# Questions tagged [polyhedra]

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### If we know the combinatorics of a polyhedron, and all but one of its dihedral angles, does that uniquely determine the remaining dihedral angle?

If we know the combinatorics of a polyhedron, and all but one of its dihedral angles, does that uniquely determine the remaining dihedral angle? I’m happy to assume the polyhedron is simply connected, ...
• 2,876
75 views

### What tools can show that (possibly irregular) dodecahedra do not fill space?

(Formerly on MSE.) Here is a fairly natural question: Can three-dimensional space be filled with convex polyhedra of the same incidence structure (if not the same geometry) as the regular dodecahedron,...
• 2,743
1k views

### Concrete works by Alexandre Grothendieck, other than Dessin d'Enfants?

For me "Dessin d'Enfants" by Alexandre Grothendieck is the more concrete research work he has done. I would like to know if there are others. When he was teaching at Montpellier University (...
91 views

### Software for computing polytopes

As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
49 views

### Polyhedra volume, faces and edges from vertices

Given a set of vertices in 3D corresponding to a convex polyhedron, what is the most efficient way to find its volume, faces, and edges? I've found some techniques using convex hulls. But I think I ...
• 101
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### Dividing a polyhedron into two similar copies

The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original). Right ...
211 views

### The realization space of non-convex polyhedra - What is known?

The space $\mathfrak R_{\mathrm c}(P)$ of convex realizations of a (3-dimensional, spherical) polyhedron $P$ is known to be well-behaved: it is a contractible manifold of dimension $\#\text{edges}+6$ (...
• 13.1k
1 vote
67 views

### Polyhedra with equal faces

It is easy to see that for isosceles tetrahedra (https://en.wikipedia.org/wiki/Disphenoid) all faces are equal acute triangles. If we consider regular tetrahedra and attach a regular triangular ...
• 681
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### Is it possible for the dihedral angles of a polyhedron to all grow simultaneously?

(Originally on MSE.) Suppose $P$ and $Q$ are combinatorially equivalent non-self-intersecting polyhedra in $\mathbb{R}^3$, with $f$ a map from edges of $P$ to edges of $Q$ under said combinatorial ...
• 2,743
1 vote
48 views

### Enumeration of uniform polyhedra

[I already asked this question on MSE (here) but got no answer so I am trying here] It is known that there are two infinite classes of polyhedra (prisms and antiprisms) together with $75$ uniform ...
• 1,101
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### Bounding distance to an intersection of polyhedra

Let $P$ and $Q$ be polyhedra in ${\mathbb R}^m$ with a non-empty intersection. I believe there should exist a constant $C_{PQ}>0$ such that for any point $x\in {\mathbb R}^m$ the following ...
• 1,559
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### Bounding distance to a polyhedron

I need to estimate the Euclidean distance from a point $x\in {\mathbb R}^m$ to a polyhedron $P\subset {\mathbb R}^m$ in terms of distances from $x$ to the tangent hyperplanes which define $P$. By a ...
• 1,559
111 views

### 4 triangular faces 6 vertices not tetrahedron [closed]

I have made a solid and would like to know its' name, volume and related formulas. It is made using a flat potato chip bag. The end opposite the factory seal is sealed perpendicular to the factory ...
987 views

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

If such a pyramid exists, could someone provide the coordinates of its vertices?
1 vote
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### Dodecahedron deformation II

(Follow-up to this question) Can a dodecahedron be deformed into a great stellated dodecahedron while maintaining the number of dimensions each element occupies?
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### Can a dodecahedron be deformed into a great stellated dodecahedron?

Can a convex regular dodecahedron be deformed into a great stellated dodecahedron while keeping all pentagons planar and all edges of nonzero length the whole time?
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348 views

### Request for an article by Jim Lawrence

Jim Lawrence has a very important paper on the topic of valuations on polyhedra called "Rational-function-valued valuations on polyhedra", published in the DIMACS volume Discrete and ...
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