2
votes
1answer
74 views

Criteria for abstract polytopes to be convex polytope

Suppose I have an abstract polytope defined by a poset. Are there any methods for determining whether the abstract polytope can be geometrically realized as convex-hull on its set of vertices?
10
votes
1answer
205 views

Why do convex polytope options constrict with dimension, rather than expand?

There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in $\mathbb{R}^4$, and then three regular polytopes in every dimension $d > 4$. There ...
7
votes
0answers
188 views

Right-angled polytopes

%This question is motivated by the little discussion here at the bottom. The following thing are known about hyperbolic right-angled polytopes: Compact hyperbolic right-angled polytopes do not ...
5
votes
1answer
280 views

Does every simplicial polytope have a topology-preserving contractible edge?

An edge of a triangulated manifold is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the edge is ...
2
votes
2answers
530 views

Cone over the Join of two topological spaces

Suppose $X$ and $Y$ are topological spaces. Let's define the join $X\ast Y$ as the quotient space $X\times Y\times [0,1]/\sim$, where $\sim$ is the equivalence relation generated by ...
11
votes
5answers
1k views

On the number of Archimedean solids

Does anyone know of any good resources for the proof of the number of Archimedean solids (also known as semiregular polyhedra)? I have seen a couple of algebraic discussions but no true proof. Also, ...
11
votes
5answers
725 views

A characterization of convexity

While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here. Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, ...
11
votes
1answer
393 views

Does a triangulation without fixed simplex property always exist?

Suppose we are given a triangulable topological space $X$. If $X$ has the fixed point property (FPP), then obviously for every triangulation $K$ of $X$ and every simplicial map $f:K\to K$ a simplex ...
5
votes
6answers
870 views

Database of polyhedra

As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, ...
3
votes
0answers
151 views

Collapsing the medial axis of a polytope

Let X be a convex polyhedron in hyperbolic 3-space. Let M be the medial axis of X. Question: Is M collapsible? It is easy to see that M is contractable. In the case of Euclidian 3-space, instead ...