Questions tagged [polyhedra]
The polyhedra tag has no usage guidance.
74
questions with no upvoted or accepted answers
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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 ...
14
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0
answers
477
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Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?
After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
13
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0
answers
461
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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 ...
11
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0
answers
294
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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 ...
11
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0
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723
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Making a convex polyhedron with two sheets of paper
Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that $S$...
10
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329
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Bi-spherical polyhedra
Bicentric polygons have been studied: a polygon all of whose vertices lie on its
circumcirle, and whose incircle is tangent to every edge:
I have not been able to find a comparable literature ...
9
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0
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530
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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 ...
8
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answers
1k
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Maximum volume cross-section of a hypercube
This is surely well known, but:
Q1. What is the $(d{-}1)$-dimensional polytope
that realizes the maximum volume cross-section of a unit hypercube
by a $(d{-}1)$-dimensional hyperplane?
...
8
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0
answers
170
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Which -icial sets produce the "standard" representations of symmetric groups?
Suppose you have a system of cell complexes (say, even convex polyhedra) $(P_n)_{n\geqslant0}$ which occur as faces of each other and are used to define the corresponding notion of "$P_*$-set". So ...
7
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answers
210
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Tiling space with supertile of hypercube unfoldings
Two students in my class
asked and answered what might be a novel question.
It is well known that the cube has exactly $11$ edge-unfoldings
(or "nets"), as shown below:
(Image from ...
6
votes
0
answers
220
views
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 ...
6
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232
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Complexity of scissors congruence?
Where is the complexity of the problem 'Given two bounded compact convex integral polyhedra in $\mathbb R^n$ presented by $O(poly(n))$ integer valued halfspaces given by linear inequalities with ...
6
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answers
113
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Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces
Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...
5
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218
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Is there a well-established terminology for polyhedra/polytopes?
I got confused lately. It seems like in the metric context a polyhedron tends to mean an intersection of a finite number of half-spaces, while a polytope is a convex hull of a finite set of points. At ...
4
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79
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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 ...
4
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0
answers
141
views
Name for facet of a cone containing all but one edge
Let $C \subseteq \mathbb R^n$ be a polyhedral cone, so generated by its edges ($1$-dimensional faces) and $F \subseteq C$ a facet (codimension $1$ face) of it containing every edge except $e$. In ...
4
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143
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Perimeters of nested convex spherical polygons
I seek a reference—not a proof—that if $P_1$ and $P_2$
are two convex polygons on a sphere composed of geodesic segments,
contained in a hemisphere, and
$P_1 \subseteq P_2$, then the ...
4
votes
0
answers
268
views
Which (polytopal) fans/polytopes are secondary?
Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.
The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional ...
4
votes
0
answers
2k
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Regular cross-sections of a dodecahedron; analogous sections of 4-polytopes
One can intersect a dodecahedron with a plane and
obtain an equilateral triangle, a square, a regular pentagon,
a regular hexagon, and a regular decagon:
&...
3
votes
0
answers
110
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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 ...
3
votes
0
answers
112
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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
votes
0
answers
82
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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}\}
...
3
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0
answers
192
views
Motivic strong bellows conjecture
There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.
Is ...
3
votes
0
answers
142
views
Can bellows make loops?
Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
3
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answers
220
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Reconstructing plane graphs from degree- and face-sequences
Let $G$ be a plane $3$-connected graph; so it partitions the plane
into regions bounded by faces.
Let $\mathrm{deg}_v$ be the sequence of vertex degrees of $G$,
and $\mathrm{deg}_f$ be the sequence of ...
3
votes
0
answers
294
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Factorization of tropical polynomials
I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here a tropical polynomial ...
3
votes
0
answers
314
views
Cubical approximation theorem for cubical complexes
A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain.
I have found a claim ...
