept of the power of a point and its applications to the solution of olympiad-level geometry problems. The discussion encompasses the classical definitions of the power of a point, the radical axis, and the radical center, as well as their various generalizations --- including the interpretation of a point as a circle of zero radius, the notion of coaxial circles, and the linearity property of power differences. Detailed examples drawn from both national and international mathematical olympiads are presented to showcase the effectiveness of these methods in addressing both classical and modern geometric problems. Furthermore, the paper considers potential extensions and applications within a broader framework of elementary geometry, with particular emphasis on their value as practical tools for olympiad training and problem solving.
Category: Geometry

Radial Signatures and Pulsars

Authors: Roberto C. M. Navacchia
Comments: 8 Pages. In Portuguese; License: CC BY-NC (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This paper presents an alternative method of analysis for identifying pulsars using data from the High Time Resolution Universe Survey neutron star catalog and the construction of radial curves derived from statistical parameters of the observed radio signals, where, through a geometric approach, highlights their unique and distinct signature for each pulsar, which differs significantly from the curves produced by noise signals. The method seeks to offer the scientific community a complementary means of classification, opening up possibilities for the creation of a ‘Geometric Atlas of Pulsars’ useful in Astrophysics, Data Science, Quantum Mechanics, and especially in Analytical Geometry.
Category: Geometry

Characterization of Perspective Affinities that Preserve Arc Length and Curvature

Authors: Florian Gashi
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references)

In this work, a precise characterization is given of those perspective affine reflections that preserve both the arc length and the magnitude of curvature for any smooth planar curve. The result shows that the only non-trivial transformation with this property is the orthogonal reflectionwith respect to the given axis, i.e., the perpendicular affine symmetry with characteristic constant k=-1.
Category: Geometry

Doubling of a Cube (the Delian Problem) with Compass and Straightedge

Authors: Sigrid M. -L. Obenland
Comments: 6 Pages.

As is generally known, the side of a cube having twice the volume of a cube with volume 1 is 2^(1/3). It has been proven to be impossible to construct the cube having twice the volume of the initial cube with compass and straightedge (ruler) alone, when starting with a cube of 1 unit. The ancient Greeks devised several methods by using additional tools1, and later Albrecht Dürer has found a method of constructing the ratio of 1 to 2^(1/3) by a method wherein two sections of a certain straight line have to be made of equal length by trial and error2. I here present a simple new method of constructing the ratio of 1 to 2^(1/3) that uses a compass, a straightedge and properties of a normal parabola that can be drawn with compass and straightedge by tackling the problem in reverse order, i.e. starting from a cube having a side length of 2^(1/3) in an arbitrary system of units and, thus, a volume of 2 in the same system, and constructing the side length of a cube with half the volume in the arbitrary system of units. By using the intercept theorem this can be converted to any desired unit, such as cm. It should be noted that a length unit as displayed on the screen, such as 1 cm, may not be preserved when this document is printed on paper.
Category: Geometry

The Morse-Witten Complex

Authors: Joa Weber
Comments: 111 Pages. In German. 1993 Diploma thesis TU Berlin; English version: http://dx.doi.org/10.1016/j.exmath.2005.09.001

Construction of the Morse-Witten complex for closed Riemannian manifolds using dynamical systems: intersections of stable and unstable manifolds.

Konstruktion des Morse-Witten Komplex für geschlossene Riemannsche Mannigfaltigkeiten mittels Dynamischer Systeme: Schnittmengen von stabilen und instabilen Mannigfaltigkeiten.
Category: Geometry

A Note On The Second Radius of Curvature of the Ellipsoid

Authors: Abdelmajid Ben Hadj Salem
Comments: 7 Pages. In French.

In this note, we study the intersection of the ellipsoid of revolution with a plane orthogonal to the tangent plane at a point on the ellipsoid to show that the second radius of curvature of the ellipsoid (denoted $N$ as the great normal) coincides with the radius of curvature of the ellipse intersecting the plane and the ellipsoid at the point in question.
Category: Geometry

The Durer Polyhedron and a Composite Three-dimensional Model of a Symmetrical Geometric Object

Authors: Andrey V. Voron
Comments: 14 Pages.

Two variants of the geometry of the Durer polyhedron, conventionally named as "traditional" and "alternative", are investigated. The geometric properties of both variants of polyhedra are revealed. According to the results of the field experiment, it was revealed that the "alternative" version of the polyhedron has the property of complementarity.
Category: Geometry

Non-Differential Geometry: Mathematical Tools Abandoning the Differential Framework

Authors: Binyin Zheng
Comments: 27 Pages.

This paper proposes and systematically elaborates a novel geometric framework—non-differential geometry—whose core lies in entirely abandoning the reliance on smoothness ($C^1$ or higher continuity) required by traditional differential geometry, demanding only $C^0$ continuity for geometric objects. By introducing new mathematical tools based on limits and infinite series, non-differential geometry overcomes the smoothness constraints in calculating geometric quantities such as curvature. Furthermore, this paper constructs a unique ``integration tool" (distinct from classical integration theory) specific to non-differential geometry, providing a novel approach for analyzing geometric objects. Current research focuses on Euclidean space, but the theoretical framework itself is not confined to any specific spatial structure and is applicable across low- to high-dimensional spaces. Non-differential geometry completely resolves the contradiction between continuity and differentiability and offers potential support for theoretical innovations in fields such as physics.
Category: Geometry

The Cubic Curve Know as Witch of Agnesi

Authors: Rolando Zucchini
Comments: 7 Pages.

This type of cubic curve, in the United Kingdom know as the Witch of Agnesi, it is approached from a geometric, trigonometric and analytical point of view; using the scheme: model, algorithm, resolution, graph.
Category: Geometry

Vertex-Edge-Combinatorial Polytopes: A Class Defined by the Local Structure at Each Vertex

Authors: Miquel Piñol
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We introduce a class of geometric bodies, which we call vertex-edge-combinatorial polytopes, defined by a local structure in which each vertex is connected to a number of edges equal to the dimension of the body, and where any subset of those edges belongs to a face whose dimension equals the subset’s cardinality. These polytopes satisfy an empirical formula for the number of vertices, from which a general combinatorial expression for the number of faces can be deduced. The class includes simplices, hypercubes, and the dodecahedron, and excludes the octahedron, the icosahedron, and any higher-dimensional polytopes derived from them. In some cases where the formula diverges, such as the hexagonal tiling, an infinite regular structure does exist, although this is not always the case.
Category: Geometry

Platonic Solids as Structured Geometric Objects

Authors: Andrey V. Voron
Comments: 15 Pages.

The possibility of constructing Platonic solids from structural elements is shown — Kepler triangles (ratio of legs 1:√1.618..) and Fibonacci (ratio of legs 1:1.618...) — provided that the area of these elements remains unchanged. The number of elements (or pairs of elements) that make up the structure of the "tetrahedron", "octahedron", "cube" increases, thus, by two times, and the "icosahedron" — by five times in relation to the number of elements of the tetrahedron, while the indicator "area of all structural elements of the figure" and radius (r=3) remain unchanged inscribed in the Platonic bodies of the sphere. In addition, the area of the structural elements of two dodecahedra (S=√959325) is equal to the area of the structural elements of any 5 Platonic solids, for example, 5 tetrahedra (or octahedra, cubes, icosahedra) (S=√38373). The possibility shown is in accordance with the text of Plato's work Timaeus, according to which Platonic bodies can "transform into each other...".
Category: Geometry

A Right Triangle as an Adder of the Area of Regular Two-dimensional and Three-dimensional Geometric Shapes

Authors: Andrey V. Voron
Comments: 2 Pages. (Note by viXra Admin: Please don't use all CAPS in the article title)

The article shows the possibility of using a right triangle and the logic of the Pythagorean theorem to find the areas of regular two-dimensional and three-dimensional geometric shapes (in particular, for Platonic solids).
Category: Geometry

An Acute-angled Triangle as an Adder of the Volume of Regular Three-dimensional Geometric Shapes

Authors: Andrey V. Voron
Comments: 3 Pages. (Note by viXra Admin: Please don't use all CAPS in the article title)

The article shows the possibility (using an acute-angled triangle, the logic of the Pythagorean theorem) of finding the volume of regular three-dimensional geometric shapes based on the mathematical equation c=3√(a3+b3). A number of theorems have been formulated that complement the Pythagorean theorem.
Category: Geometry

Geometric Rationale for the Riemann Hypothesis

Authors: Shanzhong Zou
Comments: 4 Pages.

This paper constructs a "three-dimensional complex coordinate system" and proposes that the non-trivial zeros of the Riemann Zeta function lie on demarcation lines within the critical region. By infinitely projecting the singular point s=1 between complex planes in the 3D complex space and the standard complex plane, we derive regions where non-trivial zeros cannot exist. The boundaries of these regions (including Re(s) =1/2) are identified as potential loci for zeros.
Category: Geometry

Drawing Bent to Straight a New Solution to Geometric Problems

Authors: Hongfa Zi, Lei Zi
Comments: 6 Pages.

This article proposes an innovative method based on geometric transformation and limit construction, which successfully solves the problem of "drawing curves as straight". By introducing a linear function of fixed arc length and isosceles trapezoid, we prove that the transformation between a circle and an equal area square can achieve geometric equivalence in finite steps, and provide specific graphical steps and mathematical proof. This study reveals the limitations of traditional ruler drawing constraints and achieves accurate area conversion through an extended toolkit. Finally, the paper discusses the mathematical significance of this solution, including the algebraic treatment of the transcendental number and its supplementation to the Euclidean geometry system.
Category: Geometry

A Refined Symmetric Mean Integral Approach to Bounding the Perimeter of an Ellipse

Authors: Harun Abdul Rohman
Comments: 6 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We refine the symmetric mean integral method for estimating the perimeter of an ellipse by restricting the integration limits to [0,π/4]. This approach allows the application of the squeeze theorem by leveraging the extremal behavior of the integrand, yielding explicit upper and lower bounds. The results provide a foundation for further research to derive improved perimeter estimates for ellipses.
Category: Geometry

Integrabily And Cohomology And 6-Sphere

Authors: Jun Ling
Comments: 6 Pages.

We construct a differential from Nijenhuis tensor of any almost complex structure on a differentiable manifold, and show a relationship between the integrability of the almost complex structure and the cohomology of the manifold. For the case of 6-sphere, we first show that this form does not vanish for a special almost complex structure, and then show that this form does not vanish for any almost complex structure on the 6-sphere. Therefore all almost complex structures on 6-sphere are not integrable.
Category: Geometry

Translation of the Kimberling's Glossary Into Barycentrics ("le Glossaire de Pierre")

Authors: Pierre L. Douillet
Comments: 503 Pages.

Described at https://faculty.evansville.edu/ck6/encyclopedia/ETC.html by "if you're unsure of a term, click Glossary or Pierre Douillet's much expanded and very useful version".
Category: Geometry

The Method of Dividing the 60° Angle Into Three Equal Parts

Authors: Hongfa Zi, Hongyun Zi
Comments: 5 Pages.

The problem of dividing a 60 ° angle into three equal parts in modern mathematics has not yet been solved. This involves the infinite extension of this trigonometric function in a generalized perspective. After research, it was found that the solution is located between r and two-thirds of r. The former represents a curve, while the latter represents a horizontal line. This article aims to utilize the relationships between various shapes to divide the 60 ° angle into three equal parts, which can then be extended to any angle less than 180 ° between r and two-thirds of r.
Category: Geometry

Surface Area of the Mobius Strip

Authors: Richard J. Mathar
Comments: 6 Pages.

The (half) area of the surface of the Mobius strip is the expected product of the length of the circular spine times the width of the sweep line times a positive correction factor. The manuscript writes down this factor as a Taylor series of the ratio of width over circle radius; it approachesone if that ratio approaches zero.
Category: Geometry

A Geometric Algebra (GA) Solution to a Multiple-Tangency Problem

Authors: James A. Smith
Comments: 8 Pages.

Using GA's capacities for formulating reflections, we solve an interesting multiple-tangency problem. The solution is obtained in two ways, the easiest of which transforms the relevant reflections into a single rotation. The solution is validated via a GeoGebra worksheet.
Category: Geometry

Reducing Uncertainty Through the Application of Empirical Symbolism: Delos and Adelos

Authors: Pier Paolo Conti
Comments: 37 Pages.

Throughout history, humanity has sought to limit error in order to gain more precise insights and optimize its tools, be they physical or conceptual. This work explores the intricate relationship between geometry, mathematics, and the propagation of uncertainty, with particular attention to methods that either mitigate or avoid the amplification of uncertainty in mathematical, geometric, and applied contexts. The study begins by examining the foundational concepts underlying uncertainty in mathematical models, exploring how various geometric and topological structures can be leveraged to better understand and control the flow of uncertainty across different domains. A new number convention, the so-called empirical number, is introduced, enabling a more accessible assessment of uncertainty propagation. Particular focus is placed on those constructions that actively work to reduce uncertainty, offering insights into techniques that prevent the cascading effect of errors, a challenge often encountered in both theoretical and applied mathematics.Through the use of geometric principles, this work provides novel approaches to managing the inherent uncertainties in complex systems, ranging from simple algebraic problems to intricate applications. It highlights methods such as error propagation reduction, geometrically optimized models, and innovative adaptations to traditional methods that reduce computational or conceptual uncertainty. These techniques are of significant theoretical importance and are also crucial in practical applications, where precision and reliability are paramount, particularly in applied mathematics.By addressing both the philosophical and practical dimensions of uncertainty, this work paves the way for a refined understanding of the interaction between mathematical theory and real-world applications, offering tools to navigate complex, uncertain environments with greater confidence and precision. The exploration of these ideas provides new insights into the role of uncertainty in mathematical systems, particularly in those constructions that prioritize stability and error mitigation over mere approximation.The exposition begins with the definition of the empirical number, followed by the wide application of the Monte Carlo method to assess the functionality presented. It also recalls the theory of error propagation, addresses basic segment operations, and reexamines the Pythagorean theorem, which plays a crucial role in limiting error propagation. Further, the work discusses the applications of these ideas in calculus, specifically in the propagation of uncertainty through differentiation and integration.
Category: Geometry

Floerfolds and Floer Functions

Authors: Urs Frauenfelder, Joa Weber
Comments: 19 Pages.

In this article we introduce the notion of Floer function which has the property that the Hessianis a Fredholm operator of index zero in a scale of Hilbert spaces. Since the Hessian has a complicated transformation under chart transition, in general this is not an intrinsic condition. Therefore we introduce the concept of Floerfolds for which we show that the notion of Floer function is intrinsic.
Category: Geometry

Supportive Intersection (Without Analyticity)

Authors: Bin Wang
Comments: 22 Pages.

