All Questions
8 questions
6
votes
1
answer
539
views
Proofs of Euler's characteristic formula for n-Dim polytopes
Twenty proofs of Euler's formula V - E + F - 1 = 1, which applies to convex polyhedrons, i.e., 3-dimensional polytopes, are presented at the Geometry Junkyard.
I'm interested in proofs of the more ...
- 10.5k
8
votes
1
answer
497
views
Higher dimensional scutoids?
The recent discovery of scutoids in biological structures is fascinating. Two scutoids are depicted below (from Scientists Have Discovered an Entirely New Shape, And It Was Hiding in Your Cells), each ...
- 10.5k
8
votes
1
answer
498
views
Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes)
From "The multiple facets of the associahedra" by Loday:
Let us consider the formal power series
$$f(x) = x+a_1 x^2 +a_2 x^3 + \cdots+ a_n x^{n+1} + \cdots$$
and let
$$ g(x) = x+b_1 x^2 + ...
- 10.5k
3
votes
1
answer
158
views
Cyclic polytopes whose boundary is a flag complex
A cyclic polytope $C(n, d)$ is defined as the convex hull of $n$ distinct points on the moment curve in $\mathbb{R}^d$ (here $n>d$). This is a simplicial polytope so its boundary $\partial C(n, d)$ ...
- 1,017
4
votes
1
answer
388
views
What are the 4 convex simplicial 4-polytopes that have 6 vertices?
In Convex polytopes and related complexes by Klee and Kleinschmidt they state the number of $d$-polytopes with $d+2$ vertices is $\lfloor \frac{d^2}{4}\rfloor$.
I was wondering what the four $4$-...
- 43
11
votes
0
answers
352
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 exist ...
- 1,268
15
votes
3
answers
1k
views
Classification of Platonic solids
My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula $v-e+...
- 955
12
votes
1
answer
651
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 $\...
- 2,104