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

Generalized Hodge Conjecture on 3-Folds

Authors: Bin Wang
Comments: 8 Pages.

The generalized Hodge conjecture can be viewed as a prediction on the coniveau filtration at all levels $$degree-2 coniveau, $$ an integer associated to each cohomological subgroup in the filtration (see (1.3) below). At the level 0, the conjecture coincides with the millennium Hodge conjecture on Hodge classes, which remains outstanding. At the non negative levels, Voisin has worked out the detail of Grothendieck's vision: the conjecture at the level 1 is implied by the conjecture at the level 0. In this paper, we would like to introduce a method with a new interpretation of the coniveau filtration in general. In case of 3-folds, the new method shows that Voisin's proof for the level 1 could work in the same way without the millennium Hodge conjecture.
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 Bound for the Isotropic Constant

Authors: Johan Aspegren
Comments: 4 Pages.

We obtain a dimension independent bound for the isotropic constants for the convex bodies.
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

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

HCR's Theorem

Authors: Harish Chandra Rajpoot
Comments: 9 Pages. Original Research Work

The author derives mathematical formula to analytically compute the V-cut angle (δ) required for rotating through the same angle (θ) the two co-planar planes, initially meeting at a common edge bisecting the angle (α) between their intersecting straight edges, about their intersecting straight edges until their new straight edges (generated after removing V-shaped planar region) coincide. As a result, we get a point (apex) where three planes intersect one another out of which two are original planes (rotated) & third one is their co-plane (fixed). This theorem is very useful for making specific pyramidal flat containers with polygonal (regular or irregular) base using sheet of paper, cardboard, polymer, metal or alloy which can be easily bent and butt-joined at the mating edges, closed right pyramids/bi-pyramids & polyhedrons having two regular n-polygonal & 2n congruent trapezoidal faces. The author has also presented some paper models for making pyramidal flat containers with regular pentagonal, heptagonal and octagonal bases.
Category: Geometry

Conformally Isometric Applications

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

We show that a conformally isometric application of the space is an isometry composed with an homothety.
Category: Geometry

How and Why to Use my Basic Scheme to make Polygonal Spirals

Authors: Dante Servi
Comments: 29 Pages. Copyright by Dante Servi.

Lo schema di base a cui mi riferisco è descritto nel mio articolo "Spirali Poligonali con Inclinazione Gestibile Versione Completa della Trattazione" pubblicato su vixra.org. Questo articolo aggiorna la terza pagina del mio precedente “Come e Perché Utilizzare il mio Metodo Grafico” non più aggiornabile. Nell’ultima pagina ho inserito il link dove si trovano su GeoGebra.org raccolte le attività che ho realizzato partendo dagli articoli che ho pubblicato su viXra.org ed anche di questi ho messo il link che li raccoglie. The basic scheme to which I refer is described in my article "Polygonal Spirals with Manageable Inclination Full Version of the Discussion" published on vixra.org. This article updates the third page of my previous "How and Why to Use my Graphic Method" which is no longer updatable. On the last page I inserted the link where you can find on GeoGebra.org collected the activities I have carried out starting from the articles I published on viXra.org and also of these I put the link that collects them.
Category: Geometry

The Center and the Barycenter

Authors: Volker Thürey
Comments: 7 Pages.

In the first part we deal with the question which points we have to connect to generate a non self-intersectioning polygon. Afterwards we introduce polyholes, which is a generalization of polygons. Roughly spoken a polyhole is a big polygon, where we cut out a finite number of small polygons. In the second part we introduce two 'centers', which we call center and barycenter. In the case that both centers coincide we call this polygon nice. We show that if a polygon has two symmetry axes, it is nice. We yield examples of polygons with a single symmetry axis which are nice and which are not nice. In a third part we introduce the Spieker center and the Point center for polygons. We define beautiful polygons and perfect polygons. We show that all symmetry axes intersect in a single point.
Category: Geometry

Polygonal Stars

Authors: Dante Servi
Comments: 4 Pages.

