A triangle in a triple system is a collection of three edges isomorphic to {123, 124, 345}. A triple system is triangle-free if it contains no three edges forming a triangle. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is triangle-free. We prove that almost all trian...

Starting with any nondegenerate triangle, we can use an interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show that starting with any arbitrary triangle, the resulting set of triangles formed by this process contains triangles arbitrarily close (up to similarity) to any given tria...

Complex hyperbolic triangle groups, originally studied by Mostow in building the first nonarithmetic lattices in PU(2, 1), are a natural generalization of the classical triangle groups. A theorem of Takeuchi states that there are only finitely many Fuchsian triangle groups that determine an arithmetic lattice in PSL2(R), so triangle groups are generically nonarithmetic. We prove similar finiten...

In this paper we study a special class of monodromy evolving deformations (MED), which represents Halphen’s quadratic system. In 1996, Chakravarty and Ablowitz [4] showed that a fifth-order equation (DH-V) ω 1 = ω2ω3 − ω1(ω2 + ω3) + φ , ω 2 = ω3ω1 − ω2(ω3 + ω1) + θ , ω 2 = ω3ω1 − ω2(ω3 + ω1)− φθ, (1) φ = ω1(θ − φ) − ω3(θ + φ), θ = −ω2(θ − φ)− ω3(θ + φ), which arises in complex Bianchi IX cosmol...

Multiresolution modelling of polygonal surface meshes has been presented as a solution for the interactive visualisation of scenes formed by hundreds of thousands of polygons. On the other hand, it has been shown that representing surfaces using sets of triangle strips or fans greatly reduces visualisation time and provides an important memory savings. In this paper we present a new method to m...

Recently the study and construction of quad/triangle subdivision schemes have attracted attention. The quad/triangle subdivision starts with a control net consisting of both quads and triangles and produces finer and finer meshes with quads and triangles. The use of the quad/triangle structure for surface design is motivated by the fact that in CAD modelling, the designers often want to model c...

Armed only with a triangle and a dream, Marion Walter, a professor of mathematics at the University of Oregon, embarked on a quest for mathematical enlightenment. She battled rough angles and contemplated numerous strategies to find answers. In the end, Walter discovered characteristics of a triangle when each of its sides is trisected. Okay, Walter’s “quest” was not that exciting or epic, but ...

Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct Triangle Logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRLs. In this paper, we prove that the so-called pseudo-prelinear tri...