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 ...
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Smooth toric variety which is a cube is a bott tower (reference request)
According to Lee, Masuda and Park (page 3), the following result is "well-known in toric topology". I've found a proof, but I would like a published reference.
Let $X$ be a toric variety. ...
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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, ...
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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,...
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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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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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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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?
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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,\...
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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 ...
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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 ...
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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 ...
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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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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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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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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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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 ...
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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 ...
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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
$$\begin{array}{ccc}
\text{8 vertices} & \left(0,0,0,\pm1\right) & \text{all permutations,}\\
\text{16 vertices} & \frac{1}{2}\left(\pm1,\...
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What is the difference between a simple polyhedron and a triangulated graph?
On a famous website I've seen the following:
The skeletons of the simple polyhedra correspond to the triangulated graphs, the smallest of which are illustrated above. That "illustration above&...
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Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?
Suppose that $P$ is a polyhedron represented by
$$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 ...
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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 ...
2
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253
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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 $...
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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 ...
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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 ...
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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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