I describe the geometric basis of particular stars formed by a single closed polygonal, it is not the banal perimeter; the polygonal completely realizes the star. As I will show this can only be achieved for stars with an odd number of points and at least equal to five.
Category: Geometry

The Exterior Connections

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

We define the notion of exterior connection which is a connection for the exterior algebra.
Category: Geometry

Medial Triangle and Irrational Numbers

Authors: Satyam Roy
Comments: 1 Page. [Corrections made by viXra Admin to conform with scholarly norm]

If the co-ordinates of a triangle is rational, then there is at least one irrational number present that will satisfy the equation of sides of its medial triangle.
Category: Geometry

The Generalized Smooth Functions

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

We define the generalized smooth functions for a manifold and we propose applications for the Poisson brackets and the cohomology.
Category: Geometry

The Infinite Manifolds

Authors: Antoine Balan
Comments: 3 Pages. In English

We define the infinite manifolds and the infinite bundles which are spaces of dimension the cardinality of the continuum.
Category: Geometry

Gauss-Bonnet Theorem for Surfaces Embedded in a Four Dimensional Space

Authors: Vincenzo Nardozza
Comments: 3 Pages.

We study the Gauss-Bonnet theorem applied to a specific example.
Category: Geometry

Proving Basic Theorems about Chords and Segments via High-School-Level Geometric Algebra

Authors: James Smith
Comments: Pages.

We prove the Intersecting-Chords Theorem as a corollary to a relation-ship, derived via Geometric Algebra, about the product of the lengths of two segments of a single chord. We derive a similar theorem about the product of the lengths of a secant a chord.
Category: Geometry

Using Geometric Algebra: A High-School-Level Demonstration of the Constant-Angle Theorem

Authors: James A. Smith
Comments: 8 Pages.

Euclid proved (Elements, Book III, Propositions 20 and 21) proved that an angle inscribed in a circle is half as big as the central angle that subtends the same arc. We present a high-school level version of Hestenes' GA-based proof ([1]) of that same theorem. We conclude with comments on the need for learners of GA to learn classical geometry as well.
Category: Geometry

Simple Close Curve Gagnetization and Application to Bellman's Lost in the Forest Problem

Authors: Theophilus Agama
Comments: 7 Pages.

In this paper we introduce and develop the notion of simple close curve magnetization. We provide an application to Bellman's lost in the forest problem assuming special geometric conditions between the hiker and the boundary of the forest.
Category: Geometry

One Property of The Geodesic Lines of The Ellipsoid of Revolution

Authors: Abdelmajid Ben Hadj Salem
Comments: Pages.

In this note, we give one propriety of the geodesic lines of the ellipsoid of revolution.
Category: Geometry

The Continuous Tensor Calculus

Authors: Antoine Balan
Comments: 2 Pages. In french

We define a continuous tensor calculus when the index are continuous instead of being discreet. The spaces are called the infinite manifolds.
Category: Geometry

Remarks on the Circle Arising from Laurent Expansion

Authors: Hiroshi Okumura
Comments: 2 Pages.

We consider the notable circle for the arbelos arising from Laurent expansion appeared in [1], [2,3] in detail.
Category: Geometry

The Spinorial Flow

Authors: Antoine Balan
Comments: 1 Page.

We define a flow over connections of the spinor fiber bundle of a manifold.
Category: Geometry

About the "Addition" of Scalars and Bivectors

Authors: James A. Smith
Comments: 5 Pages.

Unfortunately, some students, teachers, and even physicists object to the "addition" of scalars and bivectors, on the basis that we cannot add things that are not of the same type. Perhaps the objection is not so much to the operation itself, as to the use of the name "addition" for this operation. Still, the operation interacts with others in unfamiliar ways that might 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 dening not only the operations themselves, but their interactions with each other. The document nishes with an example of how this "addition" proves useful.
Category: Geometry

Division by Zero Calculus and Euclidean Geometry - Revolution in Euclidean Geometry

Authors: Hiroshi Okumura, Saburou Saitoh
Comments: 14 Pages.

In this paper, we will discuss Euclidean geometry from the viewpoint of the division by zero calculus with typical examples. Where is the point at infinity? It seems that the point is vague in Euclidean geometry in a sense. Certainly we can see the point at infinity with the classical Riemann sphere. However, by the division by zero and division by zero calculus, we found that the Riemann sphere is not suitable, but D\"aumler's horn torus model is suitable that shows the coincidence of the zero point and the point at infinity. Therefore, Euclidean geometry is extended globally to the point at infinity. This will give a great revolution of Euclidean geometry. The impacts are wide and therefore, we will show their essence with several typical examples.
Category: Geometry

The Cohomology with Values in an Algebra

Authors: Antoine Balan
Comments: 2 Pages.

