Questions tagged [polyhedra]
For questions related to polyhedra and their properties.
270 questions
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The smallest set of polygonal regions that can all together form 2 different convex polyhedrons
We add a little to On reconstructing convex polyhedrons from disconnected faces that are all mutually non congruent
We call a set of polygonal regions that all together form a convex polyhedron a ‘...
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On reconstructing convex polyhedrons from disconnected faces that are all mutually non congruent
Ref: https://arxiv.org/pdf/1307.3472
It is well known that given only a set of convex polygonal regions (call this set of polygons a 'face set') and no further information, one cannot uniquely ...
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Twisted Rupert property
It has recently been proven that the 2017 conjecture that all
convex polyhedra $P$ are Rupert is false:
"A convex polyhedron without Rupert's property,"
Jakob Steininger, Sergey Yurkevich. ...
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A tetrahedron's vertices are random points on a sphere. What is the probability that the tetrahedron's four faces are all acute triangles?
This question resisted attacks at MSE.
A tetrahedron's vertices are independent uniformly random points on a sphere. What is the probability that the tetrahedron's four faces are all acute triangles?
...
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Random tetrahedron inscribed in a sphere: expectation of angle between faces?
The vertices of a tetrahedron are independent and uniform random points on a sphere.
What is the expectation of the internal angle between faces?
Simulation suggests $\frac{3\pi}{8}$
I simulated $10^...
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Prove that at least two edges of a polyhedron does not intersect a given plane
The same question was asked on SE
https://math.stackexchange.com/questions/5075489/prove-that-at-least-two-edges-of-a-polyhedron-does-not-intersect-a-given-plane
Let $P$ be a polyhedron (not ...
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Triangulation with prescribed vertices
It seems that the following statement can be proved using a Voronoi--Delaunay-type argument.
Is there a reference?
Let $P \subset \mathbb{R}^n$ be a compact subset (it is OK to assume that $P$ is a ...
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Does a peeling sequence always exist for noncrossing perfect matchings on 2n points?
Let ${M} = \{M_1, M_2, \dots, M_N\}$ be the set of all noncrossing perfect matchings on a circle with $2n$ labeled points arranged clockwise. Then $N = \frac{1}{n+1} \binom{2n}{n}$ is the $n$-th ...
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Presentation of the symmetry group of a regular star polyhedron from its Coxeter diagram
Here is my question:
As we know, for a string-type Coxeter diagram such as
$$\circ\overset{p}{---}\circ \overset{q}{---}\circ \overset{r}{---}\circ \cdots \circ $$
where $p,q,r,\ldots$ are integers ...
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Using Euler's characteristic formula to classify 3-uniform hypergraph?
For a 3-uniform hypergraph $H$ on a finite vertex set $V$, i.e., $H\subseteq \binom{V}{3}$, we assume $H$ has no isolated vertices and is connected (no non-trivial partition of $V$ such that each edge ...
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Are there convex polytopes with 1-skeleton given by 4-valent Cayley graphs of $S_{n}$ with generators long cycle, and n-1 cycle?
Consider a 4-valent Cayley graph generated by long cycle $(1,2,3,...,n)$ and n-1 cycle $(1,2,3,...n-1)$. (See the beautiful image from Wikipedia below for n=4).
Question 1: Are there convex ...
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Growth polynomial of the Associahedron graph ? (Is it approximately Gaussian ?)
Consider Associahedron, consider graph build from its vertices and edges. Choose some vertex. Let us count the number of vertices on distances $k$ from the selected vertex. Write a generating ...
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Tiling with one of each 3D shape
Encouraged by the positive solutions to my question,
Tiling with one of each shape,
I'd like to pose the $\mathbb{R}^3$ equivalent:
Q. Is there a tiling of $\mathbb{R}^3$ by (bounded) polyhedra, one ...
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Sum of Simplex Volumes with Corners from Points in Convex Configuration
Question:
given $k,\,k>n$ points in convex configuration and general position in $n$ dimensional Euclidean space, i.e. no $n+1$ points of which are co-hyperplanar,
what can be said about how the ...
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What is the "schematic" point of view for regular polyhedra?
Last week, I read Wikipedia's article on Alexander Grothendieck. It lists his twelve greatest contributions to mathematics as accounted for in Grothendieck's own Récoltes et Semailles. The final item ...