Part I. On a manifold, we apply the analysis in Part II below to define an intersection called supportive intersection for singular cycles. It has a topological descend to the cup-product. The result is motivated by a problem in cohomology theory. The tool is the notion of currents. A current which is a functional was first introduced by de Rham in 1955. Ever since then, currents played a central role in geometry. However, the part about the support has not been in focus. For instance, the cup-product has been extensively studied in the past. Yet, there is no adequate control on the support of cohomological classes. So, we would like to introduce the supportive intersection that will catch this property. The purpose of this paper is to build the foundation for exploring further.In the end, we'll give an application in this direction.

Part II. This is the technical foundation for the geometry above, but it may have an independent interest. It consists of a functional analysis on a very specific type of convergence of currents. In terms of classical analysis, it is an extension of mollifiers. Classically, mollifier is mostly applied as a smoother for a distribution which is usually viewed as a current of degree $0$. We extend the mollifier to currents where the degrees are positive.
Category: Geometry

On Curves in N-Dimensional Euclidean Spaces

Authors: Emanuels Grinbergs
Comments: 20 Pages. Translated (from Latvian) and submitted by Dainis Zeps

This work examines curves in n-dimensional spaces, as well as varieties contained in such spaces, with the main focus on curves and osculating linear and spherical varieties. Absolute dierential calculus a method almost exclusively used in n-dimensional dierential geometry in recent times is convenient in systematic terms because it enables the determination of all dierential invariantsusing classical techniques. However, it is cumbersome and inconvenient. Since I have sought to examine purely geometric properties and their relationships, I have consistently used vector analysis both independently and in conjunction with direct geometric reasoning, as wellas with Cartesian coordinates.
Category: Geometry

Introduction to Differential Geometry of Space Curves and Surfaces

Authors: Taha Sochi
Comments: 252 Pages.

This book is about differential geometry of space curves and surfaces. The formulation and presentation are largely based on a tensor calculus approach. It can be used as part of a course on tensor calculus as well as a textbook or a reference for an intermediate-level course on differential geometry of curves and surfaces. The book is furnished with an index, extensive sets of exercises and many cross references, which are hyperlinked for the ebook users, to facilitate linking related concepts and sections. The book also contains a considerable number of 2D and 3D graphic illustrations to help the readers and users to visualize the ideas and understand the abstract concepts. We also provided an introductory chapter where the main concepts and techniques needed to understand the offered materials of differential geometry are outlined to make the book fairly self-contained and reduce the need for external references.
Category: Geometry

Trisecting an Arbitrary Angle Problem Solved

Authors: Joseph Musonda
Comments: 5 Pages.

Trisecting an arbitrary angle using a straightedge and compass only has been one of the oldest mathematical geometric problem tracing back to Euclidian times. This problem was never solved until 1837 when it was proven impossible by French Mathematician Pierre Wantzel. As stated by Pierre Laurent Wantzel (1837), the solution of the angle trisection problem corresponds to an implicit solution of the cubic equation x cubed minus 3x minus 1 equals 0, which is algebraically irreducible, and so is the geometric solution of the angle trisection problem. The step-by-step instructions are given which are used to trisect an acute or obtuse angle( a reflex angle can be trisected by first trisecting 180 degrees angle and then the remaining one is trisected by using the authors steps, finally a 60 degrees is added to a corresponding other angle that results from trisection). This method explained here can trisect any arbitrary angle. Only a compass and straightedge is used. The formal proof is later given after a practical illustration. For practical sake and to prove the possibility of trisecting an arbitrary angle, the author used the most common angle of 60 degrees that mathematicians uses to explain the proof for impossibility. The author believes that this proof will act as a basis for further research in geometry in future.Keywords: trisecting, arbitrary angle, geometry, straightedge and compas, implicit solution
Category: Geometry

On the Spectral Flow Theorem of Robbin-Salamon for Finite Intervals

Authors: Urs Frauenfelder, Joa Weber
Comments: 79 Pages. 6 figures

In this article we consider operators of the form ∂sξ + A(s)ξ where s lies in an interval [−T , T ] and s → A(s) is continuous. Without boundary conditions these operators are not Fredholm. However, using interpolation theory one can define suitable boundary conditions for these operators so that they become Fredholm. We show that in this case the Fredholm index is given by the spectral flow of the operator path A.
Category: Geometry

Rainbow, Great Pyramid, Icosahedron: Mathematics in an Entertaining Way

Authors: Hans Hermann Otto
Comments: 6 Pages.

The rainbow angle of about 42° is comparable with the golden mean based angle between edge and base of the Great Pyramid. It allows bringing together different areas of knowledge in an amusing way using simple geometry besides laws of optics. The mathematical exercise may encourage students to understand spectacles of nature in a simple and didactical manner.
Category: Geometry

Cycloid, Semi-cycloid, Elliptic Cycloid and Elliptic Semi-cycloid

Authors: Tai-Choon Yoon
Comments: 15 Pages.

Cycloid, semi-cycloid, elliptic cycloid, and elliptic semi-cycloid are all types of trochoids, and parts of roulette. They all refer to curves that trace their paths as they roll along a straight line, a circular orbit, or an elliptical orbit. Unlike cycloids, semi-cycloids are the curves traced by a point on a bicycle wheel as it rolls around the bicycle axle. Elliptic cycloids and elliptic semi-cycloids are the curves traced by ellipses as they roll along a straight line, but ellipses do not roll as smoothly as circles on a straight line. I also investigated the curves traced by circles and ellipses as they roll along circular or elliptical paths.
Category: Geometry

The Star Flare Method

Authors: George William Tokarsky
Comments: 56 Pages.

We find a finite neighbourhood of the star flare (10,20).
Category: Geometry

The Equality of the Values of Area and Perimeter for Two-Dimensional Shapes, Volume and Area for Three-Dimensional Ones

Authors: Andrey V. Voron
Comments: 4 Pages.

Possible variants of the equality of the values of the area and perimeter of a number of two—dimensional figures (square, circle, rectangular, obtuse and equilateral triangles), volume and area - three-dimensional (Platonic bodies, cone, cylinder, pyramid and sphere) are considered.
Category: Geometry

Area of any Quadrilateral from Side Lengths

Authors: Alan Michael Gómez Calderón
Comments: 7 Pages.

In this paper we show that area of any quadrilateral can be estimated from the four lengths sides. With the Triangle Inequality Theorem and a novel provided diagonal's formula, the boundaries of quadrilateral diagonals are found. Finally Bretschneider's formula can be applied to find a set of possible areas.
Category: Geometry

A Translation of Euclid’s Elements Book One into Ewe

Authors: Makafui Apedo
Comments: 49 Pages. (Correction made by viXra Admin to conform with schoarly norm - Please conform!)

In this paper, I have translated Euclid’s Element Book One into the Ewe language of Ghana, Togo and the Republic Benin. It is my hope that this translation will inspire researchers across Africa to translate Euclid into their respective languages.
Category: Geometry

An Inequality for a Slice of Any Compact Set

Authors: Johan Aspegren
Comments: 3 Pages.

In this paper we will prove that if a compact $A$ in $R^n$ belongs to the unit ball in $R^n$, then $A$ has a slice of measure greater than a calculable constant times the measure of $A$. Our result is sharp.
Category: Geometry

Growth of Eigenvalues of Floer Hessians

Authors: Urs Frauenfelder, Joa Weber
Comments: 50 pages, 2 figures

In this article we prove that the space of Floer Hessians has infinitely many connected components.
Category: Geometry

Mathematics of Skateboard Quarter Pipe Construction

Authors: Thomas Günther
Comments: 14 Pages. Published in International Journal of Engineering and Applied Physics - Vol. 4, no. 3, pp. 1041—1054, 2024.

The present article provides a detailed mathematical treatise on the geometry of skateboard transition ramps. These usually form a circular segment shape and are used in skateboarding for transitioning from a horizontal plane to another angle of incline. Transitions play a crucial role in the flow of skateparks and are required, for example, for quarter pipes, mini ramps, vert ramps, or jump ramps. Skateboarding has been an Olympic sport since 2023, yet many publicly funded skateparks still do not meet the demands of the sport. On one hand, there is often a lack of willingness from authorities to engage with the athletes beforehand; on the other hand, the people involved in the planning process often lack the necessary experience and mathematical expertise to perfectly fulfill the users' needs. Thoughtful planning is particularly crucial for ramps with curved surfaces to ensure the flow of the skatepark. In addition to the mathematical analysis, source codes for programs are provided to make the calculations as convenient as possible for all users.
Category: Geometry

Classification of Pythagorean Triples Based on Differences in the Tangents of Their Triangles

Authors: Andrey V. Voron
Comments: 10 Pages.

The classification of Pythagorean triples is based on differences in the tangents of their triangles in a certain group, the value of which tends to one divided by a prime number and has the form of a fraction. The corresponding tables containing primitive and non-primitive Pythagorean triples are constructed in accordance with the proposed visual classification. Based on the presented classification, the concept of the "parent" Pythagorean triple is introduced — a primitive triple underlying a series (table) of Pythagorean triples derived from it. The first "parent" Pythagorean triple among the set of primitive tangent values of their triangles is defined by us as "119, 120, 169". It is shown that the first "parent" Pythagorean triple on the basis of the increase of the smaller legs of their triangles — 3, 4, 5 — can be considered as the initial structural unit of a two—dimensional planar construction of a figure — a right triangle, and the product of the numbers 3, 4, 5 — equal to 60 - it is advisable to consider, in turn, as a structural unit of a three-dimensional A three—dimensional figure is a parallelogram.
Category: Geometry

A New Look at Pythagorean Triples and Its Extension

Authors: Vladislav Koshchakov
Comments: 9 Pages. In Russian (An abstract is required in the article)

A geometric approach to the formation of Pythagorean triples is proposed, which allows not only to limit oneself to the second degree of the equation, but also to expand it to any value.
Category: Geometry

On the Geometry of Axes of Complex Circles of Partition

Authors: Berndt Gensel, Theophilus Agama
Comments: 16 Pages.

In this paper we continue the development of the circles of partition by introducing a certain geometry of the axes of complex circles of partition. We use this geometry to verify the condition in the squeeze principle in special cases with regards to the orientation of the axes of complex circles of partition.
Category: Geometry

Optimization of Energy Numbers Continued

Authors: Parker Emmerson
Comments: 12 Pages.

In this paper, we explore the properties and optimization techniques related to polyhedral cones and energy numbers with a focus on the cone of positive nxn semidefinite matrices and efficient computation strategies for kernels. In Part (a), we examine the polyhedral nature of the cone of positive semidefinite matrices, S^n , establishing that it does not form a polyhedral cone for due to its infinite dimensional characteristics. In Part (b), we present an algorithm for efficiently computing the kernel function K(x, x prime) = (latex formatting in wysiwig)on-the-fly, leveraging a polyhedral description of the convex hull generated by the feature mappings phi and phi prime. By restructuring the problem and using gradient-based optimization techniques, our approach minimizes memory usage and computational overhead, thus enabling scalable computation. Through examples and visualizations, we demonstrate the practical applications and efficiency of the proposed algorithm in optimizing these kernel computations.
Category: Geometry

There Are no Extraordinary Cycles in Collatz Conjecture

Authors: Chao Chen
Comments: 3 Pages.

In this paper, through the careful study and thinking of the Collatz conjecture, We give a short proof for any positive integer odd number x, when the odd number x is equal to 1, the cyclesequence is: 1, 1, 1, · · · , When the odd number x is greater than 1, there is no Extraordinary cyclic sequence.
Category: Geometry

Coraz Conjecture that there is no Non-Trivial Cycle

Authors: Chao Chen
Comments: 4 Pages. In Chinese

This paper studies whether there is a non-trivial cyclic sequence in Koraz's conjecture. Using the proof by contradiction method, it is obtained that for any positive integer odd number x0, when x0 is equal to 1, a trivial cycle will occur, and the cyclic sequence is: 1, 1, 1, · · · , the conclusion that no non-trivial cyclic sequence will occur when x0 is a positive integer odd number greater than 1.

本文研究了考拉兹猜想是否存在非平凡循环序列问题. 利用反证法, 获得了对于任意一个正整数奇数x0, 当x0 等于1 时产生平凡循环, 循环序列为:1, 1, 1, · · · , 当x0 为大于1 的正整数奇数时不产生非平凡循环序列的结论.
Category: Geometry

A Problem of Seiyo Sampo Revisited with 1/0=0

Authors: Hiroshi Okumura, Saburou Saitoh, Jun Ozone
Comments: 2 Pages.

We show that the result in [1] holds in the limiting cases using 1/0.
Category: Geometry

Beyond Archimedes: A Pythagorean Theorem Solution to Pi as Lone Unknown Triangle Side

Authors: G. Freeman
Comments: 18 Pages.

We explore the practical application of isoperimetric inequality (L² ≥ 4πA) to classical methods of circle measurement. Exampling Archimedes' n-gon approach, we compare it to a real-world kinematic scenario of a unit diameter circle rolling on a flat plane surface. Using annular geometry, we demonstrate that π can be derived algebraically by solving for the linear distance its centre travels per full revolution. Unlike exhaustive methods involving non-circular figures, our annular approach begins with isoperimetric equality by deriving a right triangle (with π its lone unknown side) & applying the Pythagorean theorem to it. This algebraic approach to π reveals unexpected yet significant connections between it and the golden ratio. We further explore more assumptions underlying 3.14159... discovering its embedment in an unbounded plane to be catastrophic & remedy with a bounded one. Finally, we close with a fresh new perspective on the notoriously unsolved Riemann Hypothesis problem. Our result suggests both a need for physical experimentation, as well as a need to re-evaluate the general reliability of non-circular methods in rigorously bounding and/or converging on the circle constant π.
Category: Geometry

A Solution to the "Snellius-Pothenot" Problem via Rotations and Reflections in Geometric Algebra (GA)

Authors: James A. Smith
Comments: 9 Pages.

Using GA's capacities for rotating and reflecting vectors, we solve the classic 2-D version of the Snellious-Pothenot surveying problem. The method used here provides two solutions, which can be averaged to better estimate the location of the unknown point P. A link to a GeoGebra worksheet of the solutions is provided so that the reader may test the validity of the method.
Category: Geometry

A Method for Calculating Euler Parallelepipeds Based on the Values of Pythagorean Triples

Authors: Andrey V. Voron
Comments: 4 Pages.

