We present a very symmetric triangulation of the 3-sphere, where every edge is in at most five facets but which is not the boundary of a polytope. This shows that not every triangulation of a sphere, where angles around faces of codimension two are less than 2π in the metric pieced together by regular Euclidean simplices, is polytopal. The counterexample presented here is the smallest triangula...

An exclusion region for a triangulation is a region that can be placed around each edge of the triangulation such that the region can not contain points from the set on both sides of the edge. We survey known exclusion regions for several classes of triangulations, including Delaunay, Greedy, and Minimum Weight triangulations. We then show an exclusion region of larger area than was previously ...

Given a set S of n points in the plane, a triangulation is a maximal set of non-intersecting edges connecting the points in S. The weight of the triangulation is the sum of the lengths of the edges. In this paper, we show that for ¿ 1=sin , the -skeleton of S is a subgraph of a minimum weight triangulation of S, where = tan−1(3= √ 2 √ 3) ≈ =3:1. There exists a four-point example such that the -...

Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of triangulations which are not full dimensional is surface triangulations in three dimensions. In this paper we address the question of converting a surface t...

AIM To describe the use of methodological triangulation in a study of how people who had moved to retirement communities were adjusting. BACKGROUND Methodological triangulation involves using more than one kind of method to study a phenomenon. It has been found to be beneficial in providing confirmation of findings, more comprehensive data, increased validity and enhanced understanding of stu...

0-efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold M can be modified to a 0-efficient triangulation orM can be shown to be one of the manifolds S3,RP3 or L(3, 1). Similarly, any triangulation of a compact, orientable, irreducible, ∂-irreducible 3-manifold can be modified to a 0-efficient triangula...

Let Σ be a surface whose boundary consists of a set of disjoint simple closed curves. A Catalan triangulation of Σ is a triangulation of Σ such that all vertices are on the boundary. An edge in a Catalan triangulation is called separating if the removal of its end vertices disconnects the triangulation. The number of Catalan triangulations of a disk is given by the well-known Catalan numbers. C...

This study examined the association between youths' triangulation in marital conflict and three aspects of their peer relations in 416 families during early adolescence. A 4-wave, longitudinal research design was used. As hypothesized, triangulation was associated negatively with perceived support from friends and positively with perceived peer rejection. Triangulation was a risk factor for bot...

We show that any triangulation on n vertices can be transformed into a 4connected one using at most b(3n−7)/5c edge flips. We also give an example of a triangulation that requires d(3n− 10)/5e flips to be made 4-connected, showing that our bound is tight. In addition, we improve the upper bound on the number of flips required to transform any 4-connected triangulation into the canonical triangu...

A random Eulerian triangulation is a random triangulation where an even number of triangles meet at any given vertex. We argue that the central charge increases by one if the fully packed O(n) model is defined on a random Eulerian triangulation instead of an ordinary random triangulation. Considering the case n → 0, this implies that the system of random Eulerian triangulations equipped with Ha...