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

### A curious relation between angles and lengths of edges of a tetrahedron

Euclidean case
Using the formula for the tan of the half solid angle that Robin Houston quotes, and expressing everything in terms of edge lengths by using the cosine law to convert the dot products, ...

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 ...

30
votes

### A curious relation between angles and lengths of edges of a tetrahedron

This is not an answer to the question, but an experimental observation that suggests a sharper conjecture: it’s only written as an answer because I’d like to flesh it out a bit more than there’s room ...

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, ...

21
votes

Accepted

### Is there a pyramid with all four faces being right triangles?

This picture of a quadrirectangular tetrahedron is from István Lénárt, `The Right Triangle as the Simplex in 2D Euclidean Space, Generalized to $n$ Dimensions'

20
votes

Accepted

### Great polyhedra: What does "great" signify?

There are two things “great” can refer to. The first, as Sam explained, is a specific kind of stellation. The second is to distinguish conjugates. Since they’d otherwise have the same name, one of ...

19
votes

### A curious relation between angles and lengths of edges of a tetrahedron

Here one can find an algebra-geometric proof of the trigonometric relations. The main point is that given a (say, spherical) tetrahedron $T$ one can construct a rational elliptic surface $X_T$ with ...

18
votes

### The space of triangles that fit inside a given triangle, parametrized by edge lengths

Here's the abstract of K.A. Post, "Triangle in a triangle: On a problem of Steinhaus", Geom Dedicata (1993) 45: 115; this paper was cited in the one given in the comment by Nemo.
A necessary and ...

18
votes

### A curious relation between angles and lengths of edges of a tetrahedron

Here goes the elementary proof of the claim by Robert Houston that the quadraples $(P_1^{-1},P_2^{-1},P_3^{-1},P_4^{-1})$ and $(\cot \frac{\Omega_1}2,\cot \frac{\Omega_2}2,\cot \frac{\Omega_3}2,\cot \...

17
votes

Accepted

### Why do some uniform polyhedra have a "conjugate" partner?

There is nothing particularly mysterious here. Roughly, fix the lattice (incidence relations of all faces) and the edge lengths. Then there are several realizations of this structure, typically a ...

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 ...

15
votes

### Great polyhedra: What does "great" signify?

Quoting Wikipedia: https://en.wikipedia.org/wiki/Stellation#Naming_stellations
John Conway devised a terminology for stellated polygons, polyhedra and polychora (Coxeter 1974). In this system the ...

14
votes

### Dodecahedral rolling distance

Here are a few trivial lemmas. I won't use anything about the rolling motion, just that the distance is defined by gluing pentagons edge-to-edge:
The $dd$-circle of radius $k$, which I'll call $C_k$, ...

13
votes

### Dividing a polyhedron into two similar copies

Definitely not tetrahedra, see this paper. The proof generalizes to other families of polytopes, actually. For example, by Sydler's theorem these cannot be polytopes with nonzero Dehn invariant ($\...

12
votes

### What is Kept Fixed for Flexible Spheres

Victor Alexandrov and Robert Connelly (http://arxiv.org/abs/0905.3683) constructed a counterexample to the Strong Bellows Conjecture i.e. a flexible polyhedral surface with non-constant Dehn invariant....

11
votes

Accepted

### The $32$-deg polynomial for the tetrahedron inscribed in the icosahedron?

I guess it would be difficult to prove that the answero your question is "no", since proving that "no a priori reason exists" might be hard.
More modestly, I can say that I don't really know a good ...

11
votes

### If I have zeros at the vertices of an icosahedron, where should the poles go?

MR1032073
Doyle, Peter; McMullen, Curt,
Solving the quintic by iteration.
Acta Math. 163 (1989), no. 3-4, 151–180.

10
votes

### What is Kept Fixed for Flexible Spheres

I think the answer to Q3 is negative.
Take a flexible polyhedron and choose two adjacent faces $F_1$, $F_2$ such that the angle between them changes during the deformation. Now attach to $F_2$ a ...

10
votes

### Why do some uniform polyhedra have a "conjugate" partner?

Igor Pak's answer suggests that conjugate polyhedra in the sense of the question should be different realizations of the same abstract polyhedron (defined by a face lattice); so in this answer I'll ...

10
votes

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

$\newcommand\Z{\mathbb{Z}}$
This is an elementary answer to the warmup question.
There are precisely $10$ ways of inscribing a tetrahedron inside a dodecahedron. The symmetry group $G \subset O(3)$ ...

10
votes

### Are there Monohedra with odd numbers of faces?

This is too long for a comment.
The figure shows how to construct an $11$-face convex polyhedron whose faces all quadrilaterals, but not congruent. I think it's only interesting because it shows that ...

9
votes

### The intersection of two $l_1$ balls

There is an exponential upper bound of $9^n$, since every vertex of $B_1 \cap B_2$ is the intersection of a $k$-face of $B_1$ and a $(n-k)$-face of $B_2$ for some $k$, and the $\ell_1$-ball has $3^n$ ...

9
votes

Accepted

### Alexandrov's generalization of Cauchy's rigidity theorem

The following is Theorem 27.2 of Igor Pak's book Lectures on Discrete and Polyhedral Geometry (which in general is a very nice resource for these sorts of questions):
Let $P,Q\subset\mathbb{R}^d$ (...

8
votes

### Visualizing polyhedra from their 1-skeletons

By Steinitz's theorem, your graph will not be the 1-skeleton of a polyhedron unless it is 3-connected.
As indicated on the Wikipedia page, one way to prove Steinitz' theorem is to use circle ...

8
votes

### Embedding an icosahedron

The answer to your question is actually "yes", but maybe not in the way you wanted. Indeed, the full (oriented) symmetry group of the icosahedron is isomorphic to $A_5$. The stabilizer of a ...

7
votes

### Database of integer edge lengths that can form tetrahedrons

Here is an example of what you can do in your head.
Tabulate some triples representing edge lengths of small triangles. I have y,x,x for y less than 2x,
x,x,x , and 2,3,4. Now consider ...

7
votes

### Solid angles of a tetrahedron

One can show that the face opposite the largest solid angle has the largest perimeter. This follows directly from here (see the answer of Greg Egan with an explicit linear relation).

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