A non-formulaic method has been found for calculating Euler parallelepipeds of the second family based on the values of Pythagorean triples of Euler parallelepipeds of the first family, the largest common divisors. To do this, three triangles with integer values of the sides are allocated in the figure. Next, Pythagorean triples are determined from the obtained triangles by selecting the values of their greatest common divisors. These triples are entered in the table. By using a cross-arrangement in the table of two values (out of three) of Pythagorean triples (using the described algorithm of mathematical operations), the values of the three sides of the "derivative" Euler parallelepiped are calculated.
Category: Geometry

N-Complex Number, N-Dimensional Polar Coordinate and 4D Klein Bottle with 4-Complex Number

Authors: Kuan Peng
Comments: 21 Pages.

While a 3D complex number would be useful, it does not exist. Recently, I have constructed the N-complex number, which has demonstrated high efficiency in computations involving high-dimensional geometry. The N-complex number provides arithmetic operations and polar coordinates for N-dimensional spaces, akin to the classic complex number. In this paper, we will explain how these systems work and present studies on 4D Klein bottles and hyperspheres to illustrate the advantages of these systems.
Category: Geometry

Gaps and Overlappings

Authors: Volker W. Thürey
Comments: 5 Pages.

In the first part, we investigate the tiling of the plane by convex polygons, and we introduce many constants. At the end, we calculate one. We provide an example, where we cover the plane with convex 8-gons. In a second part, we take other curves and convex polygons.
Category: Geometry

Rectangle is Also a Parallelogram, and Square Too Has Parallel Sides

Authors: Arjun Dahal, Bipeen Singh Kunwar
Comments: 2 Pages.

We present a short theoretical proof to show that rectangle is also a parallelogram, and square too has parallel sides.
Category: Geometry

A Simple Approximation of pi

Authors: Wolfgang Sturm
Comments: 1 Page.

To approximate pi, the area of a circle segment is extrapolated to the full circle area and divided by the square radius.
Category: Geometry

The Method of Layered Separation of N-Cubes Along the Main Diagonal and Its Application in Geometry

Authors: Vladislav Koshchakov
Comments: 9 Pages.

The non-obvious possibility of decomposing any n-cube consisting of n-cubes (including visually perceptible 2D and 3D) into layers of these cubes sequentially placed along the main diagonal of this n-cube is presented. At the same time, the number of n-cubes in each layer turned out to be closely related to the numbers of Pascal's triangle. The coefficients of cutting each n-cube from the last (n-1) layers of them with a section of dimension (n-1)D are calculated. Examples are given that allow us to outline some ways to further explore this possibility. In Addition, the possibility of using this method to prove the tetrahedron volume formula without using infinitesimal methods is shown.
Category: Geometry

Quadrature of the Circle with Compass and Straightedge and a Surprising Result for the Value of π in π R^2

Authors: Sigrid Obenland
Comments: 3 Pages.

It is general believe and deemd to be proven that the value of π in the formula for calculating the area of a circle; i.e. π r^2, is identical to the value of π in theformula for calculating the circumference of a circle; i.e. 2π r, which is irrational.Therefore, quadrature (or squaring) of the circle with compass and straightedge (or ruler) has been deemed to be impossible. We show that this was a prejudice and proof that quadrature is possible and clearly delivers π = 3 in the formula π r2 for calculating the area of the circle. We also show a physical experiment thatunambiguously proofs this result.
Category: Geometry

Deriving Curves from Points in the Cartesian Plane

Authors: Kohji Suzuki
Comments: 20 Pages.

We derive curves from predetermined points in the Cartesian plane and obtain the elliptic.
Category: Geometry

A Geometric Algebra Solution to a Contest Problem

Authors: James A. Smith
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references!)

We show how to use rotations of vectors in GA to solve the following problem: "The following are known about a triangle: The ratio of the lengths of two sides; the angle formed by those sides; and the length of that angle’s bisector. Find the length of the side opposite that angle."
Category: Geometry

Constructing Legendrian Links from Chiral Reeb Chords

Authors: Ryan J. Buchanan
Comments: 8 Pages.

Some theorems on the construction of links (defined herein) from Reeb chords are established and proved. We relate these concepts to physics by constructing an arbitrary spacetime from a background knot using Reidemeister moves.
Category: Geometry

A Geometric Algebra Solution to a "Divided Triangle" Problem

Authors: James A. Smith
Comments: 6 Pages.

We show how to use properties of Geometric Algebra bivectors to solve the following problem: "A triangle is divided into three smaller triangles and a quadrilateral by two lines drawn from vertices to the opposite sides. Given only the areas of the three triangles, find the area of the quadrilateral."
Category: Geometry

Complementary Elements of The Theory of The Surfaces

Authors: Abdelmajid Ben Hadj Salem
Comments: 63 Pages. In French

In this fascicle, we give some complementary elements concerning the theory of surfaces like the lines of curvature, the asymptotic lines.
Category: Geometry

The Physical Mathematics and Geometry of Dialectical Materialism Versus the Euclidean "Mathematics" and "Geometry" of Philosophical Idealism

Authors: Ángel Blanco Nápoles
Comments: Spanish, Russian and English versions. 14 pages each. 14 drawings.

This work reveals the antagonistic and unsolvable internal contradictions of the Euclidean "geometry" of Philosophical Idealism with itself and with the mathematics that derives from it, also providing the definitive solution of Dialectical Materialism, which not only solves the aforementioned contradictions, but many others in the field of mathematics, physics, astronomy and cosmology.
Category: Geometry

Supportive Intersection

Authors: Bin Wang
Comments: 19 Pages.

Let $X$ be a differentiable manifold. Let $mathscr D'(X)$ be the space of currents, and $S^infty(X)$ the Abelian group freely generated by $C^infty$ cells, i.e. the maps from polyhedrons to $X$ can be extended defferentiablelly to a neighborhoods of the polyhedrons. In this paper, we define a bilinear map begin{equation}begin{array}{ccc}S^infty(X)times S^infty(X) &ightarrow & mathscr D'(X) (sigma_1, sigma_2) &ightarrow & [sigma_1wedge sigma_2]end{array}end{equation} such that1) the support of $[sigma_1wedge sigma_2]$ is contained in the set-intersection of the supports of $sigma_1, sigma_2$; 2) if $sigma_1, sigma_2$ are closed, $[sigma_1wedge sigma_2]$ is also closed and its cohomology class is the cup-product of the cohomology classes of $sigma_1, sigma_2$. We call the current $[sigma_1wedge sigma_2]$ the supportive intersection of $sigma_1, sigma_2$.
Category: Geometry

More Than 'The Chromatic Number of the Plane'

Authors: Volker W. Thürey
Comments: 4 Pages.

We generalize the famous `Chromatic Number of the Plane'. For every finite metric space we define a similar question. We show that 15 colors suffice togenerate a coloring of the plane without monochromatic distances 1 or 2.
Category: Geometry

An Analytical Treatment of Rotations in Euclidean Space

Authors: Archan Chattopadhyay
Comments: 8 Pages.

An analytical treatment of rotations in the Euclidean plane and 3-dimensional Euclidean space, using differential equations, is presented. Fundamental geometric results, such as the linear transformation for rotations, the invariance of the Euclidean norm, a proof of the Pythagorean theorem, and the existence of a period of rotations, are derived from a set of fundamental equations. Basic Euclidean geometry is also constructed from these equations.
Category: Geometry

Incenter-Orthocenter-Centroid Triangle Operator

Authors: Yuly Shipilevsky
Comments: 4 Pages. (Note by viXra Admin: Please list scientific references in future submissions)

We consider a mapping from the set of triangles on the same plane onto its- elf, wherein each triangle is being mapped to the triangle, having vertices, which are the orthocenter, the centroid and the incenter of the parent triangle and we consider the corresponding inverse mapping as well.
Category: Geometry

Local Gluing

Authors: Urs Frauenfelder, Joa Weber
Comments: 45 Pages. 3 Figures. Bull. Braz. Math. Soc. (N.S.) 56, 44 (2025). https://doi.org/10.1007/s00574-025-00464-5

In the local gluing one glues local neighborhoods around the critical point of the stable and unstable manifolds to gradient flow lines defined on a finite time interval [−T,T] for large T. If the Riemannian metric around the critical point is locally Euclidean, the local gluing map can be written down explicitly. In the non-Euclidean case the construction of the local gluing map requires an intricate version of the implicit function theorem.In this paper we explain a functional analytic approach how the local gluing map can be defined. For that we are working on infinite dimensional path spaces and also interpret stable and unstable manifolds as submanifolds of path spaces. The advantage of this approach is that similar functional analytical techniques can as well be generalized to infinite dimensional versions of Morse theory, for example Floer theory.A crucial ingredient is the Newton-Picard map. We work out an abstract version of it which does not involve troublesome quadratic estimates.
Category: Geometry

Infinity-Cosmoi and Fukaya Categories for Lightcones

Authors: Ryan J. Buchanan
Comments: 17 Pages.

We propose some questions about Fukaya categories. Given a class of isomorphisms $0 sim tau$, where $tau$ represents the truth value of a particle, and $0$ is a $0$ object in a Fukaya category, what are its spectral homology theories? This is a variation on the works of P. Seidel and E. Riehl.
Category: Geometry

One Tile Suffices

Authors: Volker W. Thürey
Comments: 3 Pages.

We have found for all k larger than two a possibility to tile the plane completely with k-gons. We use infinite many copies of a single tile. The proofs are not by written words, but by pictures. Amongst others, we use the well-known tiling with hexagons. We show for k larger than 4 new ways to cover the plane.
Category: Geometry

Complex Curvature and Complex Radius

Authors: Cavași Ioan Abel
Comments: 3 Pages.

I define the notions of complex curvature and complex radius and prove that one of these complex numbers is exactly the inverse of the other.
Category: Geometry

Exact Sines and Cosines Including a Small Table

Authors: Claude Michael Cassano
Comments: 12 Pages.

Using half angle formulas and other trigonometric identities sines and cosines for exact angles may be established and such table produced.
Category: Geometry

Finding Rational Points of Circles, Spheres, Hyper-Spheres via Stereographic Projection and Quantum Mechanics}

Authors: Carlos Castro
Comments: 14 Pages.

One of the consequences of Fermat's last theorem is the existence of a countable infinite number of rational points on the unit circle, which allows in turn, to find the rational points on the unit sphere via the inverse stereographic projection of the homothecies of the rational points on the unit circle. We proceed to iterate this process and obtain the rational points on the unit $S^3$ via the inverse stereographic projection of the homothecies of the rational points on the previous unit $S^2$. One may continue this iteration/recursion process ad infinitum in order to find the rational points on unit hyper-spheres of arbitrary dimension $S^4, S^5, cdots, S^N$. As an example, it is shown how to obtain the rational points of the unit $ S^{24}$ that is associated with the Leech lattice. The physical applications of our construction follow and one finds a direct relation among the $N+1$ quantum states of a spin-N/2 particle and the rational points of a unit $S^N$ hyper-sphere embedded in a flat Euclidean $R^{N+1}$ space.
Category: Geometry

Equality of the Values of the Area and Perimeter of a Number of Two—dimensional Figures, Volume and Area

Authors: Andrey VORON
Comments: 3 Pages.

Possible variants of the equality of the values of the area and perimeter of a number of two—dimensional figures (square, circle, rectangular, obtuse and equilateral triangles), volume and area - three-dimensional (Platonic bodies, cone, cylinder, pyramid and sphere) are considered.
Category: Geometry

Levelwise Accessible Equivalence Classes of Fibrations

Authors: Ryan J. Buchanan
Comments: 6 Pages.

For a space of directed currents, geometric data may be accessible by means of a certain $frac{1}{n}$-type functor on a sheaf of germs. We investigate pointwise periodic homeomorphisms and their connections to foliations.
Category: Geometry

Linear-Time Estimation of Smooth Rotations in ARAP Surface Deformation

Authors: Mauricio Cele Lopez Belon
Comments: 10 Pages.

In recent years the As-Rigid-As-Possible with Smooth Rotations (SR-ARAP [5]) technique has gained popularity in applications where an isometric-type of surface mapping is needed. The advantage of SR-ARAP is that quality of deformation results is comparable to more costly volumetric techniques operating on tetrahedral meshes. The SR-ARAP relies on local/global optimisation approach to minimise the non-linear least squares energy. The power of this technique resides on the local step. The local step estimates the local rotation of a small surface region, or cell, with respect of its neighbouring cells, so a local change in one cell’s rotation affect the neighbouring cell’s rotations and vice-versa. The main drawback of this technique is that the local step requires a global convergence of rotation changes. Currently the local step is solved in an iterative fashion, where the number of iterations needed to reach convergence can be prohibitively large and so, in practice, only a fixed number of iterations is possible. This trade-off is, in some sense, defeating the goal of SR-ARAP. We propose a linear-time closed-form solution for estimating the codependent rotations of the local step by solving a sparse linear system of equations. Our method is more efficient than state-of-the-art since no iterations are needed and optimised sparse linear solvers can be leveraged to solve this step in linear time. It is also more accurate since this is a closed-form solution. We apply our method to generate interactive surface deformation, we also show how a multiresolution optimisation can be applied to achieve real-time animation of large surfaces.
Category: Geometry

Totally Lossless Projections

Authors: Ryan J. Buchanan
Comments: 6 Pages.

In this brief note, we discuss projective morphisms of perfect categories which are fully faithful, i.e., totally lossless.
Category: Geometry

Application of Rational Representation in Euclidean Geometry

Authors: Bo Zhang
Comments: 310 Pages. In Chinese

This book focuses on the application of rational representations to plane geometry. Most plane geometry objects, such as circles, triangles, quadrilaterals, conic curves, and their composite figures, can be represented almost exclusively in terms of rational parameters, which makes the process of computation and proof straightforward.
Category: Geometry

Frenet's Trihedron of the Second Order

Authors: Abel Cavași
Comments: 9 Pages.

Based on the remarkable property of the Darboux vector to be perpendicular to the normal, I define a new trihedron associated with curves in space and prove that this trihedron also satisfies Frenet's formulas. Unlike the previous paper, where I used the trigonometric form of Frenet's formulas for simplicity, in this paper I construct a proof based only on curvature and torsion, respectively, darbuzian and lancretian.
Category: Geometry

Triedrul Lui Frenet de Ordinul al Doilea (Frenet's Trihedron of the Second Order)

Authors: Abel Cavași
Comments: 9 Pages. In Romanian

Bazându-mă pe proprietatea remarcabilă a vectorului lui Darboux de a fi perpendicular pe normală, definesc un nou triedru asociat curbelor din spațiu și demonstrez că și acest triedru satisface formulele lui Frenet. Spre deosebire de lucrarea anterioară, unde am folosit pentru simplitate forma trigonometrică a formulelor lui Frenet, în această lucrare construiesc o demonstrație bazată doar pe curbură și torsiune, respectiv, pe darbuzian și lancretian.

