The polyhedra tag has no wiki summary.

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### Equiprojective polyhedra

Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open,
and which some might find intriguing.
Define an ...

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**1**answer

275 views

### Standard (special) spines and hyperbolic structure on 3-manifolds

My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...

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### Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear ...

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**1**answer

290 views

### Complexity of the union of randomly rotated unit cubes

It is a remarkable fact that the union of congrent cubes
has only at most near-quadratic combinatorial complexity,
$O^*(n^2)$ for $n$ cubes, known to be almost tight.
This contrasts with the union of ...

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**2**answers

219 views

### Volumes of convex vs non-convex polyhedra with prescribed facets areas

It is known that given a set of Areas $A_f$ and normals $\vec{n}_f$ if $\sum_f A_f \vec{n}_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov ...

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**1**answer

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### non-convex Polytope definition

I have a simple question. I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$,
we can define a convex polytope in the following way:
$$P:= \Big\{ \sum_{u\in S} \mu_u u \,\Big| \, ...

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**1**answer

354 views

### Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...

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### Polyhedra Classification

The following is inspired by this question. From time to time I search the web for tables of polyhedra, but without much success. Part of the problem is that there are many non-equivalent questions ...

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**3**answers

707 views

### Solid angles of a tetrahedron

This is a problem I have had for a while. For a triangle, the side opposite the largest angle has the largest length (and similarly for smallest angle). For a tetrahedron, the question is whether the ...

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**0**answers

91 views

### Convex Polyhedra with Largest Constant Dihedral Angle for Given Number of Faces

Which $20$ faced convex polyhedron has the largest constant dihedral angle?
Which $24$ faced convex polyhedron has the largest constant dihedral angle?
Also, what about $30$ or $32$ faced polyhedra? ...

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**3**answers

619 views

### Visualizing polyhedra from their 1-skeletons

Except for a few simple cases (typically pyramids and prisms) I find it hard to visualize a polyhedron from its 1-skeleton embedded in the plane, e.g. the hexahedral graph 5, as can be seen here.
...

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**1**answer

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### Polyhedra that combinatorially shadow a sequence

Let $P$ be a polyhedron in $\mathbb{R}^3$.
Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if
there is a continuous rotation of $P$ such that its orthogonal-projection
shadows ...

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**2**answers

822 views

### Is the tensor product of polyhedra a polyhedron?

Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...

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533 views

### Dilogarithm, tetrahedrons, and hyperbolic space

The Bloch-Wigner function $D(z)$, gives the volume of a ideal tetrahedron in hyperbolic space $\mathbb H3$, where z is the cross-ratio (z1,z2,z3,z4) parametrizing the tetrahedron in $\mathbb CP1$.
...

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290 views

### Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb{H}^3$, has a boundary $\mathbb{CP}^1$.
A ideal tetrahedron in $\mathbb{H}^3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb{CP}^1$.
The four ...

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**2**answers

532 views

### Does Euler's formula imply bounds on the degree of vertices in a 3-polytopal graph?

A corollary of Euler's formula tells us that the edge-vertex graph of every convex 3-polyhedron must have a face with either 3, 4, or 5 edges, using an argument about the average degree of vertices.
...

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**1**answer

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### coarser than triangulations “almost partitions” into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an ...

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**1**answer

436 views

### Wrapping a convex polyhedron with string

This is a meta-question, rather than a specific mathematical question.
I am seeking a mathematical definition that captures the following physical idea.
Suppose you have a convex polyhedron $P ...

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**1**answer

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### How can pushing a vertex in a polytope lead to merging facets?

I'm trying to understand the Lemma 2.2 and Corollary 2.3 of Francisco Santos paper "A counterexample to the Hirsch conjecture": http://arxiv.org/abs/1006.2814
Corollary 2.3 is a proof of a result of ...

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**1**answer

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### Building a genus-$n$ torus from cubes

I wonder if this has been studied:
What is the fewest number of unit cubes
from which one can build an $n$-toroid?
The cubes must be glued face-to-face,
and the boundary of the resulting ...

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**2**answers

721 views

### Fractal Tiling of Rhombic Dodecahedra

Hello, this is my first question on Math Overflow...
Rhombic dodecahedra can be tiled in 3-space, leaving no gaps. This tiling corresponds to the close-packing of spheres.
Consider a "nucleus" ...