We define a cohomology with values in an algebra fiber bundle.
Category: Geometry

Proving Unproved Euclidean Propositions on a New Foundational Basis

Authors: Antonio Leon
Comments: 19 Pages.

This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid’s Postulate 1), and Axiom 8 (an extended version of Euclid’s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid's First Postulate, Euclid's Second Postulate, Hilbert's Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid's Postulate 4, Posidonius-Geminus' Axiom, Proclus' Axiom, Cataldi's Axiom, Tacquet's Axiom 11, Khayyam's Axiom, Playfair's Axiom, and an extended version of Euclid's Fifth Postulate.
Category: Geometry

The Geometric Theory of Teleportation

Authors: Franz Hermann
Comments: 9 Pages.

It is known that it is impossible to leave the hyperbolic geometry induced by the absolute as a second-order curve on the projective real plane. However, using the properties and methods of some imaginary hyperbolic geometry, it is possible to "teleport" a straight line segment located on a plane in hyperbolic geometry to another part of the same plane where the elliptic geometry is induced. Read about this in our article.
Category: Geometry

Теория геометрических преобразований, как векторных функций (The Theory of Geometric Transformations as a Vector of Functions)

Authors: Franz Hermann
Comments: 15 Pages.

Данная работа являет собой новое направление в обширном геометрическом разделе, который носит название «Геометрические преобразования». Представление геометрического преобразования в виде векторной функции позволяет рассматривать некоторые вопросы, которые ранее здесь просто не могли бы даже возникнуть. Мы имеем ввиду прежде всего алгебру некоторых геометрических преобразований и формулы их композиций, которые позволяют построить новый математический аппарат в геометрии преобразований. This work is a new direction in the extensive geometric section, which is called "Geometric transformations". The representation of a geometric transformation as a vector function allows us to consider some questions that previously could not even arise here. We mean first of all the algebra of certain geometric transformations and formulas of their compositions, which allow us to build a new mathematical apparatus in the geometry of transformations.
Category: Geometry

Введение в теорию касательных сфер (Introduction to the Theory of Tangent Spheres)

Authors: Franz Hermann
Comments: 33 Pages.

Сегодня практически в любом учебнике по аналитической геометрии есть раздел, посвящённый коническим сечениям. Великий математик древности Евклид когда-то написал сочинение «Начала конических сечений» (к сожалению до нас не дошедшее). Другой великий математик древности Аполлоний Пергский главный труд своей жизни так и озаглавил «Конические сечения». До наших дней сохранились семь из восьми книг этого сочинения. Вопрос конических сечений – кривых второго порядка – с древних времён интересовал человечество. И вопрос этот окончательно не закрыт до сих пор (например, некоторые особенности задачи Аполлония). В настоящей работе мы познакомим читателя с ещё одним взглядом на вопрос конических сечений, который мы назвали «Введение в теорию касательных сфер». Today, almost every textbook on analytic geometry has a section on conical sections. The great mathematician of antiquity Euclid once wrote the essay "Beginnings of Conical Sections" (unfortunately not extant). Another great mathematician of antiquity, Apollonius of Perga, titled the main work of his life "Conical sections". Seven of the eight books of this work have survived to this day. The question of conical sections - curves of the second order - has been of interest to mankind since ancient times. And this question has not yet been completely closed (for example, some features of the Apollonius problem). In this paper, we will acquaint the reader with another look at the issue of conic sections, which we called "Introduction to the theory of tangent spheres".
Category: Geometry

Аналитическая теория поверхностей Мёбиуса (Analytical Theory of Möbius Surfaces)

Authors: Franz Hermann
Comments: 30 Pages.

Все конечно знают, что такое лист Мёбиуса. Берётся прямоугольная полоска бумаги достаточной длины относительно её ширины, перекручивается на 180 градусов и склеивается противоположными (короткми) сторонами. Не путайте с односторонней поверхностью, о которой говорится в учебниках по дифф. геометрии. Склееный лист Мёбиуса имеет кривизну поверхности равную нулю, как и у эвклидовой плоскости. Именно такой лист мёбиуса и является частью проверхности Мёбиуса. Поверхность Мёбиуса является односторонней поверхностью с самопересечением, поэтому сделать модель этой поверхности руками довольно трудно. Проверхность Мёбиуса состоит из четырёх правильных полуконусов, плавно переходящих друг в друга. О построении и исследовании этой поверхности и рассказывается в этой работе. (Everyone knows of course what a Mobius strip is. Take a rectangular strip of paper of sufficient length relative to its width, twist it 180 degrees and glue it on opposite (short) sides. Not to be confused with the one-sided surface mentioned in the diff. geometry. The glued Möbius strip has a surface curvature equal to zero, as in the Euclidean plane. It is such a mobius leaf that is part of the Mobius surface. The Möbius surface is a one-sided self-intersecting surface, so it is rather difficult to model this surface by hand. The Mobius surface consists of four regular semi-cones, smoothly merging into each other. The construction and study of this surface is described in this work.)
Category: Geometry

