# Questions tagged [polyhedra]

The polyhedra tag has no usage guidance.

256
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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, ...

5
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0
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75
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### 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
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3
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### 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 (...

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0
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### 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 ...

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0
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### 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 ...

15
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1
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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 ...

2
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1
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### 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 $\#\text{edges}+6$ (...

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0
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### 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 ...

10
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1
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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 ...

1
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0
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### 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 ...

4
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1
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141
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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 ...

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2
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194
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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 ...

0
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1
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### 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 ...

4
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3
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987
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### 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
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0
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### A face and all its neighbors: terminology?

Suppose $F$ is a face of a 2-complex, and $F_1,\dotsc,F_n$ are the faces that are adjacent to (i.e., share an edge with) $F$. Is there a standard term for a collection of faces of the form $\{F,F_1,\...

5
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1
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### Regular polyhedral spaces

By symmetrically gluing together opposite faces of a dodecahedron together, one of three spaces can be obtained, depending on the angle the faces are rotated by before twisting. In fact, this can be ...

2
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0
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### Why is it impossible to create a numerically balanced die with more than 120 sides?

I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...

5
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1
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### Orienting the dual of the associahedron

Let $A_n$ be the dual simplicial complex to the associahedron on $n$ letters. The complex $A_n$ is thus a simplicial triangulation of an $(n-3)$-dimensional sphere. The vertices of $A_n$ correspond ...

2
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2
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### Is there a way to parametrize the configuration space of all convex polyhedra of a given combinatorial type as a convex set?

I'm sure this is easy/known, but I'm not hitting an appropriate search term for finding the answer and the coffee hasn't kicked in enough to come up with it myself:
Let $T$ be a simplicial 2-complex ...

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1
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### Which simplicial complexes are completely determined by the 1-skeleton of their dual polyhedral complexes?

Consider the following line of reasoning that shows certain simplicial complexes (of arbitrary dimension) are completely determined by corresponding graphs:
The facet complex of any simplicial ...

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0
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### Does this sequence stop?

Let $\{ X_i\}$ ($i=1,2,\ldots $) be a family finite CW-complexes such that $X_{i+1}$ is homotopy domintaed by $X_i$, i.e. there exists contionuos maps $g_i:X_i \to X_{i+1}$ and $f_i :X_{i+1} \to X_i$ ...

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0
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### Inside-out dissections of solids

We add to Inside-out dissections of polygons - a generalization. The inside-out (fully inside-out) dissections are defined on pages linked there.
How does one inside-out dissect a tetrahedron into ...

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0
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### About the number of faces of the conification of a polytope

Let $P\subset\mathbb{R}^n$ be a polytope of dimension $(n-1)$ such that the origin $\vec{0}\not\in\text{Aff}(P)$, where $\text{Aff}(P)$ denotes the affine hull of $P$ in $\mathbb{R}^n$. Now, we ...

5
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1
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### The bounded complex of a polyhedral decomposition

Let $\mathscr{P}$ be a polyhedral decomposition of a real vector space $V$. By that I mean that $\mathscr{P}$ is a finite set of polyhedra in $V$ satisfying the following three properties:
The union ...

0
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1
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### How can I find the hyperplane passing through a 600-cell

I have a 600-cell, whose coordinates are given by
$$\begin{array}{ccc}
\text{8 vertices} & \left(0,0,0,\pm1\right) & \text{all permutations,}\\
\text{16 vertices} & \frac{1}{2}\left(\pm1,\...

1
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0
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### What is the difference between a simple polyhedron and a triangulated graph?

On a famous website I've seen the following:
The skeletons of the simple polyhedra correspond to the triangulated graphs, the smallest of which are illustrated above. That "illustration above&...

1
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0
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### Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?

Suppose that $P$ is a polyhedron represented by
$$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$
and $P$ contains interior points. Moreover, the ...

2
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0
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### Is it possible to deduce Poincaré duality from duality of polytopes?

I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance:
https://math.stackexchange.com/a/14469/454016
Poincaré duality is explained through a duality of ...

1
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1
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### Intersection of conical neighbourhoods on a polyhedral space

Let $P$ be a non-negatively curved (in the Alexandrov sense) polyhedral space (of dimension 3, say), $p,q\in P$ be vertices, and let $e$ be an edge connecting $p$ and $q$. Assume $e$ has cone angle $0&...

2
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1
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### Tangent cone on polyhedral spaces

Let $X$ be an n-dimensional polyhedral space with, say, $n\geq 3.$ Let also $p\in X$ be a vertex on a triangulation $\tau$ of $X,$ so a vertex on the polyhedral space.
The tangent cone (as a metric ...

