The polyhedra tag has no wiki summary.

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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

380 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

618 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

258 views

### 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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**0**answers

289 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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**3**answers

632 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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**0**answers

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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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**2**answers

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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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**2**answers

793 views

### 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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**2**answers

205 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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**3**answers

634 views

### 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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**3**answers

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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

375 views

### 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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**1**answer

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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 ...

**2**

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 ...

**0**

votes

**1**answer

335 views

### 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 ...