Теорема Пифагора (Pythagorean Theorem)

Authors: Franz Hermann
Comments: 25 Pages.

Мы используем теорему Пифагора минимум на четверть, а может быть и меньше Это заявление, скорее всего, поставит вас в тупик. Все помнят ещё со школьной скамьи утверждение: «a квадрат плюс b квадрат равно c квадрат». И сразу понимаем – это теорема Пифагора: «Квадрат гипотенузы равен сумме квадратов катетов». Конечно же, теорема эта связана с прямоугольным трегольником, где есть гипотенуза и два катета и речь идёт об их длинах. Однако это не всегда так. Вы убедитесь в этом, прочитав нашу заметку. (We use the Pythagorean theorem for at least a quarter, and maybe less. This statement will most likely confuse you. Everyone remembers from school the statement: “a square plus b square is equal to c square”. And we immediately understand - this is the Pythagorean theorem: "The square of the hypotenuse is equal to the sum of the squares of the legs." Of course, this theorem is connected with a rectangular triangular triangle, where there is a hypotenuse and two legs, and we are talking about their lengths. However, this is not always the case. You will be convinced of this by reading our article.)
Category: Geometry

A Mystery Circle Arising from Laurent Expansion

Authors: Hiroshi Okumura
Comments: 4 Pages.

For a parametric equation of circles touching two externally touching circles, we consider its Laurent expansion around one of the singular points. Then we can find an equation of a notable circle and the equations of the external common tangents of the two circles from the coefficient of the Laurent expansion. However it is a mystery why we can find such things.
Category: Geometry

Haga's Theorems in Paper Folding and Related Theorems in Wasan Geometry Part 2

Authors: Hiroshi Okumura
Comments: 19 Pages.

We generalize problems in Wasan geometry which nvolve no folded figures but are related to Haga's fold in origamics. Using the tangent circles appeared in those problems with division by zero, we give a parametric representation of the generalized Haga's fold given in the first part of these two-part papers.
Category: Geometry

Biinvariant Distance Vectors

Authors: Jan Hakenberg
Comments: 10 Pages.

We construct biinvariant vector valued functions of relative distances using the influence matrix, and the Mahalanobis distance defined by scattered sets of points on Lie groups. The functions are invariant under all group operations. Distance vectors define an ordering of the points in the scattered set with respect to a group element. Applications are classification, inverse distance weighting, and the construction of generalized barycentric coordinates for the purpose of deformation, and domain transfer.
Category: Geometry

On a Special Family of Right Triangles

Authors: Juan Moreno Borrallo
Comments: 5 Pages.

In this brief paper they are studied the right triangles (a,b,c) such that c-b=b-a, showing the maths behind their most remarkable special property.
Category: Geometry

Pythagoras Theorem is an Alternate Form of Ptolemy's Theorem

Authors: Radhakrishnamurty Padyala
Comments: 5 Pages.

Generally, the proofs given to demonstrate Ptolemy’s theorem prove Pythagoras theorem as a special case of Ptolemy’s theorem when certain special conditions are imposed. We prove in this article, that Pythagoras theorem follows from Ptolemy’s theorem in all cases. Therefore, we may say that Pythagoras rediscovered Ptolemy’s theorem.
Category: Geometry

Triple Cosines Lemma and π-Sums of Arccosines

Authors: Yuly Shipilevsky
Comments: 5 Pages.

We obtain a relationship between cosines of two independent angles and cosine of the angle that depends on them in 3D space and then we use that relationship to obtain π-sums of Arccosines.
Category: Geometry

On the Existence of Triangles

Authors: Volker Thürey
Comments: 2 Pages.

We formulate criterions about the existence of triangles depending on its sidelengths.
Category: Geometry

Pappus Chain and Division by Zero Calculus

Authors: Hiroshi Okumura
Comments: 6 Pages.

