# Tagged Questions

432 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 ...
346 views

### 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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### 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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### 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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### Polar duality and -1 [closed]

We define the polar dual of a polytope $P$ as the set $$\{x\in \mathbb{R}^n: x \cdot a\geq -1 \text{ for all } a\in P\}$$ Why do we require $-1$ instead of $-2$ or any other constant?
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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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### 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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### 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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### 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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### 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 ...
### Pach's “Animals”: What if genus $> 0$ ?
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 ...