Based on the remarkable property of the Darboux vector to be perpendicular to the normal, I define a new trihedron associated with curves in space and prove that this trihedron also satisfies Frenet's formulas. Unlike the previous paper, where I used the trigonometric form of Frenet's formulas for simplicity, in this paper I construct a proof based only on curvature and torsion, respectively, darbuzian and lancretian.
Category: Geometry

The Geometric Collatz Correspondence

Authors: Darcy Thomas
Comments: 17 Pages.

The Collatz Conjecture is a math puzzle that has stumped experts and beginners for a long time. Atfirst glance, it seems simple, but looks can be deceiving. It has become one of the most famousunsolved problems in math. One of the biggest challenges is that there’s nothing quite like it in terms of comparison. This makes it hard for many to figure out where to start when trying to analyze and explore the conjecture. However, in my journey to understand this puzzle, I’ve found two exciting links: one connects the Collatz orbits for odd numbers with a certain type of triangle called a PrimitivePythagorean Triple, and the other ties it to another famous number called the golden ratio. On the way to explain these connections, we develop a framework for treating the Collatz Function as aprocess that maps integers into a space similar to computer RAM (Randomly Accessible Memory).Each orbit can be represented as a unique location in "Collatz Memory" which is specified by a tupleof three numbers: the stopping time, the page, and the offset into the page. This gives us a new wayto investigate the inner structure of Collatz Orbits.
Category: Geometry

Quick Tiling

Authors: Volker W. Thürey
Comments: 4 Pages.

In the first part, we tile the plane with k-gons for natural numbers k which have the rest three if we devide it by four. The proof is by pictures. In a second part, we extend the result to all natural numbers larger than two. The foundation is the tiling of the plane by rectangles or hexagons. We use at most two different tiles for the covering.
Category: Geometry

Electrostatic Polyhedron

Authors: Domenico Oricchio
Comments: 7 Pages.

I minimize the N charges electric potential on a sphere, the minimum potential optimize the distance between the charges and it is possible to obtain the polyhedrons from the N charge positions
Category: Geometry

Sines and Cosines of Any Angles May be Determined to Any Degree of Accuracy and a Relativistic Non-Doppler Effect

Authors: Claude Michael Cassano
Comments: 3 Pages.

The unit circle yields an exact half-angle formulas for sines, cosines, tangents, etc. of ANY angles, with examples.
Category: Geometry

Every Convex Pentagon Has Some Vertex Such that the Sum of Distances to the Other Four Vertices is Greater Than Its Perimeter

Authors: Juan Moreno Borrallo
Comments: 4 Pages.

In this paper it is solved the case n = 5 of the problem 1.345 of the Crux Mathematicorum journal, proposed by Paul Erdös and Esther Szekeres in1988. The problem was solved for n ≥ 6 by János Pach and the solution published by the Crux Mathematicorum journal, leaving the case n = 5open to the reader. In september 2021, user23571113 posed this problem at the post https://math.stackexchange.com/questions/4243661/prove-thatfor-one-vertex-of-a-convex-pentagon-the-sum-of-distances-to-the-othe/4519514#4519514,and it has finally been solved.
Category: Geometry

One Piece

Authors: Volker W. Thürey
Comments: 4 Pages.

At first we tile the plane by 8-gons. Then we present a way to tile the plane by k-gons for a every fixed k for all natural numbers k larger than two. We use an infinite number of equal tiles to cover the plane.
Category: Geometry

Triedrul Lui Frenet de Ordinul al Doilea

Authors: Cavași Ioan Abel
Comments: 7 Pages.

Bazându-mă pe proprietatea remarcabilă a vectorului lui Darboux de a fi perpendicular pe normală, definesc un nou triedru asociat curbelor din spațiu și demonstrez că și acest triedru satisface formulele lui Frenet. Spre deosebire de lucrările anterioare, unde am folosit pentru simplitate forma trigonometrică a formulelor lui Frenet, în această lucrare construiesc o demonstrație bazată doar pe curbură și torsiune, respectiv, pe darbuzian și lancretian.
Category: Geometry

Evrende Dik Açı Kavramı Üzerine (On the Concept of Right Angles in the Universe)

Authors: Mesut Kavak
Comments: 8 Pages.

Alan, herhangi bir boyutta ve yönde hareketin oluşması için zorunlu ön koşullardan biridir. Hareketin hızı ve bunun bir süre içerisinde yapıldığı zamana bağlıdır. Herhangi bir boyutta hareket kabiliyeti, yalnızca buna müsade eden bir fiziksel ortam olduğu sürece olanaklı olur. Aksi halde diğer hareket unsurları da oluşmayacaktır. Peki alanı hangi kurallar yönetir?

Field is one of the necessary prerequisites for movement of any size and direction to occur. It depends on the speed of the movement and the time it is done within a period of time. Mobility in any dimension is possible only as long as there is a permissive physical environment. Otherwise, other movement elements will not occur. So what rules govern the space?
Category: Geometry

Deriving the Spherical Pythagorean Theorem Using Infinitesimal Area

Authors: Russell P. Patera
Comments: 3 Pages.

The Spherical Pythagorean Theorem is derived by performing an infinitesimal rotation of a spherical right triangle about a vertex not containing the right angle. The infinitesimal areas swept out by the sides of the spherical triangle are easily computed and used to derive the Spherical Pythagorean Theorem.
Category: Geometry

Location and Radius of a Triangle's Incircle Via Geometric Algebra

Authors: James A. Smith
Comments: 11 Pages.

We show how to use the GA concept of the "rejection" of vectors, and also the related outer product, to derive equations for the location and radius of a triangle's incircle.
Category: Geometry

Algorithmic Computation of Multivector Inverses and Characteristic Polynomials in Non-Degenerate Clifford Algebras

Authors: Dimiter Prodanov
Comments: 9 Pages.

Clifford algebras provide the natural generalizations of complex, dual numbers and quaternions into the concept of non-commutative Clifford numbers.The paper demonstrates an algorithm for the computation of inverses of such numbers in a non-degenerate Clifford algebra of an arbitrary dimension.The algorithm is a variation of the Faddeev--LeVerrier--Souriau algorithm and is implemented in the open-source Computer Algebra System Maxima.Symbolic and numerical examples in different Clifford algebras are presented.
Category: Geometry

Regular N-gonal Right Antiprism: Application of HCR’s Theory of Polygon

Authors: Harish Chandra Rajpoot
Comments: 34 Pages. Original Research Work

A regular n-gonal right antiprism is a semiregular convex polyhedron that has 2n identical vertices all lying on a sphere, 4n edges, and (2n+2) faces out of which 2 are congruent regular n-sided polygons, and 2n are congruent equilateral triangles such that all the faces have equal side. The equilateral triangular faces meet the regular polygonal faces at the common edges and vertices alternatively such that three equilateral triangular faces meet at each of 2n vertices. This paper presents, in details, the mathematical derivations of the generalized and analytic formula which are used to determine the different important parameters in terms of edge length, such as normal distances of faces, normal height, the radius of the circumscribed sphere, surface area, volume, dihedral angles between adjacent faces, solid angle subtended by each face at the center, and solid angle subtended by polygonal antiprism at each of its 2n vertices using HCR’s Theory of Polygon. All the generalized formulae have been derived using simple trigonometry, and 2D geometry which are difficult to derive using any other methods.
Category: Geometry

Via Geometric Algebra: A Solution to the Snellius-Pothenot Resection (Surveying) Problem

Authors: James A. Smith
Comments: 8 Pages.

Using geometric algebra (GA), we derive a solution to the classic Snellius-Pothenot problem. We note two types of cases where that solution does not apply, and present a GA-based solution for one of those cases.
Category: Geometry

Illustrative Proof of Time=2

Authors: Yuji Masuda
Comments: 1 Page.

The purpose of this chapter is to prove time by explaining 4-dimensional space-time through Pascal's famous triangle.
Category: Geometry

New proof of Pythagorean Theorem

Authors: Junyoung Jang
Comments: 2 Pages.

We found a new proof of Pythagorean Theorem by using trigonometry. We induced double angle formula of sine and cosine functions in non-circular way.
Category: Geometry

On the Geometry of Axes of Complex Circles of Partition Part 1

Authors: Theophilus Agama
Comments: 12 Pages. This paper advances complex circles of partition by introducing a particularly innate geometry.

In this paper we continue the development of the circles of partition by introducing a certain geometry of the axes of complex circles of partition. We use this geometry to verify the condition in the squeeze principle in special cases with regards to the orientation of the axes of complex circles of partition.
Category: Geometry

The Regular Hexagon

Authors: Volker W. Thürey
Comments: 1 Page. (Abstract added to Article by viXra Admin - Please conform!)

We provide coordinates of a regular hexagon.
Category: Geometry

Semi-Stable Quiver Bundles Over Gauduchon Manifolds

Authors: Dan-Ni Chen, Jing Cheng, Xiao Shen, Pan Zhang
Comments: 12 Pages.

In this paper, we prove the existence of the approximate $(sigma,tau)$-Hermitian Yang--Mills structure on the $(sigma,tau)$-semi-stable quiver bundle $mathcal{R}=(mathcal{E},phi)$ over compact Gauduchon manifolds. An interesting aspect of this work is that the argument on the weakly $L^{2}_1$-subbundles is different from ['{A}lvarez-C'{o}nsul and Garc'{i}a-Prada, Comm Math Phys, 2003] and [Hu--Huang, J Geom Anal, 2020].
Category: Geometry

Two Notes on Regular Polygons: Geometric Motivation of the π Constant

Authors: Irakli Dochviri, Ana Chokhonelidze
Comments: 3 Pages.

In the paper we prove that the ratio between the circumference of the incircle of the regularn-gon and its perimeter is equivalent to the ratio of their areas, respectively. These ratios are constantsfor regular n-gons. Also, it is shown that the ratio of circumference of the excircle and perimeter ofthe regular n-gon is not the same as the ratio of areas of the excircle and this regular n-gon.
Category: Geometry

Connections Between the Plastic Constant, the Circle and the Cuspidal Cubic

Authors: Marc Schofield
Comments: 8 Pages.

The unit circle and the cuspidal cubic curve have been found to intersect at coordinates that can be defined by the Plastic constant, which is defined as the solution to the cubic function x^3 = x + 1. This report explores the connections between the algebraic properties of the Plastic constant and the geometric properties of the circle and this curve.
Category: Geometry

Tomography Geometric Algorithm to Reconstruct Image

Authors: Guillermo Ayala-Martinez
Comments: 5 Pages. In Spanish

Computed tomography CT is an im portant diagnostic imaging methodo used in medicine, consists of appliyng an X-ray scan to a flat section to obtain its imagen, it is a non-invasive and non-destructive procedure. The imagin CT is obtained whit multiple projections, for this reason it is necessary to use a computer. A simple geometric algorithm is proposed to program the computer, it is no necessary to use successive approximations or discretize the image, as in other procedures, in this case the image is a point map.
Category: Geometry

A Novel Formula for Ellipse Perimeter Approximation Giving Absolute Relative Error Less Than 3.85 Ppm.

Authors: K. Idicula Koshy
Comments: 6 Pages.

In this article, the author communicates a novel formula for Ellipse Perimeter Approximation. The algebraic form of the formula is unique in the sense that no other formula published so far has this form. It has achieved the objective of entering the elite club of very few single-expression formulae yielding Absolute Relative Error less than 10 ppm for any ellipse. The Absolute Relative Error obtained with this formula is less than 3.85 ppm, (less than 3.85 millimeter per kilometer), for any ellipse, and less than 2 ppm for about 80% of them.
Category: Geometry

Improved Bound for the Number of Integral Points in a Circle of Radius R Larger Than 1

Authors: Theophilus Agama
Comments: 9 Pages.

Using the method of compression, we prove an inequality related to the Gauss circle problem. Let $mathcal{N}_r$ denotes the number of integral points in a circle of radius $r>0$, then we have $$2r^2bigg(1+frac{1}{4}sum limits_{1leq kleq lfloor frac{log r}{log 2}floor}frac{1}{2^{2k-2}}bigg)+O(frac{r}{log r}) leq mathcal{N}_r leq 8r^{2}bigg(1+sum limits_{1leq kleq lfloor frac{log r}{log 2}floor}frac{1}{2^{2k-2}}bigg)+O(frac{r}{log r})$$ for all $r>1$.
Category: Geometry

La Neutro-Geometría Y la Anti-Geometría Como Alternativas Y Generalizaciones de Las Geometrías no Euclidianas
Neurometrics and Anti-Geometry as Alternatives and Generalizations of Non-Euclidean Geometries

Authors: Florentin Smarandache
Comments: 14 Pages. In Spanish

In this paper we extend Neutro-Algebra and Anti-Algebra to geometric spaces, founding Neutro/Geometry and AntiGeometry. While Non-Euclidean Geometries resulted from the total negation of a specific axiom (Euclid's Fifth Postulate), AntiGeometry results from the total negation of any axiom or even more axioms of any geometric axiomatic system (Euclidean, Hilbert, etc. ) and of any type of geometry such as Geometry (Euclidean, Projective, Finite, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.), and Neutro-Geometry results from the partial negation of one or more axioms [and without total negation of any axiom] of any geometric axiomatic system and of any type of geometry. Generally, instead of a classical geometric Axiom, one can take any classical geometric Theorem of any axiomatic system and of any type of geometry, and transform it by Neutrosophication or Antisofication into a Neutro-Theorem or Anti-Theorem respectively to construct a Neutro-Geometry or Anti-Geometry. Therefore, Neutro-Geometry and Anti-Geometry are respectively alternatives and generalizations of Non-Euclidean Geometries. In the second part, the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra and Anti-Algebra, then to Neutro-Geometry and Anti-Geometry, and in general to Neutro-Structure and Anti-Structure that arise naturally in any field of knowledge is recalled. At the end, applications of many Neutro-Structures in our real world are presented.
Category: Geometry

Neutrogeometry & Antigeometry Are Alternatives and Generalizations of the Non-Euclidean Geometries (Revisited)

Authors: Florentin Smarandache
Comments: 22 Pages.