We consider circles touching two of three circles forming arbeloi with division by zero and division by zero calculus.
Category: Geometry

L’inverso Della Sezione Aurea e la Sorella Speculare Della Spirale Aurea. the Reverse of the Golden Ratio and the Mirror Sister of the Golden Spiral.

Authors: Dante Servi
Comments: 6 Pages.

Le spirali logaritmiche (r=ae^bθ), partendo da un punto di distanza (a) dalla loro origine si possono sviluppare allontanandosi (se b > 0) oppure avvicinandosi (se b < 0) ad essa, questo provo a dire che vale anche per la spirale aurea. The logarithmic spirals (r=ae^bθ), starting from a point of distance (a) from their origin, can develop by moving away (if b > 0) or approaching (if b < 0) to it, this I try to say that also applies to the golden spiral.
Category: Geometry

A Formula for the Number of (n − 2)-Gaps in Digital N-Objects

Authors: Angelo Maimone, Giorgio Nordo
Comments: 11 Pages.

We provide a formula that expresses the number of (n − 2)-gaps of a generic digital n-object. Such a formula has the advantage to involve only a few simple intrinsic parameters of the object and it is obtained by using a combinatorial technique based on incidence structure and on the notion of free cells. This approach seems suitable as a model for an automatic computation, and also allow us to find some expressions for the maximum number of i-cells that bound or are bounded by a fixed j-cell.
Category: Geometry

Using a Common Theme to Find Intersections of Spheres with Lines and Planes via Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: Pages.

After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenient, we identify a common theme through which those types of calculations can be used to find the intersections of spheres with lines, planes, and other spheres.
Category: Geometry

3D Polytope Hulls of E8 4_21, 2_41, and 1_42

Authors: J Gregory Moxness
Comments: 15 pages, 18 figures, 6 equations, and 9 citations

Using rows 2 through 4 of a unimodular 8X8 rotation matrix, the vertices of E8 4_21, 2_41, and 1_42 are projected to 3D and then gathered & tallied into groups by the norm of their projected locations. The resulting Platonic and Archimedean solid 3D structures are then used to study E8's relationship to other research areas, such as sphere packings in Grassmannian spaces, using E8 Eisenstein Theta Series in recent proofs for optimal 8D and 24D sphere packings, nested lattices, and quantum basis critical parity proofs of the Bell-Kochen-Specker (BKS) theorem.
Category: Geometry

Objective Mathematics (OM), Mathematics Built Оn A Circle And A Sphere

Authors: Souren E. Karapetian
Comments: 17 Pages.

The author of this article asked himself: why is mathematics built on a mythical infinite straight line and plane, and what happens if it is built on a circle and a sphere - the most perfect objects of nature? The author called the resulting theory Objective Mathematics (OM), bearing in mind that this theory operates on natural objects that exist in nature (a circle and a sphere) and does not use the axiomatic approach, in particular the infinite parallel straight lines and planes present in traditional mathematics (TM). The constructs and proofs in this article are first made on a circle (one-dimensional OM), and then the resulting law is generalized to a sphere (two-dimensional OM) and 3-sphere (three-dimensional OM). The results obtained 4 empirical laws and 21 laws. The paper gives definitions of such concepts as: • A harmonic four (quartet) on a circle, • A logical three-dimensional count, • Preliminary mathematics (Premathematics) in OM, • Non-Euclidean geometry in OM, • One-dimensional, two-dimensional, three-dimensional OM, etc. As a result of the analysis, the author concludes that the universe is a three-dimensional sphere, where a ray of light is a large circle of this sphere.
Category: Geometry

Postulations on the Behaviour Exhibited by the Circumscribing Center of a Triangle Alongside the Perpendicular Heights

Authors: Ebuka Precious Iwuagwu
Comments: 29 Pages.

The whole of the postulations made in this paper simply aim at describing the positioning and occurrence of the circumscribing center of a triangle so much so that given any specifications and orientation for a particular triangle, the position could be sketched to exact precision and accurate dimensions without a single construction. With these postulations we are able to to envision clearly and describe where the circumscribing center of a triangle will be located without a single construction detail, all stemming from the fact that by the postulations we are able to study the circumscribing center's behavior with respect to the angles in the triangle given a particular orientation. Contained also in this paper are the mathematical justifications for each postulation made. A rule analogous to the sine rule is also observed but here pertains to the three 'perpendicular heights' obtainable respectively from the three vertices in the triangle, wherein the other two maybe obtained when only one is given alongside all the angles in the triangle.
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

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: 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

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 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 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

How and Why to Use my Basic Scheme to make Polygonal Spirals

Authors: Dante Servi
Comments: 31 Pages. Copyright by Servi Dante.

