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40 votes
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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 ...
Moritz Firsching's user avatar
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. ...
Igor Rivin's user avatar
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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 ...
Gjergji Zaimi's user avatar
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, ...
DPKR's user avatar
  • 351
25 votes
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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 ...
Nate's user avatar
  • 2,122
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 ...
Tony Huynh's user avatar
  • 31.8k
23 votes
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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 ...
Adam P. Goucher's user avatar
23 votes
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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-...
Moritz Firsching's user avatar
22 votes
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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. ...
Igor Pak's user avatar
  • 16.8k
22 votes
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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 ...
David E Speyer's user avatar
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 ...
Matthias Wendt's user avatar
21 votes
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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 ...
Cheuk Ting Li's user avatar
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, ...
Ivan Izmestiev's user avatar
19 votes
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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.
Pietro Majer's user avatar
  • 57.9k
18 votes
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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 ...
M. Winter's user avatar
  • 12.8k
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 ...
Nicholas Kuhn's user avatar
17 votes
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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). ...
Guillaume Aubrun's user avatar
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 $xy$-...
Wlodek Kuperberg's user avatar
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. ...
Moritz Firsching's user avatar
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 ...
Sam Hopkins's user avatar
  • 23.1k
14 votes
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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-...
Fedor Petrov's user avatar
14 votes
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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 ...
Alexandre Eremenko's user avatar
14 votes
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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 ...
M. Winter's user avatar
  • 12.8k
14 votes
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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 ...
Mohammad Ghomi's user avatar
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 ...
Geva Yashfe's user avatar
  • 1,356
13 votes
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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 ...
Ivan Izmestiev's user avatar
13 votes
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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 $2i+7$", ...
David E Speyer's user avatar
12 votes
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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 ...
j.c.'s user avatar
  • 13.6k
12 votes
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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), ...
Guillaume Aubrun's user avatar
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 ...
Timothy Chow's user avatar

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