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and the NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world.
Category: Geometry

An Idea of Geometry Described by Set Theory

Authors: Antoine Warnery
Comments: 23 Pages. In French

The purpose of this paper is to present an idea of geometry described by set theory. This method can describe the axioms of the different geometric representations of space. The axioms of Euclid will be described through straight, segment or sphere subsets, for example the axiom of parallels will be described through a straight set and the definition of the acute angle. The axioms of algebra will be described in the same way using subsets of space with original properties. This description by set theory makes it possible to make a theoretical link between geometry and algebra, and to make a practical link between formulas from different mathematical universes such as trigonometry, algebra and geometry. Apart from the description of axioms, this paper makes it possible to reformulate and explain the meaning of theorems (trigonometric formula, Euler formula, etc.) in an original way, in order to find coherent and efficient methods of describing space.
Category: Geometry

A Somewhat Intuitive Visual Representation of the Formulae for Pi^3 and Ramanujan’s Pi^4

Authors: Janko Kokosar
Comments: 9 Pages.

In the article, I show a visual representation of the formula $pi^3=31.00627..$, respectively, visualization how $pi^3$ is close to 31. I show this using the area of a circle with radius $pi$ that is compared with the area that is quite simply composed of squares and triangles. In the same way, the Ramanujan formula $pi^4=97.5-1/11+1.2491..x10^{-7}$ is visualized. At the end, I mention once again the challenge to explain the Ramanujan formula for $pi^4$.
Category: Geometry

Continuidad Limitada en el Cubo (Limited Continuity in the Cube)

Authors: Carlos Alejandro Chiappini
Comments: 2 Pages. Email: carloschiappini@gmail.com

He intentado construir una antena con forma de cubo. La idea era construirla con un alambre continuo, sin cortes ni interrupciones, que recorriese las aristas una a una. Me sorprendió el hecho siguiente. Sin cortar el alambre y sin pasar más de una vez por alguna arista me resultó imposible formar más de 9 aristas. Muchos intentos me convencieron de la imposibilidad práctica. Eso despertó el interés por averiguar algo respecto a eso en términos teóricos, sea por geométría, por topología o por combinación de ambas.Mis habilidades en geometría son escasas y en topología nulas. Me gustaría recibir la noticia de una demostración teórica del hecho mencionado. Por eso he redactado esta nota y porque puede brindar entretenimiento a personas amantes de la geometría y de la topología.

I have tried to build a cube shaped antenna. The idea was to build it with a continuous wire, without cuts or interruptions, that ran through the edges one by one. I was surprisedthe following fact. Without cutting the wire and without going over any edge more than once, was impossible to form more than 9 edges. Many attempts convinced me of thepractical impossibility. That aroused interest in finding out something about it intheoretical terms, either by geometry, by topology or by a combination of both.My skills in geometry are poor and in topology null. I would like to receive the news of a theoretical proof of the mentioned fact. That is why I have written this note and because it can provide entertainment for people who love geometry and topology.
Category: Geometry

Use Geometric Algebra to Identify the Planes that are Tangent to Three Given Spheres

Authors: James A. Smith
Comments: 14 Pages.

To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we derive the equation for a plane that is tangent to three given planes. The approach that we use determines the unit bivector of the tangent plane from the interior and exterior products of the vectors that connect the centers of the given spheres. A more-general version of this approach is presented in an appendix.
Category: Geometry

Lagrange Multipliers and Adiabatic Limits I

Authors: Urs Frauenfelder, Joa Weber
Comments: 60 Pages.

Critical points of a function subject to a constraint can be either detected by restricting the function to the constraint or by looking for critical points of the Lagrange multiplier functional. Although the critical points of the two functionals, namely the restriction and the Lagrange multiplier functional are in natural one-to-one correspondence this does not need to be true for their gradient flow lines. We consider a singular deformation of the metric and show by an adiabatic limit argument that close to the singularity we have a one-to-one correspondence between gradient flow lines connecting critical points of Morse index difference one. We present a general overview of the adiabatic limit technique in the article [FW22b].The proof of the correspondence is carried out in two parts. The current part I deals with linear methods leading to a singular version of the implicit function theorem. We also discuss possible infinite dimensional generalizations in Rabinowitz-Floer homology. In part II [FW22a] we apply non-linear methods and prove, in particular, a compactness result and uniform exponential decay independent of the deformation parameter.
Category: Geometry

Lagrange Multipliers and Adiabatic Limits II

Authors: Urs Frauenfelder, Joa Weber
Comments: 47 Pages. 2 figures

In this second part to [FW22a] we finish the proof of the one-to-one correspondence of gradient flow lines of index difference one between the restricted functional and the Lagrange multiplier functional for deformation parameters of the metric close to the singular one. In particular, we prove that, although the metric becomes singular, we have uniform bounds for the Lagrange multiplier of finite energy solutions and all its derivatives. This uniform bound is the crucial ingredient for a compactness theorem for gradient flow lines of arbitrary deformation parameter. If the functionals are Morse we further prove uniform exponential decay. We finally show combined with the linear theory in part I that if the metric is Morse-Smale the adiabatic limit map is bijective. We present a general overview of the adiabatic limit technique in the article [FW22b].
Category: Geometry

On a Proof π ≠ 3.14159...

Authors: J. F. Meyer
Comments: 1 Page.

By comparing the square of the approximated pi's quarter to the actual geometric width of a plotted pi annulus, this paper resolutely proves π ≠ 3.14159... while/as discovering the presence of a (reciprocal of the) so-called ' golden ratio ' contained in/as the pi annulus' uniform width.
Category: Geometry

Tiling the Plane with K-Gons

Authors: Volker W. Thürey
Comments: 3 Pages.

We present a way to tile the plane by k-gons for a fixed k. We use usual regular 6-gons by putting some in a row and fill them with k-gons. We use only one or two or four different k-gons.
Category: Geometry

A New Home for Bivectors in Three Dimensions

Authors: Norm Cimon
Comments: 9 Pages. A proposal for a poster about this research has been submitted to the International Conference of Advanced Computational Applications of Geometric Algebra.

The impetus for the work is this quote:"...as shown by Gel’fand’s approach, we can only abstract a unique manifold if our algebra is commutative." (Hiley and Callaghan, 2010)Geometric algebra is non-commutative. Components of different grades can be staged on different manifolds. As operations on those elements proceed, they will effect the promotion and/or demotion of components to higher and/or lower grades, and thus to different manifolds. This paper includes imagery that visually displays bivector addition and rotation on a sphere.David Hestenes interpreted the vector product or rotor in two-dimensions:"as a directed arc of fixed length that can be rotated at will on the unit circle, just as we interpret a vectoras a directed line segment that can be translated at will without changing its length or directionu2026" (Hestenes, 2003)Rotors can be used to develop addition and multiplication of bivectors on a sphere. For those rotational dynamics, rotors of length pi/2 are the basis elements. The geometric algebra of bivectors — Hamilton’s "pure quaternions" — is thus shown to transparently reside on a spherical manifold.
Category: Geometry

Mathematical Analysis of 2D Packing of Circles on Bounded and Unbounded Planes: Analytic Formulation and Simulation

Authors: Harish Chandra Rajpoot
Comments: 41 Pages. Original Research Work

This paper encompasses the mathematical derivations of the analytic and generalized formula and recurrence relations to find out the radii of n umber of circles inscribed or packed in the plane region bounded by circular arcs (including sectors, semi and quarter circles) & the straight lines. The values of radii obtained using analytic formula and recurrence relations have been verified by comparing with those obtained using MATLAB codes. The methods used in this paper for packing circles are deterministic unlike heuristic strategies and optimization techniques. The analytic formulae derived for plane packing of tangent circles can be generalized and used for packing of spheres in 3D space and packing of circles on the spherical surface which is analogous to distribution of non-point charges. The packing density of identical circles, externally tangent to each other, the most densely packed on the regular hexagonal and the infinite planes have been formulated and analysed. This study paves the way for mathematically solving the problems of dense packing of circles in 2D containers, the packing of spheres in the voids (tetrahedral and octahedral) and finding the planar density on crystallographic plane.
Category: Geometry

Make 3D Vectors Parallel by Rotating Them Around their Distinct Axes

Authors: James A. Smith
Comments: 8 Pages.

To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we show how to calculate the angle through which two unit vectors must be rotated in order to be parallel to each other. Among the ideas that we use are a transformation of the usual GA formula for rotations, and the use of GA products to eliminated variables in simultaneous equations. We will show the benefits of (1) examining an interactive GeoGebra construction before attempting a solution, and (2) considering a range of implications of given information.
Category: Geometry

Extension of an Imaginary Triangle Through Complex Variables

Authors: Thomas Halley
Comments: 1 Page.

Complex Variables has a link to general geometry in placing the geometry of squares. Given is a problem in geometry where a short-cut is taken to solve what the angle is in the given situation.
Category: Geometry

Via Geometric Algebra: Rotating a Vector to Locate its Endpoint at a Specific Distance d from a Given Point P

Authors: James A. Smith
Comments: 11 Pages. (Note by viXra Admin: The name on the Submission Form and article in pdf should be the same - Please conform in the future)

To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we show how to calculate the angle through which a given vector must be rotated in order that its endpoint be at a given distance d from a specified point P. The three solution methods that are employed here start from a trigonometric equation is derived from GA’s formula for rotating vectors. The first two solutions use methods that are "automatic", but produce formulas that are not readily interpreted. In contrast, the third method —which does produce a readily interpreted formula— is based upon an examination of the geometric significance of terms in the initial trigonometric equation.
Category: Geometry

On a Related Thompson Problem in Rk

Authors: Theophilus Agama
Comments: 9 Pages.

In this paper we study the global electrostatic energy behaviour of mutually repelling charged electrons on the surface of a unit-radius sphere. Using the method of compression, we show that the total electrostatic energy $U_k(N)$ of $N$ mutually repelling particles on a sphere of unit radius in $mathbb{R}^k$ satisfies the lower boundbegin{align} U_k(N)gg_{epsilon}frac{N^{2}}{sqrt{k}}.onumberend{align}.
Category: Geometry

The Barycenter of a 4-Gon

Authors: Volker Thürey
Comments: 6 Pages.

We provide a new formula for the barycenter of a 4-gon.
Category: Geometry

On the Number of Points Included in a Plane Figure with Large Pairwise Distances

Authors: Theophilus Agama
Comments: 7 Pages.

Using the method of compression we show that the number of points that can be placed in a plane figure with mutual distances at least $d>0$ satisfies the lower bound \begin{align} \gg_2 d^{d-1+\epsilon}\nonumber \end{align}for some small $\epsilon>0$.
Category: Geometry

For GA Newcomers: Demonstrating the Equivalence of Different Expressions for Vector Rotations

Authors: James A. Smith
Comments: 5 Pages.

Because newcomers to GA may have difficulty applying its identities to real problems, we use those identities to prove the equivalence of two expressions for rotations of a vector. Rather than simply present the proof, we first review the relevant GA identities, then formulate and explore reasonable conjectures that lead, promptly, to a solution.
Category: Geometry

The Ehrhart Volume Conjecture Is False in Sufficiently Higher Dimensions in $\mathbb{R}^n$

Authors: Theophilus Agama
Comments: 7 Pages.

Using the method of compression, we show that volume $Vol(K)$ of a ball $K$ in $\mathbb{R}^n$ with a single lattice point in it's interior as center of mass satisfies the lower bound \begin{align} Vol(K)\gg \frac{n^n}{\sqrt{n}}\nonumber \end{align}thereby disproving the Ehrhart volume conjecture, which claims that the upper bound must hold \begin{align} Vol(K) \leq \frac{(n+1)^n}{n!}\nonumber \end{align}for all convex bodies with the required property.
Category: Geometry

On the Average Number of Integer Powered Distances in $\mathbb{r}^k$

Authors: Theophilus Agama
Comments: 7 Pages.

Using the method of compression we obtain a lower bound for the average number of $d^r$-unit distances that can be formed from a set of $n$ points in the euclidean space $\mathbb{R}^k$. By letting $\mathcal{D}_{n,d^r}$ denotes the number of $d^r$-unit distances~($r>1$~fixed) that can be formed from a set of $n$ points in $\mathbb{R}^k$, then we obtain the lower bound \begin{align} \sum \limits_{1\leq d\leq t}\mathcal{D}_{n,d^r}\gg n\sqrt[2r]{k}\log t.\nonumber \end{align}for a fixed $t>1$.
Category: Geometry

On the Number of Integral Points on the Boundary of a K-Dimensional Sphere

Authors: Theophilus Agama
Comments: 7 Pages. This results uses the method of compression to lower bound the number of integral points on the boundary of a sphere with a fixed radius.

Using the method of compression, we show that the number of integral points on the boundary of a $k$-dimensional sphere of radius $r$ satisfies the lower bound \begin{align} \mathcal{N}_{r,k} \gg r^{k-1}\sqrt{k}.\nonumber \end{align}
Category: Geometry

On the General Distance Problem in $\mathbb{r}^k$

Authors: Theophilus Agama
Comments: 7 Pages. This is a result of a general version of distance problem in a Euclidean space of arbitrary dimension.

Using the method of compression we obtain a generalized lower bound for the number of $d$-unit distances that can be formed from a set of $n$ points in the euclidean space $\mathbb{R}^k$. By letting $\mathcal{D}_{n,d}$ denotes the number of $d$-unit distances that can be formed from a set of $n$ points in $\mathbb{R}^k$, then we obtain the lower bound \begin{align} \mathcal{D}_{n,d}\gg \frac{n\sqrt{k}}{d}.\nonumber \end{align}.
Category: Geometry

An Analysis of the Barber Pole System of my Definition

Authors: Yuji Masuda
Comments: 2 Pages.

The purpose of this chapter is to publish some analysis of the Barber Pole.
Category: Geometry

On the Number of Integral Points Between a K Dimensional Sphere and a Grid

Authors: Theophilus Agama
Comments: 7 Pages.

Using the method of compression we show that the number of integral points in the region bounded by the $2r\times 2r \times \cdots \times 2r~(k~times)$ grid containing the sphere of radius $r$ and a sphere of radius $r$ satisfies the lower bound \begin{align} \mathcal{N}_{r,k} \gg r^{k-\delta}\times \frac{1}{\sqrt{k}}\nonumber \end{align}for some small $\delta>0$.
Category: Geometry

Square Equations Represented with Gnomon

Authors: Juan Elias Millas Vera
Comments: 5 Pages.

