41 votes
Accepted

Visibility of vertices in polyhedra

There are many points in the interior of this polyhedron, constructed (independently) by Raimund Seidel and Bill Thurston, that see no vertices. Interior regions are cubical spaces with "beams" from ...
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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

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, ...
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  • 2,822
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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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 ...
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27 votes

Can the unsolvability of quintics be seen in the geometry of the icosahedron?

This is an old question, but I wanted to write a proof that $A_5$ is simple via symmetries of the icosahedron, using as little group theory as possible. I don't think that it can lead to a proof of ...
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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
23 votes
Accepted

Building a genus-$n$ torus from cubes

Theorem. Let $c(g)$ be the minimum number of cubes such that the boundary of some configuration of $c(g)$ cubes is a genus $g$ surface. Then $c(g)/g \to 2$ as $g \to \infty$. Proof. We write $\chi(...
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  • 8,479
22 votes

Visibility of vertices in polyhedra

I have a simpler example and I see that its idea is similar to the above one. Cut the vertices of a cube to form 8 small triangles and suppose the triangles are rigid but the faces are not. Then ...
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22 votes
Accepted

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
19 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 ...
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18 votes
Accepted

Polyhedron not circumscribed about a sphere

This is exercise 21.3, of Mathematical Omnibus: Thirty Lectures on Classic Mathematics, by D.B. Fuks and Serge Tabachnikov. The solution is given on page 448: Assume that $P$ is circumscribed. ...
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17 votes
Accepted

What's that shape? Inferring a 3D shape from random shadows

This is very similar to the cryo-electron microscopy problem: You want to image a certain macromolecule, and the scale of the macromolecule requires the use of an electron microscope. Unfortunately, ...
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17 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 ...
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  • 13.1k
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 ...
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  • 15.2k
17 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 \...
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16 votes
Accepted

What was the Question that led Euler to his Investigations on Polyhedra?

May I suggest this article by Joseph Malkevitch, an AMS Feature Column. Note particularly his point below (screen snapshot--not searchable) that polyhedra were simply not viewed, at the time, in terms ...
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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
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 ...
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  • 18.8k
14 votes

What's that shape? Inferring a 3D shape from random shadows

If you have no information about the vertices and allow nonconvexity there could be problems first you will only get information about the surface of the 3d shape but even there you will have problems....
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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 $C_k$, ...
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  • 13.1k
14 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 ...
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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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12 votes

Cut and Fold Polyhedron!

The general answer is sometimes Yes, various unfold/refold pairs of convex polyhedra exist, but not always, depending on what is meant by "cut." Here is an example that Erik Demaine, Marty Demaine, ...
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12 votes

Visibility of vertices in polyhedra

Note that the answer is yes in 2 dimensions, since any polygon can be triangulated (without adding additional vertices). Thus, every point in the interior sees at least 3 vertices of $P$. One can ...
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  • 28.7k
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....
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10 votes
Accepted

Computionally efficient vertex enumeration for (convex) polytopes

cddlib is rather old; a much more efficient implementation of the double description method is in PPL (Parma Polyhedra Library). One frontend to PPL can be found in Sagemath: http://www.sagemath.org/...
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10 votes
Accepted

Convex Polyhedra Scissors Congruence Problem

If the conjecture is true then the for the unit tetrahedron it will have a unit tetrahedron decomposible into a smaller regular tetrahedron and a regular octahedron. The regular octahedron of side $a$ ...
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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 ...
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10 votes

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 ...
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