23
votes
Accepted
Can every simple polytope be inscribed in a sphere?
Not all simple polytopes are incribable, e.g. the dual of the cyclic polytope $C_4(8)$ is simple and not inscribable, as shown recently in Combinatorial Inscribability Obstructions for Higher-...
- 10.5k
22
votes
Accepted
Why are we interested in permutahedra, associahedra, cyclohedra, ...?
Philosophical questions deserve philosophical answers, so I am afraid no amount of references and specific results will probably satisfy you. Let me try to explain it in a somewhat generic way.
...
- 16.2k
17
votes
Why are we interested in permutahedra, associahedra, cyclohedra, ...?
The origins of associahedra go back to the thesis work in homotopy theory of Jim Stasheff in the early 1960's.
He did graduate work at Oxford, working with Ioan James, who in the mid 1950's had ...
- 10.5k
13
votes
Convex lattice polygons with equal area and perimeter
Pick's theorem says these two convex lattice polygons have area
$$i+\frac{b}{2}-1 = 4 + 10/2 -1 = 8 \;,$$
and they both have perimeter $8 + 2 \sqrt{2}$.
You can see I've "bumped out" two ...
- 149k
12
votes
Why are we interested in permutahedra, associahedra, cyclohedra, ...?
In my opinion there are two answers to this question.
The first is that these particular classes of polytopes have fascinating combinatorial properties and structure. Presumably you're aware of the ...
- 76.8k
11
votes
Why are we interested in permutahedra, associahedra, cyclohedra, ...?
There are remarkable combinatorial formulas for the face numbers and the volumes (of certain geometric realizations of) of these polytopes and a more general family ("generalized permutohedra" a.k.a. "...
- 21.5k
9
votes
Accepted
Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?
The answer is No, there are no other such polytopes. The proof is quite laborious in parts, and I wrote it down in this article.
Theorem. In dimension $d\ge 4$, an edge-transitive polytope is vertex-...
- 11.9k
9
votes
Why are we interested in permutahedra, associahedra, cyclohedra, ...?
I'm not an expert, but perhaps this could help:
Hohlweg, Christophe. "Permutahedra and associahedra: Generalized associahedra from the geometry of finite reflection groups." arXiv:1112.3255 (2011): ...
- 149k
8
votes
Accepted
Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?
There are other polytopes. To construct one let's do the following. Remember first that in the hyperbolic $4$-space there exists a regular compact right-angled 120-cell. Here, right-angled means that ...
- 28.6k
8
votes
Accepted
Is a polytope that has in-spheres for faces of all dimensions already regular?
This is true in all dimensions, and can be proved by induction (on $d$) applied to the following (slightly stronger) hypothesis:
Theorem: If $P$ is a convex $d$-polytope with $k$-in-spheres for all $...
- 12.1k
7
votes
Accepted
Realizing spherical complexes as convex polytope
Another way to phrase your question is "whether every complete fan is (combinatorially equivalent to) the face fan (or the normal fan) of a convex polytopes". The answer is No in dimension $...
- 11.9k
6
votes
Accepted
Triangulation of a simplex
You are looking for the edgewise subdivision:
Edelsbrunner, H.; Grayson, D. R., Edgewise subdivision of a simplex, Discrete Comput. Geom. 24, No. 4, 707-719 (2000). ZBL0968.51016.
The basic idea is to ...
- 924
6
votes
A rational polytope that is not a 01-polytope?
Rather obvious in retrospect, but all 01-polytopes are inscribed (the vertices lie on a sphere). So if $P$ is non-inscribable, then it can't be a 01-polytope. There are non-inscribable rational ...
- 11.9k
5
votes
What does it mean polynomials share Newton polytope?
Newton polytopes and the polynomials they support
We will use the standard notion $\mathbf{x}^{\mathbf{a}} := \prod_{i=1}^{n} x_i^{a_i}$ to represent monomials in a multivariate (Laurent) polynomial ...
- 974
5
votes
“Totally transitive” polytopes which are not regular
It is possible to have an abstract polytope where the automorphism group acts transitively on the faces of each rank (fully transitive) but does not act transitively on flags (not regular).
For any ...
- 7,640
5
votes
Approximation of convex hull in high dimension
This recent paper
Sartipizadeh, Hossein, and Tyrone L. Vincent. "Computing the Approximate Convex Hull in High Dimensions." arXiv:1603.04422 (2016).
includes a summary of previous work on ...
- 149k
5
votes
Accepted
Edges of the contact polytope of the Leech lattice
Using the unimodular scaling of the Leech lattice, the length of each minimal vector is $\sqrt{4}$. Fixing a particular minimal vector $u$, the remaining minimal vectors $v$ are:
1 vector $v$ with $\...
- 12.1k
5
votes
4-polytopes with only one kind of regular facet
This is Theorem 1 (actually, Satz 1) of Roswitha Blind, Konvexe Polytope mit kongruenten regulären $(n- 1)$-Seiten im $\Bbb{R}^n$ $(n \ge 4)$, Comment. Math. Helvetici 54 (1979) 304--308. The short ...
- 4,054
5
votes
Solid angles at points in an orthosimplex
Note that $x+\varepsilon y\in K$ for small $\varepsilon$ if and only if $y$ satisfies the inequalities $y_i\leqslant y_{i+1}$ if $x_i=x_{i+1}$; $y_1\geqslant 0$ if $x_1=0$; $y_n\leqslant 0$ if $x_n=0$....
- 98.4k
5
votes
Accepted
Convex lattice polygons with equal area and perimeter
This answers question 2 as well. But I think both questions are way more suitable for math.stackexchange.
I add another example because, unlike the first, it has the property that multiplied by $I^{n-...
- 4,663
4
votes
Accepted
Are there any more polytopes whose 2-faces are identical 4-gons?
In fact, there are many to be found on Wikipedia under isogonal figures, even in three dimensions.
Examples in dimension four are obtained as dual polytope of runcinated 4-simplex or runcinated 24-...
- 11.9k
4
votes
Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?
If you consider a tiling of 3-space to be a 4-dimensional polytope, then the Rhombic dodecahedral honeycomb would work.
Other possibilities are limited by the potential 3-faces.
Because every edge ...
- 580
4
votes
Derive a vertex representation of a permutohedron from its linear-inequalities form
Observe that your set of inequalities is $S_n$-invariant, hence your polytope is, hence your set of vertices is. So it's enough to understand the case $x_1 \leq \ldots \leq x_n$.
Now you don't need ...
- 27.4k
4
votes
Is a polytope that has in-spheres for faces of all dimensions already regular?
In $R^3$, since the spheres are concentric, not only all faces are regular, but also all edges are of the same length, and all faces are inscribed in circles of the same radius, hence are congruent. ...
- 7,270
4
votes
Accepted
A polytope with congruent facets and an insphere that is not facet-transitive?
First, a simple remark: If a polytope with congruent facets is inscribed in a sphere, then it is circumscribed about a sphere as well, and the two spheres are concentric.
Next, there is a series of ...
- 7,270
4
votes
Do Bernoulli polynomials know about face vectors?
Following up on i9Fn's suggestion, there does seem to be at least one more case along these lines: $$660F_{11}(n) = 60n^{11} + 330n^{10} + 550n^9 - 660n^7 + 660n^5 - 330n^3 + 50n.$$
This might be ...
- 4,054
4
votes
What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?
$\color {red} {\textbf {WARNING}}$
I believe @M.Winter has shown in the comments that the following answer is, in fact, $\color {red} {\textbf {incorrect}}$. It would appear that I have merely ...
- 179
4
votes
Accepted
Quasiconformal map from a subset of $\mathbb{C}$ to a polytope
Yes, there exists a quasiconformal map (even a homeomorphism) from the Riemann sphere
to a polytope. Embedding to $R^3$ is irrelevant here, all we need is the intrinsic
metric, which is a flat metric ...
- 87.1k
4
votes
Can a dodecahedron be deformed into a great stellated dodecahedron?
Talking with Saul Schleimer, we came up with the following:
Orthogonally project the great stellated dodecahedron into the $z=0$ plane, choosing a direction that does not result in any zero length ...
- 1,886
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