In this paper I am going to present a soft extension of gnomon theory in geometry. The well known case for n^2 can be adjusted to (2n+1)^2 and (2n)^2. I show it with simple graphs and an algebraic explanation.
Category: Geometry

Research on Mathematical Butterfly Patterns Conducted up to 2013

Authors: Yuji Masuda
Comments: 3 Pages.

In the study of the deformation mechanism of metallic materials, this structure consisting of two metallic crystals with different crystal orientations, called the corresponding grain boundary, has been evaluated by a parameter called the ∑ value. In addition, it has been pointed out that the DSC (=Displacement of Complete pattern Shift)dislocation model may be affected by up to ∑ value 29 at the corresponding grain boundary. In this paper, we will focus only on the ∑ value, and use only the mathematical point of view.
Category: Geometry

On the General Gauss Circle Problem

Authors: Theophilus Agama
Comments: 7 Pages.

Using the method of compression we show that the number of integral points in a $k$ dimensional sphere of radius $r>0$ is \begin{align} N_k(r)\gg \sqrt{k} \times r^{k-1+o(1)}.\nonumber \end{align}
Category: Geometry

On a Variant of the Gauss Circle Problem

Authors: Theophilus Agama
Comments: 7 Pages.

Using the method of compression we show that the number of integral points in the region bounded by the $2r\times 2r$ grid containing the circle of radius $r$ and a circle of radius $r$ satisfies the lower bound \begin{align} \mathcal{N}_r \gg r^{2-\delta}\nonumber \end{align}for some small $\delta>0$.
Category: Geometry

On the Problem of Axiomatization of Geometry

Authors: Temur Z. Kalanov
Comments: 17 Pages.

An analysis of the foundations of geometry within the framework of the correct methodological basis – the unity of formal logic and rational dialectics – is proposed. The analysis leads to the following result: (1) geometry is an engineering science, but not a field of mathematics; (2) the essence of geometry is the construction of material figures (systems) and study of their properties; (3) the starting point of geometry is the following system principle: the properties of material figures (systems) determine the properties of the elements of figures; the properties of elements characterize the properties of figures (systems); (4) the axiomatization of geometry is a way of construction of the science as a set (system) of practical principles. Sets (systems) of practice principles can be complete or incomplete; (5) the book, “The Foundations of Geometry” by David Hilbert, represents a methodologically incorrect work. It does not satisfy the dialectical principle of cognition, “practice theory practice,” because practice is not the starting point and final point in Hilbert’s theoretical approach (analysis). Hilbert did not understand that: (a) scientific intuition must be based on practical experience; intuition that is not based on practical experience is fantasy; (b) the correct science does not exist without definitions of concepts; the definitions of geometric concepts are the genetic (technological) definitions that shows how given material objects arise (i.e., how a person creates given material objects); (c) the theory must be constructed within the framework of the correct methodological basis: the unity of formal logic and rational dialectics. (d) the theory must satisfy the correct criterion of truth: the unity of formal logic and rational dialectics. Therefore, Hilbert cannot prove the theorem of trisection of angle and the theorem of sum of interior angles (concluded angles) of triangle on the basis of his axioms. This fact signifies that Hilbert’s system of axioms is incomplete. In essence, Hilbert’s work is a superficial, tautological and logically incorrect verbal description of Figures 1-52 in his work.
Category: Geometry

About the "Addition" of Scalars and Bivectors in Geometric Algebra

Authors: James A. Smith
Comments: 5 Pages.

“You can’t add things that are of different types!” This objection to the “addition” of scalars and bivectors—which is voiced by physicists as well as students—has been a barrier to the adoption of Geometric Algebra. We suggest that the source of the objection is not the operation itself, but the expectations raised in critics’ minds by the term “addition”. Indeed, the ways in which this operation interacts with others are unlike those of other “additions”, and might well cause discomfort to the student. This document explores those potential sources of discomfort, and notes that no problems arise from this unusual “addition” because the developers of GA were careful in choosing the objects (e.g. vectors and bivectors) employed in this algebra, and also in defining not only the operations themselves, but their interactions with each other. The document finishes with an example of how this “addition” proves useful.
Category: Geometry

An Essential History of Euclidean Geometry

Authors: Saburou Saitoh
Comments: 6 Pages.

In this note, we would like to refer simply to the great history of Euclidean geometry and as a result we would like to state the great and essential development of Euclidean geometry by the new discovery of division by zero and division by zero calculus. We will be able to see the important and great new world of Euclidean geometry by Hiroshi Okumura.
Category: Geometry

Revisiting Quadrature, Infinity, and the Numbers

Authors: Gerasimos T. Soldatos
Comments: 9 Pages.

This article tackles the problem of quadrature through reductio ad impossibile in the form of proof by contradiction. The general conclusion is that an irrational number is irrational on the real plane, but in the three-dimensional world, it is as a vector the image of one at least constructible position vector, and through the angle formed between them, constructible becomes the “irrational vector” too, as a right-triangle side.
Category: Geometry

Seven Archimedean Circles with Six-Fold Symmetry for the Arbelos

Authors: Hiroshi Okumura
Comments: 3 Pages.

We show that there are seven Archimedean circles with 6-fold symmetry for the arbelos.
Category: Geometry

The Chessboard Puzzle

Authors: Volker Thürey
Comments: 5 Pages.

We introduce compact subsets in the plane and in R 3,which we call Polyorthogon and Polycuboid, respectively. We ask whether we can represent these sets by congruent bricks or mirrored bricks.
Category: Geometry

Orthogonality of Two Lines and Division by Zero Calculus

Authors: Hiroshi Okumura, Saburou Saitoh
Comments: 4 Pages.

In this paper, we will give a pleasant representation of the orthogonality of two lines by means of the division by zero calculus. For two lines with gradients $m$ and $ M$, they are orthogonal if $ m M = - 1. $ Our common sense will be so stated. However, note that for the typical case of $x,y$ axes, the statement is not valid. Even for the high school students, the new result may be pleasant with surprising new results and ideas.
Category: Geometry

An Upper Bound for the Erd\h{o}s Unit Distance Problem in the Plane

Authors: Theophilus Agama
Comments: 8 Pages.

In this paper, using the method of compression, we prove a stronger upper bound for the Erd\H{o}s unit distance problem in the plane by showing that\begin{align}\# \bigg\{||\vec{x_j}-\vec{x_t}||:\vec{x}_t, \vec{x_j}\in \mathbb{E}\subset \mathbb{R}^2,~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j \leq n\bigg\}\ll_2 n^{1+o(1)}.\nonumber \end{align}
Category: Geometry

Calculation of The Integrals of The Geodesic Lines of The Torus

Authors: Abdelmajid Ben Hadj Salem
Comments: 6 Pages.

In this second paper about the geodesic lines of the torus, we calculate in detail the integrals giving the length $s=s(\fii)$ and the longitude $\lm=\lm(\fii)$ of a point on the geodesic lines of the torus.
Category: Geometry

Semicircles in the Arbelos with Overhang and Division by Zero

Authors: Hiroshi Okumura
Comments: 8 Pages.

We consider special semicircles, whose endpoints lie on a circle, for a generalized arbelos called the arbelos with overhang considered in [4] with division by zero.
Category: Geometry

Enomoto's Problem in Wasan Geometry

Authors: Hiroshi Okumura
Comments: 3 Pages.

We consider Enomoto's problem involving a chain of circles touching two parallel lines and three circles with collinear centers. Generalizing the problem, we unexpectedly get a generalization of a property of the power of a point with respect to a circle.
Category: Geometry

Geometry and Division by Zero Calculus

Authors: Hiroshi Okumura
Comments: 34 Pages.

We demonstrate several results in plane geometry derived from division by zero and division by zero calculus. The results show that the two new concepts open an entirely new world of mathematics.
Category: Geometry

On the General no-Three-in-Line Problem

Authors: Theophilus Agama
Comments: 8 Pages.

In this paper we show that the number of points that can be placed in the grid $n\times n\times \cdots \times n~(d~times)=n^d$ for all $d\in \mathbb{N}$ with $d\geq 2$ such that no three points are collinear satisfies the lower bound \begin{align} \gg n^{d-1}\sqrt{d}\mathrm{min}_{\vec{x}\in n^d}\mathrm{Inf}(x_j)_{j=1}^{d}.\nonumber \end{align}This pretty much extends the result of the no-three-in-line problem to all dimension $d\geq 3$.
Category: Geometry

A Quantitative Version of the Erd\h{o}s-Anning Theorem

Authors: Theophilus Agama
Comments: 6 Pages.

Let $\mathcal{R}\subset \mathbb{R}^n$ be an infinite set of collinear points and $\mathcal{S}\subset \mathcal{R}$ be an arbitrary and finite set with $\mathcal{S}\subset \mathbb{N}^n$. Then the number of points in $\mathcal{S}$ with mutual integer distance satisfies the lower bound \begin{align} \gg |\mathcal{S}|\sqrt{n}\mathrm{min}_{\vec{x}\in \mathcal{S}}\mathrm{Inf}(x_j)_{j=1}^{n}\sum \limits_{\substack{k\leq \mathrm{max}_{\vec{x}\in \mathcal{S}}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]\\k\in \mathbb{N}}}\frac{1}{k},\nonumber \end{align}where $\mathcal{G}\circ \mathbb{V}_1[\vec{x}]$ is the compression gap of the compression induced on $\vec{x}$. This proves that there are infinitely many collinear points with mutual integer distances on any line in $\mathbb{R}^n$ and generalizes the well-known Erd\H{o}s-Anning Theorem in the plane $\mathbb{R}^2$.
Category: Geometry

Deriving the Pythagorean Theorem Using Infinitesimal Area

Authors: Russell P. Patera
Comments: 3 Pages.

The Pythagorean Theorem is derived by performing an infinitesimal rotation of a right triangle and using the equation for arc length and the equation for the area of a triangle.
Category: Geometry

A Rotor Problem from Professor Miroslav Josipovic

Authors: James A. Smith
Comments: 8 Pages.

We present two Geometric-Algebra (GA) solutions to a vector-rotation problem posed by Professor Miroslav Josipovic. We follow the sort of solution process that might be useful to students. First, we review concepts from GA and classical geometry that may prove useful. Then, we formulate and carry-out two solution strategies. After testing the resulting solutions, we propose an extension to the original problem.
Category: Geometry

The Seiberg-Witten Equations LCK

Authors: Antoine Balan
Comments: 2 pages, written in french

We define the moduli space of Seiberg-Witten LCK. We propose invariants for complex surfaces.
Category: Geometry

On a Covering Method and Applications

Authors: Theophilus Agama
Comments: 9 Pages.

In this paper we introduce and develop a method for studying problems concerning packing and covering dilemmas and explore some potential applications.
Category: Geometry

The Bilinski Dodecahedron is a Space-Filling (Tessellating) Polyhedron

Authors: Xavier Gisz
Comments: 8 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

These are currently four well known isohedral space-filling convex polyhedra: parellelepiped (the most symmetric form being the cube), rhombic dodecahedron, oblate octahedron (also known as the square bipyramid) and the disphenoid tetrahedron. In this paper it is shown that a Bilinski dodecahedron is an isohedral space-filling tessellating polyhedron, thus bringing the number of these to five.
Category: Geometry

Infinitely Algebraic Classes

Authors: Bin Wang
Comments: 15 Pages.

We show that on a complex projective manifold $X$, for $mathbb G=mathbb R$ or $mathbb Q$, a class in $H^{p, p}(X;mathbb Z)otimes mathbb G$ is represented by a convergent infinite series of integration currents over algebraic cycles with real coefficients. It implies that a Hodge class is represented by an algebraic cycle with rational coefficients.
Category: Geometry

Metamathematics

Authors: Dan Howitt
Comments: 9 Pages. © 2025 Dan Howitt

The following metamathematical analyses of the mathematical concepts and language "zero dimensional", "one dimensional", "two dimensional", "extra dimensional", "infinity", "zero", "nothing", "pure geometry", and "curved space", demonstrate that they are arrived at via one or more of conceptual dissociation, conceptual association, and variations of linguistic alteration, and that they, as such, cannot represent facets of the universe. (The above and below language that are quoted are so because their meanings are in question).
Category: Geometry

Metamathematics: Spatial Dimensionality, Infinity, Zero, Pure Geometry

Authors: Dan Howitt
Comments: 7.5 Pages. © 2025 Dan Howitt

Via metamathematical analyses, proven is that mathematics are inaccurate that include discussion of "zero dimensional", "one dimensional", "two dimensional", "four dimensional", et seq, "infinity", and "zero". The concepts "zero dimensional", "one dimensional", and "two dimensional" are arrived at via dissociation, and as such cannot represent facets of the universe. Moreover, the language of each are of linguistic alteration, and as such are absent of mathematic meaning. "Infinity" is an antonymic linguistic alteration. "Extra Spatial Dimensionality" is linguistic additive alteration. "Zero" ("zero itself") is a dissociation from zero particular existents. Of the three dimensional, the three dimensions that comprise it are inextricable, such that none exist without the others: It cannot be constructed by, nor deconstructed into, "lower dimensions". "Pure geometry" is a dissociation of the geometry of existents from the existents. Any mathematical and non-mathematical statement that is arrived at via dissociation cannot represent a facet of the universe, because the dissociation entails ignoring what is the case.
Category: Geometry

Metamathematics: Spatial Dimensionality, Infinity, Zero, Pure Geometry

Authors: Dan Howitt
Comments: 9 Pages. © 2025 Dan Howitt

Via metamathematical analyses, proven is that mathematics are inaccurate that include discussion of "zero dimensional", "one dimensional", "two dimensional", "four dimensional" et seq, "infinity", and "zero": The concepts of each, except the concept "infinity", are arrived at via conceptual dissociation, and as such cannot represent facets of the universe, and the language of each are of linguistic alteration ("infinity" is an antonymic linguistic alteration), and as such are absent of mathematic meaning. Of the three dimensional, the three dimensions that comprise it are inextricable, such that none exist without the others: It cannot be constructed by, nor deconstructed into, "lower dimensions". Regarding the concept "pure geometry": It is a conceptual dissociation of the geometry of existents from the existents; and anything that is arrived at via conceptual dissociation cannot represent facets of the universe.
Category: Geometry

Structures of Finite and Infinite Types of Non-Compact Riemann Surfaces Via Fundamental Groups

Authors: Absos Ali Shaikh, Uddhab Roy
Comments: 36 Pages.

