Questions tagged [polyhedra]
The polyhedra tag has no usage guidance.
249
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non-convex Polytope definition
I have a simple question. I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$,
we can define a convex polytope in the following way:
$$P:= \Big\{ \sum_{u\in S} \mu_u u \,\Big| \, \...
0
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1
answer
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The dimension of the normal cone of a face in a polytope
Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\
This seems to be intuitively obvious but I can't ...
5
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2
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Dimension of configuration space of triangulated convex polyhedron
The configuration space of all tetrahedra is $5$-dimensional, perhaps a non-obvious fact.
There are $12$ face angles, but the sum of each of the four faces angles is $\pi$,
reducing $12$ to $8$ ...
2
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0
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Dodecahedron deformation II
(Follow-up to this question)
Can a dodecahedron be deformed into a great stellated dodecahedron while maintaining the number of dimensions each element occupies?
6
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1
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Can a dodecahedron be deformed into a great stellated dodecahedron?
Can a convex regular dodecahedron be deformed into a great stellated dodecahedron while keeping all pentagons planar and all edges of nonzero length the whole time?
2
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0
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Polyhedron coordinate bound
Given a polyhedron
$$Ax\leq b$$
where we assume $A\in\mathbb Q^{m\times n}$ and $b\in\mathbb Q^{m}$ and it takes $L$ bits to represent the inequalities what is a good bound on the quantity $\|y\|_\...
3
votes
1
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Request for an article by Jim Lawrence
Jim Lawrence has a very important paper on the topic of valuations on polyhedra called "Rational-function-valued valuations on polyhedra", published in the DIMACS volume Discrete and ...
28
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5
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Visibility of vertices in polyhedra
Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
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Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?
A six year old question,
Which unfoldings of the hypercube tile $3$-space?, has just been answered by
Moritz Firsching:
All $261$ unfoldings tile space!
So now we know:
For $d=2$, the unfolding of ...
5
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1
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Convex polyhedra with non-congruent faces
Question: Are there convex polyhedra wherein all faces are convex polygons with same area and perimeter and no two faces are mutually congruent?
Remarks: If the answer to above is "no", then,...
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Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?
Recently Mark McClure constructed and displayed
the 261 unfoldings of the hypercube (tesseract)
in response to the question,
"3D models of the unfoldings of the hypercube?":
The first 9 unfoldings ...
1
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0
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118
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Regularity of Laplace equation on non-convex polyhedral domain
This might be a known problem, but I could not find a precise answer.
I have the following Laplace equation
\begin{equation}
\begin{cases}
-\Delta u = f & x \in \Omega;\\
\quad\: u = g & x \in ...
3
votes
0
answers
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"Slim" directed polytopes: any established name for them?
This is a "looking for context" question.
Let's say that a polytope is directed if its 1-skeleton is an oriented graph with no cycles, one source, one sink. (Edit: let us additionally assume ...
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1
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Detecting non-negativity of a single constraint by polyhedral constraints - $I$
We consider $$\langle a,x\rangle=b$$ (linear constraints) where $x\in\mathbb R_{\geq0}^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\...
1
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Counting $\bmod 2$ number of vertices of sparsely represented polyhedra
Given a polyhedron
$$Ax\geq b$$
is there an $NC^1$ or an $NC^2$ algorithm to count the number of vertices $\bmod2$?
Assume $A\in\{0,1\}^{m\times n}$ and $b\in\mathbb Z^{m}$ ($m=O(n)$) and assume rows ...
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Detecting non-negativity of a single constraint by polyhedral constraints - $II$
Let $$\langle a,x\rangle=b$$ be a linear constraint where $x\in\mathbb R^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\mathbb Z_{\...
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Number of linear inequalities describing a polyhedron with prescribed number of vertices
If a polytope has $d$ vertices in $k$ dimensions how many linear inequalities is required to describe it?
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Tetrahedra passing through a hole
Assume a plane $P\subset\mathbb R^3$ has a hole $H$, and that the hole is topologically a compact disc. Being so, $P\setminus H$ does not separate the space. A regular tetrahedron $\sigma^3$ (of edge-...
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Integrality of polyhedra
Given two polyhedra in $H$ representation $P_1:Ax\leq b$ and $P_2:Bx\leq c$ which are integral are bounded when is their intersection also integral?
Given two polyhedra in $H$ representation $P_1:Ax\...
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Number of vertices in a polyhedron
Consider polytopes
$$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$
$$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$
$$B[z_{1},z_{2},z]'\leq c$$
having vertex count $v_1,v_2$ and $v$ respectively.
We ...
1
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1
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Exactly counting number of vertices of a polyhedron
Suppose $Ax\leq b$ is a polyhedron, where the number of rows in $A$ is $r$, the vector $x$ lies in $\mathbb R^n$ and the rank of $A$ is $t$. Assume minimal number of hyperplane inequalities to define ...
2
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Distinguishable knots (with constraints) over polyhedra
I'm trying to find the number of distinguishable knot projections over certain convex regular polyhedra according to the following constraints. On each face on the polyhedron the knot will have a ...
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Homotopy domination of a wedge of two polyhedra
The topological space $A$ is called homotopy dominated by the space $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq id_A$.
Question: Suppose that $...
5
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Embedding an icosahedron
A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group.
Let me start with the example of a ...
4
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Finding Motzkin's original paper on copositive quadratic forms
I am currently in the process of writing my thesis about copositive matrices and would like to write a chronological narrative about the ascent of these matrices to the prominent place they have today ...
