62 votes
Accepted

The view from inside of a mirrored tetrahedron

Here are a couple pictures. If you'd like to do more, I created this with a simple povray script. Feel free to e-mail me and I'll send it to you. With four reflections. With 8 levels of ...
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  • 40.6k
39 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 ...
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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. ...
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  • 93.7k
37 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 ...
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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, ...
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  • 351
24 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 ...
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  • 1,849
23 votes

Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

Edit: The previous answer had an error, which I realized from a comment of @Will Sawin, and I've completely revised it. This group is a subgroup of an S-arithmetic lattice, which acts discretely on ...
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  • 61.2k
23 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 ...
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  • 28.7k
22 votes
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3D models of the unfoldings of the hypercube?

I implemented the ideas in the paper using Mathematica. I pushed it a bit further to actually generate the images below. You can download this Mathematica notebook to see the code and detailed ...
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  • 1,803
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. ...
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  • 15.2k
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 ...
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21 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-...
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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.
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18 votes

Open problems in Euclidean geometry?

W. Wernick has tabulated 139 triangle construction problems using a list of sixteen points associated with the triangle. In each case three points are given and the goal is to construct a triangle ...
18 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, ...
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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 ...
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  • 9,513
17 votes

Open problems in Euclidean geometry?

Chromatic Number of the Plane or Hadwiger–Nelson problem asks for the minimum number of colors required to color the plane such that no two points at distance $1$ from each other have the same color. ...
17 votes

Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?

For $n=3$ this question was asked in 1996 by James Propp, conjecturing that the answer is Yes. (I got the reference from this page, which concerns the very special case of a rectangular box in ${\bf ...
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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 ...
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16 votes

Uniformly Sampling from Convex Polytopes

Hit and run sampling will perform much better than rejection sampling for higher dimensions. It converges to a uniform distribution in polynomial time. The basic idea is to start at any point $x_0$ ...
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16 votes
Accepted

Convex hull of total orders

I claim that this is false for $n=6$. I find it convenient to shift the variables to $w_{ij} = 2 v_{ij} -1$. So the inequalities are $$-1 \leq w_{ij} \leq 1 \quad (1)$$ $$w_{ij} + w_{ji}=0 \quad (2)$$ ...
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16 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 ...
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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 $xy$-...
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15 votes

Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

As Ian Agol said, the group is virtually free. Now, let us note that all your quaternions are congruent to $1$ modulo $\sqrt 5$. This implies your group is torsion free (a kind of Minkowski theorem),...
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  • 1,960
15 votes
Accepted

"Database" of simplicial polytopes/spheres

You can find Frank Lutz's lists of simplicial spheres (and other manifolds) here: Let me shamelessly self-advertise my list of simplicial 4-polytopes with up to $10$ vertices and various families of ...
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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 ...
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  • 18.8k
14 votes

Open problems in Euclidean geometry?

I am reposting my own answer from Not especially famous, long-open problems which anyone can understand for it answers this question as well. Are there eight points on the plane, no three on a line,...
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 ...
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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 ...
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  • 9,513
13 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. ...
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