The purpose of this article is to introduce the notion of constructing any arbitrary finite and infinite types of non-compact hyperbolic Riemann surfaces via (non-abelian) fundamental groups equipped with various types of classical Schottky structures, with limit sets as uncountable sets (but not necessarily Cantor sets), emphasising the cases in which the surfaces are of infinite hyperbolic areas. In particular, in this paper, the primary goal is to fabricate three different types of caconical non-compact infinite area Fuchsian polygons in the hyperbolic plane endowed with various kinds of classical Schottky structures. After that, we have initiated a structure of an arbitrary finite type non-compact hyperbolic Riemann surface with genus, conformal holes, cusps, and funnel ends by using the canonical Fuchsian Schottky polygons. Furthermore, in this manuscript, we have proposed the ideas of infinite types conformally compact and semi-conformally compact hyperbolic Riemann surfaces. Indeed, we have introduced four new and interesting types of infinite type hyperbolic Riemann surfaces (we call generalized flute surfaces) that are constructed from infinite sequences of infinite area hyperbolic pair of pants, each glued to the next along a common geodesic boundary with certain strategies.
Category: Geometry

Different Perspectives on Power of a Point

Authors: Yerkebulan Bolat
Comments: 40 Pages.

This paper is devoted to an in-depth study of the concept of the power of a point and its applications to the solution of olympiad-level geometry problems. The discussion encompasses the classical definitions of the power of a point, the radical axis, and the radical center, as well as their various generalizations --- including the interpretation of a point as a circle of zero radius, the notion of coaxial circles, and the linearity property of power differences. Detailed examples drawn from both national and international mathematical olympiads are presented to showcase the effectiveness of these methods in addressing both classical and modern geometric problems. Furthermore, the paper considers potential extensions and applications within a broader framework of elementary geometry, with particular emphasis on their value as practical tools for olympiad training and problem solving.
Category: Geometry

The Cubic Curve Know as Witch of Agnesi

Authors: Rolando Zucchini
Comments: 7 Pages.

This type of cubic curve, in the United Kingdom know as the Witch of Agnesi, it is approached from a geometric, trigonometric and analytical point of view; using the scheme: model, algorithm, resolution, graph.
Category: Geometry

Translation of the Kimberling's Glossary Into Barycentrics ("le Glossaire de Pierre")

Authors: Pierre L. Douillet
Comments: 521 Pages.

Described at https://faculty.evansville.edu/ck6/encyclopedia/ETC.html by "if you're unsure of a term, click Glossary or Pierre Douillet's much expanded and very useful version".
Category: Geometry

The Method of Dividing the 60 ° Angle Into Three Equal Parts

Authors: Hongfa Zi, Hongyun Zi
Comments: 6 Pages.

In the field of modern mathematics, the 60 ° angle trisecting problem has long been a classic geometric problem that has attracted much attention, and its essence is closely related to the infinite extension of trigonometric functions in the generalized dimension. Through in-depth research, it was found that the solution to this problem exists within the two-thirds interval between r and r, where r corresponds to the shape of a curve and two-thirds of r corresponds to the shape of a horizontal straight line. This article explores the inherent connections between geometric shapes from an innovative perspective and successfully constructs a new method for accurately dividing 60 ° angles into three equal parts. This method not only breaks through the limitations of traditional geometric thinking, but also has a high degree of scalability, which can be effectively extended to the problem of trisecting at any angle less than 180 ° within the two-thirds interval of r and r. It provides a new idea and solution paradigm for the theory and practical application of angle trisecting, and is expected to promote further development in related fields.
Category: Geometry

The Method of Dividing the 60° Angle Into Three Equal Parts

Authors: Hongfa Zi; Hongyun Zi
Comments: 5 Pages.

The problem of dividing a 60 ° angle into three equal parts in modern mathematics has not yet been solved. This involves the infinite extension of this trigonometric function in a generalized perspective. After research, it was found that the solution is located between 0r and one-thirds of r. The former represents a curve, while the latter represents a horizontal line. This article aims to utilize the relationships between various shapes to divide the 60 ° angle into three equal parts, which can then be extended to any angle less than 180 ° between r and two-thirds of r.
Category: Geometry

The Method of Dividing the 60° Angle Into Three Equal Parts

Authors: Hongfa Zi, Hongyun Zi
Comments: 5 Pages. (Note by viXra Admin: Frequent and/or repeated replacement submissions will not be accepted)

The problem of dividing a 60 ° angle into three equal parts in modern mathematics has not yet been solved. This involves the infinite extension of this trigonometric function in a generalized perspective. After research, it was found that the solution is located between 0r and one-thirds of r. The former represents a curve, while the latter represents a horizontal line. This article aims to utilize the relationships between various shapes to divide the 60 ° angle into three equal parts, which can then be extended to any angle less than 180 ° between r and two-thirds of r.
Category: Geometry

The Method of Dividing the 60° Angle Into Three Equal Parts

Authors: Hongfa Zi, Hongyun Zi
Comments: 5 Pages.

The problem of dividing a 60° angle into three equal parts in modern mathematics has not yet been solved. This involves the infinite extension of this trigonometric function in a generalized perspective. After research, it was found that the solution is located between r and two-thirds of r. The former represents a curve, while the latter represents a horizontal line. This article aims to utilize the relationships between various shapes to divide the 60° angle into three equal parts, which can then be extended to any angle less than 180 ° between r and two-thirds of r.
Category: Geometry

Trisecting an Acute Arbitrary Angle Problem Solved

Authors: Joseph Musonda
Comments: 5 Pages.

Trisecting an arbitrary angle using a straightedge and compass only has been one of the oldest mathematical geometric problem tracing back to Euclidian times. This problem was never solved until 1837 when it was proven impossible by French Mathematician Pierre Wantzel. As stated by Pierre Laurent Wantzel (1837), the solution of the angle trisection problem corresponds to an implicit solution of the cubic equation x cubed minus 3x minus 1 equals 0, which is algebraically irreducible, and so is the geometric solution of the angle trisection problem. This method explained here can trisect any acute arbitrary angle. Only a compass and straightedge is used. The formal proof is later given after a practical illustration. For practical sake and to prove the possibility of trisecting an arbitrary angle, the author used the most common angle of 60 degrees that mathematicians uses to explain the proof for impossibility. The author believes that this proof will act as a basis for further research in geometry in future.Keywords: trisecting, arbitrary angle, geometry, straightedge and compas, implicit solution
Category: Geometry

On the Spectral Flow Theorem of Robbin-Salamon for Finite Intervals

Authors: Urs Frauenfelder, Joa Weber
Comments: 79 Pages. 7 Figures.

In this article we consider operators of the form ∂sξ + A(s)ξ where s lies in an interval [−T , T ] and s → A(s) is continuous. Without boundary conditions these operators are not Fredholm. However, using interpolation theory one can define suitable boundary conditions for these operators so that they become Fredholm. We show that in this case the Fredholm index is given by the spectral flow of the operator path A.
Category: Geometry

Area of any Quadrilateral from Side Lengths

Authors: Alan Michael Gómez Calderón
Comments: 7 Pages. Typo correction

In this paper we show that area of any quadrilateral can be estimated from the four lengths sides. With the Triangle Inequality Theorem and a novel provided diagonal's formula, the boundaries of quadrilateral diagonals are found. Finally Bretschneider's formula can be applied to find a set of possible areas.
Category: Geometry

Gaps and Overlappings

Authors: Volker W. Thürey
Comments: 5 Pages.

In the first part, we investigate the tiling of the plane by convex polygons, andwe introduce many constants. At the end, we calculate one. We provide an example,where we cover the plane with convex 8-gons.In a second part, we take other curves and convex polygons.
Category: Geometry

The Method of Layered Separation of N-Cubes Along the Main Diagonal and Its Application in Geometry

Authors: Vladislav Koshchakov
Comments: 9 Pages. In Russian

The non-obvious possibility of decomposing any n-cube consisting of n-cubes (includingvisually perceptible 2D and 3D) into layers of these cubes sequentially placed along the maindiagonal of this n-cube is presented. At the same time, the number of n-cubes in each layerturned out to be closely related to the numbers of Pascal's triangle. The coefficients of cutting each n-cube from the last (n-1) layers of them with a section of dimension (n-1)D are calculated. Examples are given that allow us to outline some ways to further explore this possibility. In Addition, the possibility of using this method to prove the tetrahedron volume formula without using infinitesimal methods is shown.
Category: Geometry

A Geometric Algebra Solution to a Hard Contest Problem

Authors: James A. Smith
Comments: 4 Pages.

We show how to use rotations of vectors in GA to solve the following problem: "The following are known about a triangle: The ratio of the lengths of two sides; the angle formed by those sides; and the length of that angle’s bisector. Find the length of the side opposite that angle."
Category: Geometry

A Geometric Algebra Solution to a Hard Contest Problem

Authors: James A. Smith
Comments: 4 Pages.

We show how to use rotations of vectors in GA to solve the following problem: "The following are known about a triangle: The ratio of the lengths of two sides; the angle formed by those sides; and the length of that angle’s bisector. Find the length of the side opposite that angle."
Category: Geometry

Supportive Intersection

Authors: Bin Wang
Comments: 20 Pages.

Let $X$ be a differential manifold. Let $mathscr D'(X)$ be the space of currents, and $S^{infty}(X)$ the Abelian group freely generated by regular cells, each of which is a pair of a polyhedron $Uppi$ and a ifferential embedding of a neighborhood of ${Uppi}$ to $X$. In this paper, we define a variant that is a bilinear map begin{equation}begin{array}{ccc} S^{infty}(X)times S^{infty}(X) &ightarrow & mathscr D'(X)(c_1, c_2) &ightarrow & [c_1wedge c_2]end{array}end{equation} called the supportive intersection such thatpar 1) the support of $[c_1wedge c_2]$ is contained in the intersection of the supports of $c_1, c_2$; 2) if $c_1, c_2$ are closed, $[c_1wedge c_2]$ is also closed and its cohomology class is the cup-product of the cohomology classes of $c_1, c_2$.
Category: Geometry

Supportive Intersection

Authors: Bin Wang
Comments: 21 Pages.

Let $X$ be a differentiable manifold. Let $mathscr D'(X)$ be the space of currents, and $S^{an}(X)$ the Abelian group freely generated by analytic cells, i.e. the pairs of a polyhedron $Pi$ and a real analytic map $Pito X$, that can be extended to a real analytic embedding of neighborhood of $Pi$. In this paper, we define a bilinear map begin{equation}begin{array}{ccc}S^{an}(X)times S^{an}(X) &ightarrow & mathscr D'(X) (sigma_1, sigma_2) &ightarrow & [sigma_1wedge sigma_2]end{array}end{equation} such that 1) the support of $[sigma_1wedge sigma_2]$ is contained in the set-intersectionof the supports of $sigma_1, sigma_2$;par 2) if $sigma_1, sigma_2$ are closed, $[sigma_1wedge sigma_2]$ is also closed and its cohomology class is the cup-product of the cohomology classes of $sigma_1, sigma_2$. We call elements in $S^{an}$ the chains, and the current $[sigma_1wedge sigma_2]$ the supportive intersection of the chains.
Category: Geometry

Application of Rational Representation in Euclidean Geometry

Authors: Bo Zhang
Comments: 324 Pages. If you have any problems, questions, discussions or suggestions, please contact me by mail to:creasson@163.com

This book focuses on the application of rational representations to plane geometry. Most plane geometry objects, such as circles, triangles, quadrilaterals, conic curves, and their composite figures, can be represented almost exclusively in terms of rational parameters, which makes the process of computation and proof straightforward.
Category: Geometry

Application of Rational Representation in Euclidean Geometry

Authors: Bo Zhang
Comments: 324 Pages.

This book focuses on the application of rational representations to plane geometry. Most plane geometry objects, such as circles, triangles, quadrilaterals, conic curves, and their composite figures, can be represented almost exclusively in terms of rational parameters, which makes the process of computation and proof straightforward.
Category: Geometry

Quick Tiling

Authors: Volker W. Thürey
Comments: 4 Pages.

In the first part, we tile the plane with k-gons for natural numbers k which have the rest three if we devide it by four. The proof is by pictures. In a second part, we extend the result to all natural numbers larger than two. The foundation is the tiling of the plane by rectangles or hexagons. We use at most two different tiles for the covering.
Category: Geometry

Sines and Cosines of Any Angles May be Determined to Any Degree of Accuracy and a Relativistic Non-Doppler Effect

Authors: Claude Michael Cassano
Comments: 5 Pages. fixed some errors and made some additions

The unit circle yields an exact half-angle formulas for sines, cosines, tangents, etc. of ANY angles, with examples.
Category: Geometry

Location and Radius of a Triangle's Incircle Via Geometric Algebra

Authors: James A. Smith
Comments: 11 Pages.

We show how to use the GA concept of the ``rejection" of vectors, and also the related outer product, to derive equations for the location and radius of a triangle's incircle.
Category: Geometry

The Hexagon

Authors: Volker W. Thürey
Comments: 2 Pages.

We provide coordinates of a regular hexagon.
Category: Geometry

A New Formula for Ellipse Perimeter Approximation Yielding Absolute Relative Error Less Than 1.83 Ppm

Authors: K. Idicula Koshy
Comments: 6 Pages. The article is expected to encourage further research on Ellipse Perimeter Approximation.

