Related papers: An Etude on One Sharygin's Problem
I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras. (2) The computations of…
The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences…
This paper considers an extension of origami geometry to the case of "folding" a three dimensional (3D) space along a plane. First, all possible incidence constraints between given points, lines and planes are analyzed by using the geometry…
We present generalizations of theorems on Kypert's construction and on 2nd Morley's Centre. Most of our proofs are synthetic.
Nandakumar asked whether there is a tiling of the plane by pairwise non-congruent triangles of equal area and equal perimeter. Here a weaker result is obtained: there is a tiling of the plane by pairwise non-congruent triangles of equal…
Novel geometries can be created by coupling internal states of atoms or molecules to mimic movement in real-space
Polygons are cycles embedded into the plane; their vertices are associated with $x$- and $y$-coordinates and the edges are straight lines. Here, we consider a set of polygons with pairwise non-overlapping interior that may touch along their…
Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in…
While the contents of Euclid's Elements are well-known these days, some characters of the original text have been overlooked due to interpretation by modern mathematical languages. The lens of modern mathematics once anachronistically…
To produce cartographic maps, simplification is typically used to reduce complexity of the map to a legible level. With schematic maps, however, this simplification is pushed far beyond the legibility threshold and is instead constrained by…
We study linear systems of surfaces in $\mathbb{P}^3$ singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those nonempty systems where the conditions imposed by the multiple lines…
Two-, three- and four-dimensional representations of Penrose tilings of the plane are described. The vertices that occur in these representations lie on lattices. Symmetries and methods of visualizing these representations are discussed.…
Author developed a uniform model for different spaces where distance and angle measure kinds are parameters. This model is calculus centric, but can also be used in theoretical research. It is useful in the following domains: deduction of…
The pursue of what are properties that can be identified to permit an automated reasoning program to generate and find new and interesting theorems is an interesting research goal (pun intended). The automatic discovery of new theorems is a…
Distance Geometry is based on the inverse problem that asks to find the positions of points, in a Euclidean space of given dimension, that are compatible with a given set of distances. We briefly introduce the field, and discuss some open…
We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.
In classical geometric algebra, there have been several treatments of affine and projective planes based on fields. In this thesis we approach affine and projective planes from a constructive point of view and we base our geometry on local…
The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time…
In this article we will represent some ideas and a lot of new theorems in Euclidean plane geometry.
The problem of construction of the surfaces with given sets of the points with horizontal tangential planes is considered. Such considerations are of interest in the problem of computer simulations of the waved ocean surfaces.