In a Steiner triple system of order v, STS(v), a set of three lines intersecting pairwise in three distinct points is called a triangle. A set of lines containing no triangle is called triangle-free. The minimum number of triangle-free sets required to partition the lines of a Steiner triple system S, is called the triangle chromatic index of S. We prove that for all admissible v, there exists ...

Some gyrocommutative gyrogroups, also known as Bruck loops or K-loops, admit scalar multiplication, turning themselves into gyrovector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. In classical mechanics the centroid of a triangle in velocity space is the velocity of the center of momentum of three massive ob...

A generalised tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalised triangle groups and whose edge groups are finite cyclic. We prove an improved Spelling Theorem for generalised triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalised tetrahedron groups ac...

We introduce a new class of planar graphs called triangle graphs. First, we present a formal way of constructing and characterizing triangle graphs, and then show that they characterize the adjacencies of arbitrary triangulations and they are three-colorable for a certain subclass of triangulations. Subsequently, we discuss an application of triangle graphs to the parallel nite element solution...

The quad/triangular subdivision, whose control net and refined meshes consist of both quads and triangles, provides better visual quality of subdivision surfaces. While some theoretical results such as polynomial reproduction and smoothness analysis of quad/triangle schemes have been obtained in the literature, some issues such as the basis functions at quad/triangle vertices and design of inte...

In this chapter we will be investigating the intricacies of a seemingly innocuous mathematical object Pascal’s Triangle. After a brief refresher of our understanding of the triangle, we will delve into Cellular Automata as a method of building patterns, and then relate this to a specific pattern within Pascal’s Triangle. Then we shall investigate specific divisibility identities that can aid in...

We introduce a new manifold-based construction for fitting a smooth surface to a triangle mesh of arbitrary topology. Our construction combines in novel ways most of the best features of previous constructions and, thus, it fills the gap left by them. We also introduce a theoretical framework that provides a sound justification for the correctness of our construction. Finally, we demonstrate th...

First, we apply Thurston’s construction of pseudo-Anosov homeomorphisms to grid graphs and obtain translation surfaces whose Veech groups are commensurable to (m, n,∞) triangle groups. Many (if not all) of the surfaces we construct were first discovered by Bouw and Möller, however our treatment of the surfaces differs. We construct these surfaces by gluing together polygons in two ways. We use ...

We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on the Sierpinski gaskets and the Sierpinski carpets to their boundaries, where boundaries mean the triangles and rectangles which confine gaskets and carpets. As an application, we construct diffusion processes on ...