The basic scheme to which I refer is described in my article "Polygonal Spirals with Manageable Inclination Full Version of the Discussion" published on vixra.org. The article "How and Why to Use my Graphic Method" of which this is a further revision, was created with the aim of providing further information on the basic scheme by analyzing what I had already done; this led me to discover other possibilities offered by the fixed angle step. Now with the revision [v2] I wanted to offer a selectable text to facilitate a possible translation (the original language is Italian); I took the opportunity to return to consider what was my first polygonal spiral.
Category: Geometry

The Center and the Barycenter

Authors: Volker W. Thürey
Comments: 8 Pages.

In the first part, we deal with the question of which points we have to connect togenerate a non self-intersecting polygon. Afterwards, we introduce a polyhole, whichis a generalization of a polygon. Roughly speaking a polyhole is a big polygon, wherewe cut out a finite number of small polygons.In the second part, we present two ‘centers’, which we call center and barycenter.In the case that both centers coincide, we call these polygons nice. We show thatif a polygon has two symmetry axes, it is nice. We yield examples of polygons witha single symmetry axis which are nice and which are not nice. We show that allsymmetry axes intersect at one point.In the third part, we introduce the Spieker center and the Point center for polygons.We define beautiful polygons and perfect polygons.
Category: Geometry

Gauss-Bonnet Theorem for Surfaces Embedded in a Four Dimensional Space

Authors: Vincenzo Nardozza
Comments: 3 Pages.

We study the Gauss-Bonnet theorem applied to a specific example.
Category: Geometry

The Continuous Tensor Calculus

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

We define the continuous tensor calculus which is the tensor calculus when the dimension of the spaces is the cardinality of the continuum. The spaces are called infinite manifolds
Category: Geometry

Remarks on the Circle Arising from Laurent Expansion

Authors: Hiroshi Okumura
Comments: 2 Pages.

We consider the notable circle for the arbelos arising from Laurent expansion appeared in [1], [2, 3] in detail.
Category: Geometry

About the "Addition" of Scalars and Bivectors

Authors: James Smith
Comments: 5 Pages.

Unfortunately, some students, teachers, and even physicists object to the "addition" of scalars and bivectors on the basis that we cannot add things that are not of the same type. Perhaps the objection is not so much to the operation itself, as to the use of the name "addition" for this operation. Still, the operation interacts with others in unfamiliar ways that might 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

Proving Unproved Euclidean Propositions on a New Foundational Basis

Authors: Antonio Leon
Comments: 50 Pages.

This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid’s Postulate 1), and Axiom 8 (an extended version of Euclid’s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid's First Postulate, Euclid's Second Postulate, Hilbert's Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid's Postulate 4, Posidonius-Geminus' Axiom, Proclus' Axiom, Cataldi's Axiom, Tacquet's Axiom 11, Khayyam's Axiom, Playfair's Axiom, and an extended version of Euclid's Fifth Postulate.
Category: Geometry

Proving Unproved Euclidean Propositions on a New Foundational Basis

Authors: Antonio Leon
Comments: 42 Pages.

This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid’s Postulate 1), and Axiom 8 (an extended version of Euclid’s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid's First Postulate, Euclid's Second Postulate, Hilbert's Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid's Postulate 4, Posidonius-Geminus' Axiom, Proclus' Axiom, Cataldi's Axiom, Tacquet's Axiom 11, Khayyam's Axiom, Playfair's Axiom, and an extended version of Euclid's Fifth Postulate.
Category: Geometry

Proving Unproved Euclidean Propositions on a New Foundational Basis

Authors: Antonio Leon
Comments: 19 Pages.

This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid’s Postulate 1), and Axiom 8 (an extended version of Euclid’s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid's First Postulate, Euclid's Second Postulate, Hilbert's Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid's Postulate 4, Posidonius-Geminus' Axiom, Proclus' Axiom, Cataldi's Axiom, Tacquet's Axiom 11, Khayyam's Axiom, Playfair's Axiom, and an extended version of Euclid's Fifth Postulate.
Category: Geometry

Augmented Non-Euclidean Geometry

Authors: Dmitriy Skripachov
Comments: 6 Pages.