Abstract In this article, the author presents a new formula for Ellipse Perimeter Approximation. This formula, with two parameters, is unique in form among all published formulae on Ellipse Perimeter Approximation. Of the two parameters, one is a constant and the other is a polynomial of the aspect ratio, which is dependent on the chosen constant. We were able to reduce the Absolute Relative Error to less than 1.83 parts per million (ppm) for any ellipse, by suitable choice of the parameters.
Category: Geometry

Lagrange Multipliers and Adiabatic Limits I

Authors: Urs Frauenfelder, Joa Weber
Comments: 69 Pages. v2 Reference [SX14] added. J. Symplectic Geom. 23 no.5, 1109-1177 (2025). https://dx.doi.org/10.4310/JSG.251228022011

Critical points of a function subject to a constraint can be either detected by restricting the function to the constraint or by looking for critical points of the Lagrange multiplier functional. Although the critical points of the two functionals, namely the restriction and the Lagrange multiplier functional are in natural one-to-one correspondence this does not need to be true for their gradient flow lines. We consider a singular deformation of the metric and show by an adiabatic limit argument that close to the singularity we have a one-to one correspondence between gradient flow lines connecting critical points of Morse index difference one. We present a general overview of the adiabatic limit technique in the article [FW22b]. The proof of the correspondence is carried out in two parts. The current part I deals with linear methods leading to a singular version of the implicit function theorem. We also discuss possible infinite dimensional generalizations in Rabinowitz-Floer homology. In part II [FW22a] we apply non-linear methods and prove, in particular, a compactness result and uniform exponential decay independent of the deformation parameter.
Category: Geometry

Lagrange Multipliers and Adiabatic Limits I

Authors: Urs Frauenfelder, Joa Weber
Comments: 60 Pages. Reference [SX14] added

Critical points of a function subject to a constraint can be either detected by restricting the function to the constraint or by looking for critical points of the Lagrange multiplier functional. Although the critical points of the two functionals, namely the restriction and the Lagrange multiplier functional are in natural one-to-one correspondence this does not need to be true for their gradient flow lines. We consider a singular deformation of the metric and show by an adiabatic limit argument that close to the singularity we have a one-to-one correspondence between gradient flow lines connecting critical points of Morse index difference one. We present a general overview of the adiabatic limit technique in the article [FW22b].The proof of the correspondence is carried out in two parts. The current part I deals with linear methods leading to a singular version of the implicit function theorem. We also discuss possible infinite dimensional generalizations in Rabinowitz-Floer homology. In part II [FW22a] we apply non-linear methods and prove, in particular, a compactness result and uniform exponential decay independent of the deformation parameter.
Category: Geometry

Lagrange Multipliers and Adiabatic Limits II

Authors: Urs Frauenfelder, Joa Weber
Comments: 56 Pages. 2 figures. J. Symplectic Geom. 23 no.5, 1179-1134 (2025). https://dx.doi.org/10.4310/JSG.251228022324

In this second part to [FW22a] we finish the proof of the one-to-one correspondence of gradient flow lines of index difference one between the restricted functional and the Lagrange multiplier functional for deformation parameters of the metric close to the singular one. In particular, we prove that, although the metric becomes singular, we have uniform bounds for the Lagrange multiplier of finite energy solutions and all its derivatives. This uniform bound is the crucial ingredient for a compactness theorem for gradient flow lines of arbitrary deformation parameter. If the functionals are Morse we further prove uniform exponential decay. We finally show combined with the linear theory in part I that if the metric is Morse-Smale the adiabatic limit map is bijective. We present a general overview of the adiabatic limit technique in the article [FW22b].
Category: Geometry

Quantum Impedance Networks of Dark Matter and Energy

Authors: Peter Cameron
Comments: 21 Pages.

Dark matter has two independent origins in the impedance model: Geometrically, extending two-component Dirac spinors to the full 3D Pauli algebra eight-component wavefunction permits calculating quantum impedance networks of wavefunction interactions. Impedance matching governs amplitude and phase of energy flow. While vacuum wavefunction is the same at all scales, flux quantization of wavefunction components yields different energies and physics as scale changes, with corresponding enormous impedance mismatches when moving far from Compton wavelengths, decoupling the dynamics. Topologically, extending wavefunctions to the full eight components introduces magnetic charge, pseudoscalar dual of scalar electric charge. Coupling to the photon is reciprocal of electric, inverting fundamental lengths - Rydberg, Bohr, classical, and Higgs - about the charge-free Compton wavelength $lambda=h/mc$. To radiate a photon, Bohr cannot be inside Compton, Rydberg inside Bohr,... Topological inversion renders magnetic charge `dark'.Dark energy mixes geometry and topology, translation and rotation gauge fields. Impedance matching to the Planck length event horizon exposes an identity between gravitation and mismatched electromagnetism. Fields of wavefunction components propagate away from confinement scale, are reflected back by vacuum wavefunction mismatches they excite. This attenuation of the `Hawking graviton' wavefunction results in exponentially increasing wavelengths, ultimately greater than radius of the observable universe. Graviton oscillation between translation and rotation gauge fields exchanges linear and angular momentum, is an invitation to modified Newtonian dynamics.
Category: Geometry

Make Two 3D Vectors Parallel by Rotating Them Around Separate Axea

Authors: James A. Smith
Comments: 8 Pages.

To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we show how to calculate the angle through which two unit vectors must be rotated in order to be parallel to each other. Among the ideas that we use are a transformation of the usual GA formula for rotations, and the use of GA products to eliminated variables in simultaneous equations. We will show the benefits of (1) examining an interactive GeoGebra construction before attempting a solution, and (2) considering a range of implications of the given information.
Category: Geometry

Prime Numbers in Geometric Consistencies

Authors: Thomas Halley
Comments: 48 Pages.

A basic smooth manifold and a rational smooth set is explored with variations in proving that R is not equal to i.
Category: Geometry

The Barycenter of a 4-Gon

Authors: Volker Thürey
Comments: 6 Pages.

We give a new formula for the barycenter of a 4-gon.
Category: Geometry

Simplest Integrals for the Zeta Function and its Generalizations Valid in All C

Authors: Jose Risomar Sousa
Comments: 13 Pages.

In this paper we derive the possibly simplest integral representations for the Riemann zeta function and its generalizations (the Lerch function, $\Phi(e^m,-k,b)$, the Hurwitz zeta, $\zeta(-k,b)$, and the polylogarithm, $\mathrm{Li}_{-k}(e^m)$), valid in the whole complex plane relative to all parameters, except for singularities. We also present the relations between each of these functions and their partial sums. It allows one to figure, for example, the Taylor series expansion of $H_{-k}(n)$ about $n=0$ (when $-k$ is a positive integer, we obtain a finite Taylor series, which is nothing but the Faulhaber formula). With these relations, one can also obtain the simplest integral representation of the derivatives of the zeta function at zero. The method used requires evaluating the limit of $\Phi\left(e^{2\pi\ii\,x},-2k+1,n+1\right)+\pi\ii\,x\,\Phi\left(e^{2\pi\ii\,x},-2k,n+1\right)/k$ when $x$ goes to $0$, which in itself already constitutes an interesting problem.
Category: Geometry

On the Number of Points Included in a Plane Figure with Large Pairwise Distances

Authors: Theophilus Agama
Comments: 7 Pages. This is an important replacement, as the first submission contains an error. This is amended in this new submission with the right lower bound.

Using the method of compression we show that the number of points that can be placed in a plane figure with mutual distances at least $d>0$ satisfies the lower bound \begin{align} \gg_2 d^{\epsilon}\nonumber \end{align}for some small $\epsilon>0$.
Category: Geometry

For Students of Geometric Algebra: Demonstrating the Equivalence of Different Formulas for Rotating 3D Vectors

Authors: James A. Smith
Comments: 7 Pages.

As an aid to teachers and students of introductory-level Geometric Algebra, we address a doubt that an inquiring student might (and should) ask: "Are GA’s various formulas for rotating a vector truly equivalent?" Here, we use GA identities to prove the equivalence of two rotation formulas. Rather than merely present the proof, we first review the relevant identities, then formulate and explore reasonable conjectures that lead to the conclusion. Readers are encouraged to examine, in addition, a recent publication by Verhoeff ([1]) that contrasts "GA" rotation methods with methods that use matrices or classical geometry.
Category: Geometry

For GA Newcomers: Demonstrating the Equivalence of Different Expressions for Vector Rotations

Authors: James A. Smith
Comments: 5 Pages.

As an example for newcomers to GA who may have difficulty applying its identities to real problems, we use those identities to prove the equivalence of two expressions for rotations of a vector. Rather than simply present the proof, we first review the relevant GA identities, then formulate and explore reasonable conjectures that lead, promptly, to a solution.
Category: Geometry

The Chessboard

Authors: Volker W. Thürey
Comments: 5 Pages.

We introduce compact subsets in the plane and in R3, which we call Polyorthogonand Polycuboid, respectively. We ask whether we can represent these sets by equalbricks or mirrored bricks.
Category: Geometry

The Chessboard Puzzle

Authors: Volker W. Thürey
Comments: 6 Pages.

We introduce compact subsets in the plane and in R3, which we call Polyorthogon and Polycuboid, respectively. We consider a usual chessboard. We display it by equal bricks or mirrored bricks.
Category: Geometry

An Upper Bound for the Erd\h{o}s Unit Distance Problem in the Plane

Authors: Theophilus Agama
Comments: 8 Pages. A few technicalities resolved regarding the scale of compression and the inequality in the notion of points contained in a compression ball has been made strict. This is because the case of equality is treated separately as admissible points.

In this paper, using the method of compression, we prove a stronger upper bound for the Erd\H{o}s unit distance problem in the plane by showing that \begin{align} \# \bigg\{||\vec{x_j}-\vec{x_t}||:\vec{x}_t, \vec{x_j}\in \mathbb{E}\subset \mathbb{R}^2,~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j \leq n\bigg\}\ll_2 n^{1+o(1)}.\nonumber \end{align}
Category: Geometry

An Upper Bound for the Erd\h{o}s Unit Distance Problem in the Plane

Authors: Theophilus Agama
Comments: 6 Pages. Minor tweak in the summation deducing the upper bound.

In this paper, using the method of compression, we prove a stronger upper bound for the Erd\H{o}s unit distance problem in the plane by showing that\begin{align}\# \bigg\{||\vec{x_j}-\vec{x_t}||:\vec{x}_t, \vec{x_j}\in \mathbb{E}\subset \mathbb{R}^2,~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j \leq n\bigg\}\ll_2 n^{1+o(1)}.\nonumber \end{align}
Category: Geometry

On the General no-Three-in-Line Problem

Authors: Theophilus Agama
Comments: 10 Pages. This paper has been technically and substantially improved.

In this paper we show that the number of points that can be placed in the grid $ntimes ntimes cdots times n~(d~times)=n^d$ for all $din mathbb{N}$ with $dgeq 2$ such that no three points are collinear satisfies the lower boundbegin{align}gg n^{d-1}sqrt[2d]{d}.onumberend{align}This pretty much extends the result of the no-three-in-line problem to all dimension $dgeq 3$.
Category: Geometry

On the General no-Three-in-Line Problem

Authors: Theophilus Agama
Comments: 9 Pages.

In this paper we show that the number of points that can be placed in the grid $ntimes ntimes cdots times n~(d~times)=n^d$ for all $din mathbb{N}$ with $dgeq 2$ such that no three points are collinear satisfies the lower boundbegin{align}gg n^{d-1}sqrt[2d]{d}.onumberend{align}This pretty much extends the result of the no-three-in-line problem to all dimension $dgeq 3$.
Category: Geometry

On the General no-Three-in-Line Problem

Authors: Theophilus Agama
Comments: 8 Pages. A few technicalities resolved regarding the scale of compression and the inequality in the notion of points contained in a compression ball has been made strict. This is because the case of equality is treated separately as admissible points.

In this paper we show that the number of points that can be placed in the grid $n\times n\times \cdots \times n~(d~times)=n^d$ for all $d\in \mathbb{N}$ with $d\geq 2$ such that no three points are collinear satisfies the lower bound \begin{align} \gg_d n^{d-1}\sqrt[2d]{d}.\nonumber \end{align}This pretty much extends the result of the no-three-in-line problem to all dimension $d\geq 3$.
Category: Geometry

On the General no-Three-in-Line Problem

Authors: Theophilus Agama
Comments: 8 Pages. A completely revised version in response to referee reports.

In this paper we show that the number of points that can be placed in the grid $n\times n\times \cdots \times n~(d~times)=n^d$ for all $d\in \mathbb{N}$ with $d\geq 2$ such that no three points are collinear satisfies the lower bound \begin{align} \gg_d n^{d-1}\sqrt[d]{d}.\nonumber \end{align}This pretty much extends the result of the no-three-in-line problem to all dimension $d\geq 3$.
Category: Geometry

A Quantitative Version of the Erd\h{o}s-Anning Theorem

Authors: Theophilus Agama
Comments: 8 Pages. Minor tweak in the lower bound

Let $\mathcal{R}\subset \mathbb{R}^n$ be an infinite set of collinear points and $\mathcal{S}\subset \mathcal{R}$ be an arbitrary and finite set with $\mathcal{S}\subset \mathbb{N}^n$. Then the number of points with mutual integer distances on the shortest line containing points in $\mathcal{S}$ satisfies the lower bound \begin{align} \gg_n \sqrt{n}|\mathcal{S}\bigcap \mathcal{B}_{\frac{1}{2}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]}[\vec{x}]|\sum \limits_{\substack{k\leq \mathrm{max}_{\vec{x}\in \mathcal{S}\cap \mathcal{B}_{\frac{1}{2}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]}[\vec{x}]}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]\\k\in \mathbb{N}\\k>1}}\frac{1}{k},\nonumber \end{align}where $\mathcal{G}\circ \mathbb{V}_1[\vec{x}]$ is the compression gap of the compression induced on $\vec{x}$. This proves that there are infinitely many collinear points with mutual integer distances on any line in $\mathbb{R}^n$ and generalizes the well-known Erd\H{o}s-Anning Theorem in the plane $\mathbb{R}^2$.
Category: Geometry

A Quantitative Version of the Erd\h{o}s-Anning Theorem

Authors: Theophilus Agama
Comments: 6 Pages. Main theorem has been properly stated; Detailed exposition added to the proof.

Let $\mathcal{R}\subset \mathbb{R}^n$ be an infinite set of collinear points and $\mathcal{S}\subset \mathcal{R}$ be an arbitrary and finite set with $\mathcal{S}\subset \mathbb{N}^n$. Then the number of points with mutual integer distances on the shortest line containing points in $\mathcal{S}$ satisfies the lower bound \begin{align} \gg_n |\mathcal{S}|\sqrt{n}\mathrm{min}_{\vec{x}\in \mathcal{S}}\mathrm{Inf}(x_j)_{j=1}^{n}\sum \limits_{\substack{k\leq \mathrm{max}_{\vec{x}\in \mathcal{S}}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]\\k\in \mathbb{N}\\k>1}}\frac{1}{k},\nonumber \end{align}where $\mathcal{G}\circ \mathbb{V}_1[\vec{x}]$ is the compression gap of the compression induced on $\vec{x}$. This proves that there are infinitely many collinear points with mutual integer distances on any line in $\mathbb{R}^n$ and generalizes the well-known Erd\H{o}s-Anning Theorem in the plane $\mathbb{R}^2$.
Category: Geometry

The LCK+ Seiberg-Witten Equations

Authors: Antoine Balan
Comments: 2 pages, written in french

We propose the LCK+ Seiberg-Witten equations which are the Seiberg-Witten equations for a LCK+ metric.
Category: Geometry

The LCK+ Seiberg-Witten Equations

Authors: Antoine Balan
Comments: 2 pages, written in french

We define the LCK+ Seiberg-Witten equations and we propose invariants for smooth four-manifolds.
Category: Geometry