# Tag Info

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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### 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. ...

### 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 ...

### 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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### 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 ...

### 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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### 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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### 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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### 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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### 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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### 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). ...

### 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 $xy$-...

### 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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### "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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### 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 ...