Related papers: A Hierarchy of Geometric Constructions
Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…
We consider the problem of explicitly computing Beilinson--Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes,…
By using two different invariants for the Rubik's Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually…
A way to add an extra dimension is briefly discussed.
Understanding quantum phases and phase transitions in the presence of symmetries is a central objective of quantum many-body physics. A powerful modern paradigm for investigating this problem is topological holography, which relates…
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be…
Elliptic curves over finite fields with predefined conditions in the order are practically constructed using the theory of complex multiplication. The stage with longest calculations in this method reconstructs some polynomial with integer…
This work introduces a new class of symmetric matrix structures, called harmonic structures, which enable the generation of all possible directed transitions $(x_i, x_{i+1})$ over a set of $n$ symbols, without internal repetitions. Unlike…
This paper is a general survey of literature on Goppa-type codes from higher dimensional algebraic varieties. The construction and several techniques for estimating the minimum distance are described first. Codes from various classes of…
Kirigami, art of paper cutting, enables two-dimensional sheets transforming into unique shapes which are also hard to reshape once with prescribed cutting patterns. Rare kirigami designs manipulate cuts on three-dimensional objects to…
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 use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in $\RP^n$ is maximal. That is, there exist generic configurations of real linear spaces such…
Cantor's diagonal method is traditionally used to prove the uncountability of the set of all infinite binary sequences. This paper analyzes the expressive limits of this method. It is shown that under any constructive application --…
Call a pure Hodge structure geometric if it is contained in the cohomology of a smooth complex projective variety. The main goal is to show that for any set of Hodge numbers (subject to the obvious constraints), there exists a geometric…
We construct linear codes over the finite field Fq from arbitrary simplicial complexes, establishing a connection between topological properties and fundamental coding parameters. First, we study the behaviour of the weights of codewords…
One of the best things about geometry is that it's cool! Geometry enables us to create incredible designs and astounding patterns. This article shows how to use a simple technique (iteration) to create designs that are both cool and…
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues…
A construction of polytopes is given based on integers. These geometries are constructed through a mapping to pure numbers and have multiple applications, including statistical mechanics and computer science. The number form is useful in…
Urban modeling is essential for city planning, scene synthesis, and gaming. Existing image-based methods generate diverse layouts but often lack geometric continuity and scalability, while graph-based methods capture structural relations…
In classical geometry, there is no such well-known and much-studied topic as the construction of conic sections (or briefly conics) from its five points. Its importance in many applications of mechanical engineering, civil engineering and…