Questions tagged [polyhedra]

The tag has no usage guidance.

74 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
14 votes
0 answers
377 views

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 ...
Arthur B's user avatar
  • 1,882
14 votes
0 answers
477 views

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 ...
Piotr Shatalin's user avatar
13 votes
0 answers
461 views

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 ...
RavenclawPrefect's user avatar
11 votes
0 answers
294 views

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 ...
ryu576's user avatar
  • 171
11 votes
0 answers
723 views

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 $S$...
mathlove's user avatar
  • 4,727
10 votes
0 answers
329 views

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 ...
Joseph O'Rourke's user avatar
9 votes
0 answers
530 views

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 ...
Joseph O'Rourke's user avatar
8 votes
0 answers
1k views

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? ...
Joseph O'Rourke's user avatar
8 votes
0 answers
170 views

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 ...
მამუკა ჯიბლაძე's user avatar
7 votes
0 answers
210 views

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 ...
Joseph O'Rourke's user avatar
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 ...
Paul B. Slater's user avatar
6 votes
0 answers
232 views

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 ...
Turbo's user avatar
  • 13.6k
6 votes
0 answers
113 views

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 ...
Frida Mauer's user avatar
5 votes
0 answers
218 views

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 ...
მამუკა ჯიბლაძე's user avatar
4 votes
0 answers
79 views

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 ...
Alexander Chervov's user avatar
4 votes
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 ...
Allen Knutson's user avatar
4 votes
0 answers
143 views

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 ...
Joseph O'Rourke's user avatar
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 ...
Camilo Sarmiento's user avatar
4 votes
0 answers
2k views

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:             &...
Joseph O'Rourke's user avatar
3 votes
0 answers
110 views

"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 ...
Dasha Poliakova's user avatar
3 votes
0 answers
112 views

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 ...
ARG's user avatar
  • 4,342
3 votes
0 answers
82 views

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}\} ...
The Number Theorist's user avatar
3 votes
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 ...
user avatar
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?
Denis T's user avatar
  • 4,371
3 votes
0 answers
220 views

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 ...
Joseph O'Rourke's user avatar
3 votes
0 answers
294 views

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 ...
gradstudent's user avatar
  • 2,136
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 ...
Ben Knudsen's user avatar
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 $$f\colon (|M|,|N|) \to (P,Q)$$ ...
Joao Faria Martins's user avatar
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 ...
Oliver Nash's user avatar
  • 1,404
3 votes
0 answers
133 views

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 $P$ ...
Joseph O'Rourke's user avatar
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 ...
Matthieu Nauly's user avatar
2 votes
1 answer
80 views

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 ...
John's user avatar
  • 185
2 votes
0 answers
82 views

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 ...
Alexander Praehauser's user avatar
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 $...
Joseph O'Rourke's user avatar
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 $A_1$, $A_2$...
Đào Thanh Oai's user avatar
2 votes
0 answers
90 views

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?
Daniel Sebald's user avatar
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\|_\...
Turbo's user avatar
  • 13.6k
2 votes
0 answers
128 views

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 ...
David G. Stork's user avatar
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 $\mathbb{R}^3$...
Joseph O'Rourke's user avatar
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 ...
VS.'s user avatar
  • 1,816
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 $(n-1)$ ...
Christoph's user avatar
  • 363
2 votes
0 answers
40 views

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 ...
Turbo's user avatar
  • 13.6k
2 votes
0 answers
75 views

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 ...
A.2's user avatar
  • 123
2 votes
0 answers
51 views

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 ...
Sidharth Ghoshal's user avatar
2 votes
0 answers
185 views

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, ...
Seonhwa  Kim's user avatar
2 votes
0 answers
105 views

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....
John Doe's user avatar
  • 170
1 vote
0 answers
46 views

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 ...
Martin's user avatar
  • 1,101
1 vote
0 answers
80 views

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,\...
James Propp's user avatar
  • 19.4k
1 vote
0 answers
177 views

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 $f_i :X_{i+1} \to X_i$ ...
M.Ramana's user avatar
  • 1,172
1 vote
0 answers
29 views

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 ...
Nandakumar R's user avatar
  • 5,401