A new approach in non-Euclidean geometry is obtained by adding types of deformation of the coordinate system. The concept of curvature is clarified taking into account the type of deformation. Such deformations can be stretching-compression, bending, shear, and torsion. The straightness of reference systems, traditional for non-Euclidean geometry, remains unchanged.
Category: Geometry

Pythagoras Theorem is an Alternate Form of Ptolemy’s Theorem

Authors: Radhakrishnamurty Padyala
Comments: 5 pages 4 Figures

Generally, the proofs given to demonstrate Ptolemy’s theorem prove Pythagoras theorem as a special case of Ptolemy’s theorem when certain special conditions are imposed. We prove in this article, that Pythagoras theorem follows from Ptolemy’s theorem in all cases. Therefore, we may say that Pythagoras rediscovered Ptolemy’s theorem.
Category: Geometry

Pythagoras Theorem is an Alternate Form of Ptolemy’s Theorem

Authors: Radhakrishnamurty Padyala
Comments: 5 pages 4 Figures

Generally, the proofs given to demonstrate Ptolemy’s theorem prove Pythagoras theorem as a special case of Ptolemy’s theorem when certain special conditions are imposed. We prove in this article, that Pythagoras theorem follows from Ptolemy’s theorem in all cases. Therefore, we may say that Pythagoras rediscovered Ptolemy’s theorem.
Category: Geometry

Triple Cosines Lemma and π-Sums of Arccosines

Authors: Yuly Shipilevsky
Comments: 6 Pages.

We obtain a relationship between cosines of two independent angles and cosine of the angle that depends on them in 3D space and then we use that relationship to obtain π-sums of Arccosines
Category: Geometry

L’inverso Della Sezione Aurea e la Sorella Speculare Della Spirale Aurea. the Reverse of the Golden Ratio and the Mirror Sister of the Golden Spiral.

Authors: Dante Servi
Comments: 10 Pages. Copyright by Servi Dante.

Le spirali logaritmiche (r=ae^bθ), partendo da un punto di distanza (a) dalla loro origine si possono sviluppare allontanandosi (se b > 0) oppure avvicinandosi (se b < 0) ad essa, questo provo a dire che vale anche per la spirale aurea. Ho corretto la proposta per semplificare la costruzione della spirale aurea basata sui rettangoli aurei. The logarithmic spirals (r=ae^bθ), starting from a point of distance (a) from their origin, can develop by moving away (if b > 0) or approaching (if b < 0) to it, this I try to say that also applies to the golden spiral. I corrected the proposal to simplify the construction of the golden spiral based on the golden rectangles.
Category: Geometry

L’inverso Della Sezione Aurea e la Sorella Speculare Della Spirale Aurea. the Reverse of the Golden Ratio and the Mirror Sister of the Golden Spiral.

Authors: Dante Servi
Comments: 10 Pages.

Le spirali logaritmiche (r=ae^bθ), partendo da un punto di distanza (a) dalla loro origine si possono sviluppare allontanandosi (se b > 0) oppure avvicinandosi (se b < 0) ad essa, questo provo a dire che vale anche per la spirale aurea. Ho aggiunto una proposta per semplificare la costruzione della spirale aurea basata sui rettangoli aurei. The logarithmic spirals (r=ae^bθ), starting from a point of distance (a) from their origin, can develop by moving away (if b > 0) or approaching (if b < 0) to it, this I try to say that also applies to the golden spiral. I added a proposal to simplify the construction of the golden spiral based on the golden rectangles.
Category: Geometry

L’inverso Della Sezione Aurea e la Sorella Speculare Della Spirale Aurea. the Reverse of the Golden Ratio and the Mirror Sister of the Golden Spiral.

Authors: Dante Servi
Comments: 6 Pages.

Le spirali logaritmiche (r=ae^bθ), partendo da un punto di distanza (a) dalla loro origine si possono sviluppare allontanandosi (se b > 0) oppure avvicinandosi (se b < 0) ad essa, questo provo a dire che vale anche per la spirale aurea. The logarithmic spirals (r=ae^bθ), starting from a point of distance (a) from their origin, can develop by moving away (if b > 0) or approaching (if b < 0) to it, this I try to say that also applies to the golden spiral. Ho sostituito "saranno inversamente uguali" con "possono essere reciprocamente uguali, vedi ultima immagine". I replaced "will be inversely equal" with "can be mutually equal, see last image".
Category: Geometry