26
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4
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Why do some uniform polyhedra have a "conjugate" partner?
While browsing through a list of uniform polynohedra, I noticed that the square of the circumradius $R_m$ of the small snub icosicosidodecahedron ($U_{32}$) with unit edge lengths is,
$$R_{32}^2 =\...
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Classification of pseudoregular polyhedra
In contrast to a regular polyhedron, which has one orbit of flags, I’ve been studying what I call pseudoregular polyhedra, which have two orbits of flags interchanged by conjugation (explained here). ...
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Decomposition of Polyhedral - An example
There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system
$$ \...
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Is the space of Euclidean polyhedra with a fixed $1$-skeleton connected?
Let $\mathcal{A}_\Gamma$ be the space of convex (non-degenerate) Euclidean polyhedra with $1$-skeleton a certain polyhedral graph $\Gamma$. This space can be seen as a subset of $\mathcal{Gr}_2(\...
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How many ways can you inscribe five 24-cells in a 600-cell, hitting all its vertices?
You can inscribe five tetrahedra in a dodecahedron so that each vertex of the dodecahedron is the vertex of just one tetrahedron, as drawn here by Greg Egan:
Warmup question: How many ways can you do ...
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Duality argument for elliptic regularity
M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...
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Clustering of vertices in an $n$-dimensional cube
Consider the vertices of an $n$-dimensional cube. The distance between two vertices is measured as the minimum number of edges between the two vertices. Now consider a subset of these vertices.
If we ...
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Do Bernoulli polynomials know about face vectors?
This question is grounded firmly in numerology. It originates in an observation about some Bernoulli polynomials and the regular icosahedron. Let $F_{k+1}(n)=\sum_{i=1}^n i^k$ be the sum of the ...
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What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?
Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...
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Two questions on the permutohedron
The $n$-dimensional permutohedron $P_n$ is the polytope given by the convex hull of all the possible permutations of the vector $(1,2,\dots,n+1)\in\mathbb{R}^{n+1}$. So it has $(n+1)!$ vertexes.
I ...
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Minimum number of distinct triangles for tesselating the sphere
Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
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How to define and compute the degree of congruence of two rigid polyhedra in same type with knowing vertex coordinates?
If I have two sets of points in 3-dimensional space, each sets of points are the coordinates of vertices of a polyhedron. The two polyhedra have same type, so we don't need to consider the topological ...
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What are the known convex polyhedra with congruent faces?
Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
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1
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Is Sydler's theorem concerning Dehn invariants constructive?
Sydler proved something of a converse to Dehn's negative resolution
of Hilbert's 3rd problem. To quote Wikipedia, Sydler showed that
"every two Euclidean polyhedra with the same volumes and Dehn ...
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Find tetrahedron vertex given 3 vertices of a face and the 3 opposite angles
I have the following tetrahedron:
which I know the coordinates of $P$, $Q$ and $R$ and the value of angles $\theta_0$, $\theta_1$ and $\theta_2$.
I need to find the coordinates of vertex $E$. Is that ...
2
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Products of polytopes and the normals of their facets
I need to compute the normals of the facets of certain polytopes that can be represented as products of polytopes in smaller dimensions.
Searching the bibliography I found that the facets of the ...
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Classification of space-filling (by identical copies) convex polyhedra in R^3
Is the classification of space-filling (by identical copies) convex polyhedra in R^3 is known ?
There are only 5 "parallelohedra" - filling by translation.
But if relax that property to ...
2
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1
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Do continuous motions of the vertices of convex polyhedra that maintain local convexity imply global convexity? (Reference request)
A convex polyhedron has all of its internal dihedral angles in $(0, \pi)$. However, if I start with an abstract polyhedron $P$, let's say a triangulated one, so I don't have to worry about planarity ...
3
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1
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Tilings of lattice polytopes by transformations of lattice polytopes
A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...
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0
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Untruncate permutohedron of order 5
I would like to understand commutation classes of reduced expressions of the longest element in $S_5$ a little better. For this, it makes sense to look at the permutohedron of order 5. Since I am only ...
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Visualization of higher Bruhat order B(5,2)
I made the following images of the higher Bruhat order B(5,2) (in the sense of Manin/Schechtman) with vZome:
image 1
image 2
image 3
Unfortunately, in vZome its not possible do have regular octagons,...
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Pyramids whose volume can be computed by simple cutting and glueing
Since this question remained without answers even after a bounty, I thought it might be time to ask it here.
For which pyramid can you compute the volume from simple cut-and-glue processes? The Dehn ...
3
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An exponential integral over a closed convex polytope
For any $T\geq 2$, let us define the polyhedron $S$ given by
\begin{align*}
S:=\{\underline{t}:=(t_0,t_1,t_{2},t_{3},t_{4},t_{5},t_{6},t_{7})\in [0,+\infty)^{8}:A\underline{t}\leq (\log T)\textbf{1}\}
...
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Self-intersecting path of stacked regular tetrahedra
(This question occurred to me after reading
@IanAgol's reminisces
of Conway's spiral tetrahedron billiard path.)
Let $T_i$ be a regular tetrahedron,
and $P$ a collection of regular tetrahedra
glued ...
7
votes
1
answer
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Embedding of flat surfaces
Let $S$ be a orientable compact surface with a flat euclidean structure with conical singularities (cf. [T] for instance). Let also $\mathcal P$ be a polyhedral euclidean decomposition of $S$ (with ...