3
votes
0
answers
127
views
Stellar moves on pairs of polyhedra
Let $P$ and $Q$ be polyhedra of $\mathbb{R}^n$ with $Q \subset P$. Let $(M,N)$ and $(M',N)$ be pairs of abstract simplicial complexes.
Consider two triangulations
$$f\colon (|M|,|N|) \to (P,Q)$$
...
3
votes
0
answers
102
views
Is there any connection between Lagrange points and the icosahedron?
Given the Newtonian two-body problem, one can ask if there are any orbits that allow a test particle to maintain a fixed configuration relative to the two bodies. In other words, in a frame that ...
3
votes
0
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133
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Convex polyhedra jammed in $k$ disjoint holes
For a given convex polyhedron $P \subset \mathbb{R}^3$,
I was imagining finding the optimal "fixing" of $P$ in holes (or jamming them in "mud"),
which led to the following question.
First, scale $P$ ...
2
votes
0
answers
240
views
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 ...
2
votes
1
answer
80
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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 ...
2
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0
answers
82
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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
votes
0
answers
48
views
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 $...
2
votes
0
answers
228
views
Generalization of the Napoleon equilateral triangle to higher dimention
When I researched the Fermat-Dao-Nhi equilateral triangle in preamble before points X(33602) of the Kimberling triangle center. I discovered the general result for polygon as follows:
Let $A_1$, $A_2$...
2
votes
0
answers
90
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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?
2
votes
0
answers
76
views
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\|_\...
2
votes
0
answers
128
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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 ...
2
votes
0
answers
84
views
Iterated polyhedron face twisting
Let $Q$ be a polygon in the plane. Modify $Q$ by rotating each edge about its
midpoint by $180^\circ$. The result is $Q$ again: No change.
This suggests exploring a similar operation in $\mathbb{R}^3$...
2
votes
0
answers
29
views
When do projection maps of polyhedra factor?
Given three polyhedra $P$, $Q$, and $R$ in dimensions $a$, $b$, and $c$ respectively, with $a\leq b\leq c$, with the additional condition that: $P=\pi_1(Q)=\pi_2(R)$, where $\pi_1$ and $\pi_2$ are ...
2
votes
0
answers
73
views
An implementation of Minkowski reconstruction in 3 dimensions
By a theorem of Minkowski from 1903, an $n$-dimensional polytope $P\subset \mathbb R^n$ is determined up to translation by its unit face normal $u_1,\dots,u_k\in S^{n-1}$ and the corresponding $(n-1)$ ...
2
votes
0
answers
40
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Efficient $H$ representation of matrices with distinct cyclic shift permuted entries
Given points $v_1,\dots,v_n\in\mathbb Z^n$ in codimesion $1$ hyperplane $x_1+\dots+x_n=t$ with $0\leq x_{i}$ and a cyclic shift permutation $\sigma$ where
$v_1,\dots,v_n$ when written as columns of ...
2
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0
answers
75
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Making a polyhedron integral by selecting value for a specific co-ordinate of constraint vector
I am currently trying to solve a binary integer programming(maximization) problem, where the first row of the constraint matrix corresponds to the constraints on the total number of 1's in the vector ...
2
votes
0
answers
51
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Facet counting argument for polytopes
Consider a pair of piecewise-linear cobordant $n$ dimensional polyhedra $P_1, P_2$ sitting in $\mathbb{R}^{n+2}$ (with some fixed orientation).
Let $O$ be an $n+1$ dimensional piecewise-linear ...
2
votes
0
answers
185
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Harborth conjecture and polyhedra
Harborth conjecture state that every planar graph can be drawn on a plane only using staight line segments of rational or integral edge length.
( There is a good mathoverflow page for this conjecture, ...
2
votes
0
answers
105
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Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?
Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear inequalities....
1
vote
0
answers
46
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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 ...
1
vote
0
answers
80
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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,\...
1
vote
0
answers
177
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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 $f_i :X_{i+1} \to X_i$ ...
1
vote
0
answers
29
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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 ...