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### How many ways to flatten a Tesseract onto a table?

A cube can be cut and flattened out onto a table in a way that the faces stay connected and none of them overlap. There are $384$ ways to make the cuts and $11$ distinct meshes emerge (see here). And ...

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### Endpoints of intrinsic diameter of a convex polyhedron

Let $P$ be a convex polyhedron in $\mathbb{R}^3$, and $d(P)$ its intrinsic diameter,
i.e., the longest shortest surface path between two points. Say that $P$ is of
class
$D_0$ if neither endpoint of $...

0
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0
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### Polynomial-time algorithm for exact projection to polyhedral cone

Given $c \in \mathbb{R}^d$ and $A \in \mathbb{R}^{n \times d}$, project $c$ to the polyhedral cone $\{x \in \mathbb{R}^d \mid A x \leq 0\}$. Is there an algorithm that outputs an exact solution to ...

4
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1
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### On polyhedrons with specified numbers of congruent faces

Basic question: Given 3 integers n, n1 and n2 such that n1+n2 = n, to form an n-face polyhedron such that n1 of its faces are mutually congruent and the remaining n2 faces are different but congruent ...

0
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1
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### Is $\pi_2 (X_i)$ a free $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,2$?

Let $X_1$ be the suspension of $\mathbb{R}P^2$ and $X_2=\bigvee_{1\leq i\leq n} (\vee_{r_i} \mathbb{S}^i)$.
Is $\pi_2 (X_i)$ a projective (or a free) $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,...

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### Hilbert’s third problem and what a polyhedron is [closed]

What is the definition of a polyhedron used by Hilbert’s third problem?

5
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### Are the polyhedral cones the only examples of cones that remains closed when they are added to vector subspaces?

Let $C \subset \mathbb{R}^{n}$ be a closed convex cone. If one wants to know whether the linear map $T:\mathbb{R}^{n} \to\mathbb{R}^m$ sends the closed set $C$ to another closed one, $T(C)$, it is ...

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### Secondary polytope

Given a polytope $P$, what do the points of the secondary polytope correspond to?
I know that the vertices of the secondary polytope correspond to regular triangulations of $P$.
But what do the ...

10
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2
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### Great polyhedra: What does "great" signify?

Great Cubicuboctahedron
Great Icosacronic Hexecontahedron
Great Rhombic Triacontahedron
Great Snub Icosidodecahedron
Great Stellated Dodecahedron
Great Triakis Octahedron
...
There are many polyhedra ...

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### Tiling space with supertile of hypercube unfoldings

Two students in my class
asked and answered what might be a novel question.
It is well known that the cube has exactly $11$ edge-unfoldings
(or "nets"), as shown below:
(Image from ...

0
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0
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### Explicit equation for border of the Minkowski sum of sets

Assume we have sets of the form
$$
M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}
$$
where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$.
Goal
I am looking for an (explicit) representation ...

2
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1
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### Generic infinitesimal rigidity of polyhedra

Let $M$ be a 1-skeleton of a triangulation of a sphere with $V$ vertices and $E$ edges.
Definition 1 A polyhedron is a map $M\to \mathbb R^3$ that is affine on edges (and non-degenerate on faces). The ...

3
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1
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### Well-behaved trajectories

Call trajectory any continuous function $f: \mathbb{R}_{\geq 0} \to \mathbb{R}^n$ (here, $\mathbb{R}_{\geq 0}$ is interpreted as time).
A polyhedral partition of $\mathbb{R}^n$ is a finite set of ...

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3
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### Are there Monohedra with odd numbers of faces?

A monohedron is a convex polyhedron with all faces mutually congruent but with no other symmetry necessarily needed. So obviously, this is a wide class of polyhedra that includes the Platonic solids ...

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### Dimension of configuration space of triangulated convex polyhedron

The configuration space of all tetrahedra is $5$-dimensional, perhaps a non-obvious fact.
There are $12$ face angles, but the sum of each of the four faces angles is $\pi$,
reducing $12$ to $8$ ...

0
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1
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### The dimension of the normal cone of a face in a polytope

Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\
This seems to be intuitively obvious but I can't ...

2
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0
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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?

6
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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?

3
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### 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 ...

5
votes

1
answer

243
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### Convex polyhedra with non-congruent faces

Question: Are there convex polyhedra wherein all faces are convex polygons with same area and perimeter and no two faces are mutually congruent?
Remarks: If the answer to above is "no", then,...