Related papers: Taxicab Angles and Trigonometry
Flips of diagonals in colored triangle-free triangulations of a convex polygon are interpreted as moves between two adjacent chambers in a certain graphic hyperplane arrangement. Properties of geodesics in the associated flip graph are…
By "parallelogram geometry" we mean the elementary, "commutative", geometry corresponding to vector addition, and by "trapezoid geometry" a certain "non-commutative deformation" of the former. This text presents an elementary approach via…
In this paper, we present a synthetic solution to a geometric open problem involving the radical axis of two strangely defined circumcircles. The solution encapsulates two generalizations, one of which uses a powerful projective result…
Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the…
We construct a framework within which a mathematically precise, fully covariant, and exact averaging procedure for tensor fields on a manifold can be formulated. In particular, we introduce the Weitzenb\"ock connection for parallel…
We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal triad is a modification of the previously studied and similarly defined concept of bidiagonal triple. A bidiagonal triad and a bidiagonal triple both consist of…
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
For any sequence of matrix algebras that converge to a coadjoint orbit we give explicit formulas that show that the distances between the matrix algebras (viewed as quantum metric spaces) converges to 0. In the process we develop a general…
The main result of this paper, is the complete parametric description of the family of triangles which have integer sidelengths and with one angle being sixty degrees.
We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically…
We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x…
The author's method (math-ph/9804010) that uses the Laplace transform to find exact values for a large class of convergent series is extended to trigonometric series.
Some conjectures and open problems in convex geometry are presented, and their physical origin, meaning, and importance, for quantum theory and generic statistical theories, are briefly discussed.
This paper explores and proves the one-seventh area triangle using a purely algebraic approach as opposed to a geometric one. A triangle set purely in the complex plane is used so that we can utilise features of the complex number system to…
Stack triangulations appear as natural objects when defining an increasing family of triangulations by successive additions of vertices. We consider two different probability distributions for such objects. We represent, or "draw" these…
We examine implications of angles having their own dimension, in the same sense as do lengths, masses, {\it etc.} The conventional practice in scientific applications involving trigonometric or exponential functions of angles is to assume…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
With a bird's-eye view, we survey the landscape of Calabi-Yau threefolds, compact and non-compact, smooth and singular. Emphasis will be placed on the algorithms and databases which have been established over the years, and how they have…
There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both…
This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex…