40
votes

Accepted

### Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Answer to Q1: All of the 261.
I looked at this question because of a video of Matt Parker and wrote an algorithm to find solutions. See here for an example of how a solution would look like. I dumped ...

39
votes

Accepted

### Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra

These are the so-called Regge symmetries, described by T. Regge in a 1970-ish paper. For a bit on it, with references, see the paper
Philip P. Boalch, MR 2342290 Regge and Okamoto symmetries, Comm. ...

38
votes

### How many ways can you inscribe five 24-cells in a 600-cell, hitting all its vertices?

The 600-cell can be tiled by five 24-cells in exactly ten different ways. These are written explicitly in table 2 of "Parity proofs of the Bell-Kochen-Specker theorem based on the 600-cell", where you ...

25
votes

### Tetrahedra passing through a hole

Did you ever find any answer to this?
I find it intriguing that figuring out which shapes of holes a given solid object can pass through is widely considered to be a suitable puzzle for 2 year olds, ...

25
votes

Accepted

### Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Consider the polytope in $\mathbb{R}^3$ with $8$ vertices at coordinates $(\pm 1, \pm 2, 1), (\pm 2, \pm 1, -1)$. Geometrically this looks like a cube where the top face is stretched in the direction ...

25
votes

### Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Your question is essentially about extension complexity. In general, the extension complexity of a polytope $P$ is the minimum number of facets over all polytopes $Q$ which project to $P$. You are ...

23
votes

Accepted

### Rational inscribed realization of the regular dodecahedron

An example
Yes, here is a list of rational coordinates lying on the unit sphere, the convex hull of which is combinatorially equivalent to a regular dodecahedron. This polyhedron is invariant under ...

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-...

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.
...

22
votes

Accepted

### Given a group action on a simplex, can I always find a fundamental region that is a simplex?

The answer is yes!
Notation: Let $e_1$, $e_2$, ..., $e_n$ be the vertices of the simplex.
Let $[n] = \{ 1,2,3, \ldots, n \}$. For $S$ a nonempty subset of $[n]$, let $p(S) = \tfrac{1}{|S|} \sum_{i \in ...

21
votes

### Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra

To address the scissors congruence question at the end of the post: the Regge symmetries produce tetrahedra which are scissors congruent. This is proved in Section 6 (Theorem 9 and Corollary 10) of
...

21
votes

Accepted

### Information inequalities

Yes. The set of $2^n$ (or $2^n-1$ excluding the empty set) dimensional vectors formed by entropies is called the entropic region [1]. Inequalities on the entropic region not implied by the ...

19
votes

### Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra

Let me give a geometric interpretation for the case of tetrahedra of volume zero. The statement becomes as follows:
Given four positive numbers $a,b,c,d$ that satisfy the quadrangle inequalities, ...

19
votes

Accepted

### Who first used the word "Simplex"?

According to Jeff Miller's Earliest Known Uses of
Some of the Words of Mathematics, the first known occurrence is in Schoute’s Mehrdimensionale Geometrie of 1902.

18
votes

Accepted

### Polytope where each vertex belongs to all but two facets

There are other polytopes with this property that can be obtained via the free join construction.
Given two polytopes $P_1\subset\Bbb R^{d_1}$ and $P_2\subset\Bbb R^{d_2}$, the free join $P_1\bowtie ...

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 ...

17
votes

Accepted

### Inscribed $n$-polytope with $2^n$ vertices of maximal volume

For $n=3$, the maximal volume polytope with 8 vertices is described in that paper.
Berman, J. D.; Hanes, K., Volumes of polyhedra inscribed in the unit sphere in (E^3), Math. Ann. 188, 78-84 (1970). ...

16
votes

### Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Another, combinatorially minimal, counterexample of such a polytope $P$ (with only five facets) is the convex hull of the six vertices $(\pm2, 0, 0)$, $(\pm1, \pm1, 1)$. Its projection to the $xy$-...

15
votes

### 3D models of the unfoldings of the hypercube?

I used sage to make a 3d animation of all 261 unfoldings.
Here is a screenshot of the first few:
The file cube-unfoldings.txt contains all the unfoldings, each line contains a list of 8 points.
...

15
votes

Accepted

### Two questions on the permutohedron

The number of integer points in $P_n$ is the number of forests on $[n]$; see Section 3 of Stanley's Decompositions of rational convex polytopes. In fact you can see there a simple description of its ...

14
votes

Accepted

### covering convex sets by round balls

Yes. Any point $y$ in the convex hull of $x$'s is a barycenter of some non-negative masses $m_i$ in $x_i$, $\sum m_i=1$, $y=\sum m_i x_i$. Point $y$ minimizes the moment of inertia $I(p)=\sum m_i |p-...

14
votes

Accepted

### Furthest distance half the diameter?

Denote the diameter by $d$ and distance by $|x-y|$. Then there
are $y,z$ such that $d=|y-z|$ and we have by triangle inequality for every $x$:
$$d=|y-z|\leq |y-x|+|x-z|\leq 2f(x),$$
so we obtain your ...

14
votes

Accepted

### What is the convex hull of the quaternionic symmetries of the 3 dimensional cube?

This is the disphenoidal 288-cell, which is the dual of the bitruncated 24-cell.
This is also mentioned in the "Geometry" section of the Wikipedia article on the 288-cell.
It has 48 vertices, and 336 ...

14
votes

Accepted

### Covering the unit sphere in $\mathbf{R}^n$ with $2n$ congruent disks

For $n=3$ the answer is yes, as was shown by Fejes Tóth in 1943; see the Theorem on p.34 of his book Regular Figures. For $n=4$ the answer is also positive as shown in the 2000 paper, The blocking ...

14
votes

Accepted

### Determining whether a lattice is the face lattice of a polytope - NP hard or undecidable?

There's no contradiction:
I don't know the correct complexity, but I recall hearing several times that it is at least as hard as NP.
It is not difficult to show (using Tarski's algorithm, as ...

13
votes

Accepted

### Minimal combinatorial data needed to define a polytope

I don't know conditions of existence ($d$-connectivity is necessary), but the uniqueness was proved by Blind and Mani in 1987, see also the 1988 article "A simple way to tell a simple polytope from ...

13
votes

Accepted

### Curve with no embedding in a toric surface

A generic curve of genus $5$ is not a hypersurface in a toric surface. This argument is going to use conceptual ideas from Haase and Schicho's paper "Lattice polygons and the number $2i+7$", ...

12
votes

Accepted

### Rigidity of convex polyhedrons in $\mathbb R^3$ with faces removed

Yes, this is true.
One strategy is to use the naïve approach of counting degrees of freedom and constraints. For triangulated polyhedra one can easily show with the Euler characteristic that the ...

12
votes

Accepted

### Convex hull of all rank-$1$ $\{-1, 1\}$-matrices?

That polytope $P_{m,n}$ appears under many names : correlation polytope, Bell polytope, local hidden variable polytope, local polytope (sometimes these names refer to slightly different polytopes), ...

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 ...

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