# Questions tagged [polyhedra]

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### Embedding an icosahedron

A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group. Let me start with the example of a ...
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### Tetrahedra passing through a hole

Assume a plane $P\subset\mathbb R^3$ has a hole $H$, and that the hole is topologically a compact disc. Being so, $P\setminus H$ does not separate the space. A regular tetrahedron $\sigma^3$ (of edge-...
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### Finding Motzkin's original paper on copositive quadratic forms

I am currently in the process of writing my thesis about copositive matrices and would like to write a chronological narrative about the ascent of these matrices to the prominent place they have today ...
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### Clustering of vertices in an $n$-dimensional cube

Consider the vertices of an $n$-dimensional cube. The distance between two vertices is measured as the minimum number of edges between the two vertices. Now consider a subset of these vertices. If we ...
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### Two questions on the permutohedron

The $n$-dimensional permutohedron $P_n$ is the polytope given by the convex hull of all the possible permutations of the vector $(1,2,\dots,n+1)\in\mathbb{R}^{n+1}$. So it has $(n+1)!$ vertexes. I ...
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### Minimum number of distinct triangles for tesselating the sphere

Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
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### How to define and compute the degree of congruence of two rigid polyhedra in same type with knowing vertex coordinates?

If I have two sets of points in 3-dimensional space, each sets of points are the coordinates of vertices of a polyhedron. The two polyhedra have same type, so we don't need to consider the topological ...
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### What are the known convex polyhedra with congruent faces?

Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
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### Is Sydler's theorem concerning Dehn invariants constructive?

Sydler proved something of a converse to Dehn's negative resolution of Hilbert's 3rd problem. To quote Wikipedia, Sydler showed that "every two Euclidean polyhedra with the same volumes and Dehn ...