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**1**answer

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### Grading a non-graded poset as squeezed as possible

Here is a curiosity question (motivated by the recent revamp of ranked-poset routines in Sage).
Let $P$ be a finite poset. We look for a family $\left(a_p\right)_{p\in P}$ of real numbers summing up ...

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**1**answer

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### What nets fold to polyhedra?

There is a classic (and open) problem asking whether every polyhedron can be unfolded to give a non-overlapping net. The converse problem has been studied asking which polygons can be folded in some ...

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### Acute triangulation

Assume that $S$ is a finite 2-dimensional simplicial complex equipped with a metric $d$
such that each triangle is isometric to a plane triangle (so $(S,d)$ is a polyhedral space).
Is it possible ...

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302 views

### Maximum volume convex body coverable by a unit square

Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected)
pieces (where $k=1$ means just the square). Your task is to construct the largest volume
convex body ...

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648 views

### Efficient topological triangulations of non-convex polyhedra

Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a topological triangulation with complexity $O(n)$?
Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ ...

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**1**answer

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### Update to Shephard's “Twenty Problems on Convex Polyhedra”

Forty-three years ago, Geoffrey Shephard published an influential list of open problems
on convex polyhedra.
Progress has been made on several of his problems, and perhaps some have been completely ...

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votes

**3**answers

534 views

### Number of Hyper-cube cuts

In how many ways a single hyperplane can cut a hypercube? Two "ways" are considered different, if the sets into which they divide vertices of the hypercube are different. So e.g. a line can cut ...

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### On Dehn's infinitesimal rigidity theorem

Dehn's theorem states that any simplicial strictly convex polyedron P in Euclidean 3-space is infinitesimally rigid (that is, any non-trivial first order deformation of P induces a variation of its ...

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### How many vertices/edges/faces at most for a convex polyhedron that tiles space?

I wonder if this problem has already been examined before:
Consider a convex polyhedron that tiles $\mathbb R^3$. What is the maximum of vertices/edges/faces that such a polyhedron can have?
...

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### Placing points on a sphere so that no 3 lie close to the same plane

Motivation
I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...

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209 views

### What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?

The question is mostly in the title.
Imagine I have some non-convex polyhedron $P$, and I would like to find the smallest diameter ring that it can pass through in 3-space, undergoing any necessary ...

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### When does every point in a polytope lie along a chord between its edges?

Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the simplex lies on a chord between two non-adjacent edges of the simplex. Or, ...

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### Not quite regular polyhedra

Take a naive interpretation of regular polyhedra:
All vertices (including epsilon ball) congruent
All edges congruent
All faces congruent
We can now find interesting families by removing one ...

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**1**answer

1k views

### intersection of convex and non-convex polyhedra

Hi everyone,
I am trying to find the best appropriate way to intersect polyhedra which may be non-convex.
The number of vertices that build the polyhedron is hence always small (up to 20 or so).
...

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### polyhedra with equilateral pentagons faces.

In page http://loki3.com/poly/isohedra.html around six polyhedra with equilateral pentagons as faces are shown: a pyritohedron, icositetrahedrons... Is there a complete list of this kind of polyhedra? ...

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**1**answer

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### Are there non-tiling polyhedra that pack arbitrarily well?

The fact that an upper bound on the packing density $< 1$ has only recently been exhibited for regular tetrahedra in $\mathbb{R}^3$ (see this question) suggests that proving concrete bounds of ...

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### Pach's “Animals”: What if the genus is positive?

Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...

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**2**answers

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### how to estimate a polyhedron(convex hull) classifier from data sample

Given a set of points $X\in\Re^D$, they have labels $Y\in${$-1,+1$}. I would like to separate the data labeled +1 and the data labeled -1 by a polyhedron.
$min_w \sum_i \xi_i + \frac{1}{2}\|w\|_2^2$
...

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votes

**1**answer

1k views

### Finding a minimum bounding sphere for a frustum

I have a frustum (truncated pyramid defined by six planes) and I need to compute a bounding sphere for this frustum that's as small as possible.
I can choose the centre of the sphere to be right in ...

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**1**answer

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### Tetris in 3D with 5 units [closed]

Background: There are 7 "bricks" used in the game of Tetris. These are the 7 unique combinations of 4 unit squares in which every square shares at least one edge with another square. ("unique" in this ...

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### Four Dimensional Origami Axioms

What are the axioms of four dimensional Origami.
If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded ...