Questions tagged [polyhedra]
For questions related to polyhedra and their properties.
1,156
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Is there a theory of isometrically embedded polyhedra on manifolds?
There is a book that is called embeddings in manifolds that studies topological embeddings and how they relate to each other (by homeomorphisms).
I was wondering if there is a study of isometrically ...
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What tools can show that (possibly irregular) dodecahedra do not fill space?
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, such that four ...
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A polyhedron whose faces are pairwise adjacent [closed]
I've heard there exists a polyhedron (besides tetrahedron) whose faces are pairwise adjacent but I can't find any information about this.
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Icosahedron with asymmetric coloring
I am trying to determine the number of unique solutions when placing "dots" on the sides of an icosahedron. There can be up to three dots placed symmetrically on each side. The dots are ...
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Help me make the sides and faces for some trapezohedrons (like a dice D10 or D14) with a circumsphere for EVERY desire
The goal: Make the face for a 10-sided dice or a heptagonal trapezohedron with a circumsphere, using as input one side of a face or the radius of the sphere.
Let us be on the same page:
Circumsphere: ...
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Diversity of edge numbers of space filling polyhedra
I am trying to find out if there is at least one polyhedron that tessellates for each valid edge number. I have found one for all edge numbers except 10 and 13. Here is my thought process so far.
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Is there a convex polyhedron with product of numbers of faces, vertices and edges equal to $3375$?
Is there a convex polyhedron with product of numbers of faces, vertices and edges equal to $3375$?
We are told that: $|E| \cdot |V| \cdot |F| = 3375$. From that we get $|E| = \frac{3375}{|V| \cdot |F|}...
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Can three antiprisms (uniform polyhedra) fit exactly around an edge, leaving no gaps?
Let $\tau=2\pi$ $=360^\circ$.
An $n$-gon antiprism has dihedral angles
$$\theta_n = \arccos\left(-\frac1{\sqrt3}\tan\frac\tau{4n}\right)$$
(where an $n$-gon meets a triangle) and
$$\phi_n = 2\arccos\...
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Projections of polyhedra are polyhedra?
Consider a (convex, compact, oriented) k-polyhedron $P\subset \mathbb{R}^N$, and $U\subset \mathbb{R}^N$ any $n$-dimensional linear subspace. Let $\pi_U:\mathbb{R}^n\to U$ denote the orthogonal ...
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I have discovered(?) a spiral pattern in stacked snub cubes.
I really hope I don't sound crazy.
I was messing around with different geometric shapes for inspiration for a fictional alien creature I am designing, and I noticed something odd about snub cubes that ...
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Rational vertices after affine transformations of polyhedra
Given a polyhedron $P$ with vertices $V_i[x_i,y_i,z_i]$, how can we determine if there exist an affine transformation which transforms the polyhedron into $P'$ with vertices $V'_i[x'_i,y'_i,z'_i]$, ...
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Sparse projection onto a single half-space
Let $x \in \mathbb{R}^n$ and $0 \neq a \in \mathbb{R}^n$ be a given vector and $b$ be a scalar such that the half-space defined by $a^{\top}x \leq b$ always include $0$.
Question
1- Is there any ...
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A convex polyhedron has $2$ vertices where $10$ edges meet. All its faces are quadrilateral. Show that it has $\geq 20$ vertices where $3$ edges meet.
A convex polyhedron has exactly two vertices where $10$ edges meet and all the faces of this polyhedron are quadrilateral. Show that this polyhedron has at least $20$ vertices in which three edges ...
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Dehn Invariants of two space-filling tetrahedra
Here are vertices for two space-filling tetrahedra.
$A = ((0, 0, 2),(0, 4, 2),(1, 2, 2),(2, 2, 0))$
$B = ((0, 3, 3),(1, 4, 2),(3, 2, 4),(1, 0, 2))$
Dihedral angles for A: {Pi/4, ArcCos[-(1/Sqrt[6])], ...
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A convex polyhedron has exactly $2$ decagonal faces and in each of vertices $4$ edges come together. Prove that it has at least $20$ triangular faces.
A convex polyhedron has exactly $2$ decagonal faces and in each of its vertices has exactly $4$ edges coming together. Prove that this polyhedron has at least $20$ triangular faces.
Let's call:
v = ...
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Is it possible to generate all permutations of the vertices of a polyhedron using only reflections?
I have a bunch of polyhedroa. In each of them all of the vertices are painted with different colors. For each polyhedron, I start the process with an initial permutation $\mathcal{P}_0$ and I want to ...
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References on H-representation of Convex Polyhedra
I have recently learned of the H-representation of polyhedra, which states that a polyhedron is the set of all solutions $\vec x$ of the following matrix inequality:
$$
\tag{1}
A \vec x \leq \vec b
$$
...
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Volume of tetrahedron by decomposition
Let $T$ be a tetrahedron with sidelength $s$. It's volume is proportional to $s^3$.
$$V(T)=ks^3$$
Let $T_1$ and $T_2$ be tetrahedra with sidelength $s_1=\frac 13 s$ and $s_2=\frac 23 s$
Can't we ...
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Proof that there are 4 Kepler-Poinsot solids [duplicate]
Under the extended definition of regularirty there are 9 regular polyhedra, 5 Platonic solids and 4 Kepler-Poinsot solids (where we assume them to have a finite volume to exclude honeycombes and other ...
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Can every inscribable 3-polytope be circumscribed while still preserving the maximum symmetry of its canonical form?
According to Steinitz's theorem, it is possible to prove that every 3-connected graph G(P) represents the skeleton of a convex 3-polytope P, which can be realized in R3 from its skeleton G(P). Various ...
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A name for convex polyhedra with regular polygons as faces
Is there a name for the class of polyhedra that are (a) convex, and (b) have regular polygons for faces? I don't want to invent a name if a name already exists.
Call the class P. P would include,
the ...
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Is my derivation of the tetrahedral bond angle correct?
Let $T$ be a regular tetrahedron with edge length $x$.
Let $A$ be one of the faces of $T$.
Let $P$ be the plane containing $A$.
Let $L$ be the line segment from the center of $A$ to one of the ...
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Arithmetic cost of determining whether a polyhedron is empty?
Given a polyhedron of the form $P=\lbrace x \in \mathbb{R}^n:x \geq 0, Ax=b \rbrace$, what is the arithmetic cost in order to determine whether the polyhedron is empty or not.
I know, that one can use ...
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Analog of simplicial sets for space gluing from arbitrary convex polyhedra
We can describe a space gluing from simplices as simplicial set.
When space is gluing from cubes there is notion of cubical set.
What is about some other class of polyhedra? What is known in this ...
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Need polynomial time implementation for pycddlib polyhedra representation.
Request for reference
We are using the python library pycddlib as API working with polyhedra. Here is the link
https://pycddlib.readthedocs.io/en/latest/polyhedron.html
We have set of linear equations ...
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Polya's enumeration theorem for edge and vertex coloring combined.
Let's say we have a tetrahedron labelled as such:
We want to find the number of distinct ways to color the vertices and edges, such that 2 vertices are green, 2 vertices are red, 4 edges are black ...
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Which simplicial complexes are completely determined by the 1-skeleton of their dual cell 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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How to detect if a tetrahedron has a flat-like shape
Creating tetrahedra
I have this OpenSCAD code that creates tetrahedra by getting its points:
...
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Almost regular complex polytopes
Within his 2nd edition of "Regular Complex Polytopes" Coxeter seems to have added a section about "Almost Regular Polytopes". I for one do not have access to that book, all I can ...
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How to make sure the order of a tetrahedron points is in a certain way
Tetrahedon points order
I'm going to create tetrahedra whose points should exactly be in a specific order like below:
Sample: OpenSCAD
For example, my points are logged using OpenSCAD like this:
<...
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Are the extreme points of the following polyhedron all integers?
Let
\begin{equation}
S := \left\{x\in\mathbb{R}^{m\times n}:
\begin{aligned}
& 0\leq x_{ij}\leq 1,\ i=1,2,\ldots,m,\ j=1,2,\ldots,n \\
...
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Is it ture that any zonoid (not a trivial zonotope), must be a affine image of a unit ball (l2)?
I am trying to find some literature discussing zonoid in 3-dimensional space. And a seemly simple question that I am considered specifically is what are those non-trivial (not being a zonohedron) ...
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real space embedding of complex polytopes
Because of $\mathbb C^n = \mathbb R^{2n}$ complex polytopes ought be embeddable as substructures of real space polytopes (with doubled up dimensionality).
Look, complex edges do have $m\ge2$ vertices ...
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In linear programming, given an optimal unique solution, is it possible to obtain multiple optimal solutions by adding a new constraint?
In post-optimal analysis, when I already have an optimal solution (unique solution). I can add a new constraint and obtain multiple optimal solutions, but only if the new constraint eliminates the ...
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Longest line segment in a convex polyhedron in $\mathbb{R}^3$ in direction $\mathbf{n}$
Let us consider a non-empty convex polyhedron $\mathcal{C} \subset \mathbb{R}^3$ and a vector $\mathbf{n} \in \mathbb{R}^3$. If we choose a point $\mathbf{x} \in \mathbb{R}^3$, then $ \mathbf{x}$ and $...
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operants on complex polytopes
I would assume that complex polytopes (according to Shephard and Coxeter) too could be alternated (snubbed), rectified, truncated, etc. - possibly given some further restrictions.
However I have ...
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The Rolling Petersen Graph
Take an icosahedron. Label opposing faces with 0 to 9. The resulting object is a Petersen graph. There are a few dice-makers that have sorta made these, but they always seem to switch the 6 and 9.
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Listing vs. counting perfect matchings in a graph
In his Polyhedral Computation textbook, Fukuda writes:
It is known that the counting problem [of perfect matchings] is #P-complete even for bipartite graphs. There are polynomial algorithms for the ...
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2D and 3D finite rotation groups are very well behaved, what about 4D?
I've recently read up on 2D and 3D finite rotation groups. But I'm struggling to find resources on the analogous results for 4D and I'd appreciate some help finding answers/resources.
For context: In ...
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How to find the range of possible numbers of vertices and edges of a convex polyhedron, given the number of faces?
A polyhedron has $F$ faces. Find all possible amounts of its vertices and edges.
Of course, for small values of $F$, it's possible to just list the possibilities. But how to handle the general form ...
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Transform an inequality system into the input for the simplex algorithm
I have a problem that the simplex algorithm was not discussed in the course, but a sample solution uses the simplex algorithm in order to obtain a Gomory Mixed Integer Cut. The dual problem to my ...
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Is any finite subset of $S^2$ satisfying the following properties, the vertex set of a regular convex polyhedron?
Let $S^2 \subset \mathbb{R}^3$ denote the standard unit sphere.
Let $A \subset S^2$ be a finite set satisfying each of the following properties:
For each $a,a' \in A$ there exists a rotation $R$ (the ...
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Adjacent Basic Solutions of a Standard Polyhedron
Before phrasing my question, I have to mention some definitions and theorems to make sure we stay on the same ground. However, it may become a little bit lengthy and you can take a look at the end of ...
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Is some convex combination of vectors in the same direction as all those vectors?
Let $S$ be a finite set of vectors in $\mathbb{R}^d$. Suppose there exists a vector $w$ such that $a^Tw > 0$ for all $a ∈ S$. Then does there exist a vector $\widehat{w}$ such that $a^T\widehat{w} &...
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Schlafli symbols for antiprisms, crossed antiprisms
I am familiar with Schlafli symbols for polygons and polygrams, but it gets way more complicated for antiprisms. I get that {7} describes a heptagon, {7/2} describes a heptagram with 7 edges where ...
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Find all extreme points of 4 variable polyhedral set of dimension 3
$$P = \{x\in\mathbb R^4_+ | x_1 + 2x_2 + 3x_3 + 4x_4 = 36, x_1 + x_2 + x_3 + x_4 \le 12\}$$
a) By enumerating all quadruplets of relevant constraints, find all
extreme points of $P$.
I know that the ...
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Does every pointed polyhedral cone in $\mathbb{R}^n$ generated by its $n+1$ extreme rays contain two extreme ray vectors whose sum is in the interior?
This might be a stupid question, but I have been thinking and googling for some time now and I still cannot seem to find a response.
I am interested in how many extreme rays a finitely generated (...
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Visualization of a Standard Two Dimensional Polyhedron
I am reading this book on linear programming, and the authors give an excellent exposition of the topic by the interplay between the underlying algebra and geometry. Their main approach is to motivate ...
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Extreme rays, recession cone of polyhedron
We have a polyhedron $P\subset R^2$ defined by:
$P:=\{ x\in R^2$
$4x_1-2x_2 \leq -8$
$−x_2≤2$
$-2x_1-x_2≤-4$
$−2x_1+x_2≤0$
Let X={(2,0)} Y{(1,2)}
a) Find the dimension of the smallest face $F\subset P$...
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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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