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4
votes
0answers
64 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:             ...
10
votes
0answers
617 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 ...
2
votes
1answer
36 views

Seeking criteria for “threadable” pairs of centrosymmetric polyhedra

Let $A$ and $B$ be origin-centered centrosymmetric polyhedra in $\mathbb{R}^3$: "for every point $(x, y, z)$ [...] there is an indistinguishable point $(-x, -y, -z)$." Say that $A$ and $B$ are ...
2
votes
0answers
67 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 $P$ ...
28
votes
3answers
2k views

Did ancient mathematicians know Euler's characteristic for convex polyhedra?

The formula $V-E+F=2$ is so simple that I can't believe that it was really Euler (or perhaps Descartes) who first observed it (I mean the formula itself in some generality, not necessarily a valid ...
4
votes
1answer
73 views

Polyhedra with minimal edge length

Given a fixed volume and fixed surface area I would like to construct polyhedra that minimize the total length of the edges. This seems like a straight-forward problem to solve by brute force for ...
15
votes
2answers
407 views

“Derived” polyhedra and polytopes

The notion of derived polygon is natural and leads to remarkable convergence. Start with a polygon, and replace it by locating a point on every edge a fraction $\alpha$ between the two endpoints. For ...
6
votes
0answers
83 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 ...
8
votes
1answer
120 views

Convex deltahedra in higher dimensions

There are eight convex polyhedra whose faces are equilateral triangles, so-called deltahedra:        (Image from here) Q. Have the equivalent higher-dimensional ...
6
votes
0answers
61 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 ...
1
vote
0answers
77 views

method to construct platonic solids from the circumscribed sphere [closed]

I am trying to find a geometrical method of building the platonic solids starting from the circumscribed sphere. I do have a class of generative 3d modeling where we create 3d object programmatically ...
24
votes
3answers
887 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$. ...
0
votes
1answer
79 views

A question about rational convex cone

Edit: Fix a lattice $N = \mathbb{Z}^n$ and let $N_{\mathbb{R}} = N \otimes \mathbb{R}$. Let $C(S)$ be a strongly convex rational cone in $N_{\mathbb{R}}$ generated by a finite set $S \subset N$, with ...
5
votes
1answer
108 views

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 ...
3
votes
3answers
320 views

Intersection of Polyhedra

I'm writing a collision detection algorithm, and so far I've been using Joseph O'Rourke's book "Computational Geometry in C" as reference. It outlines an algorithm to determine whether a point is ...
7
votes
0answers
178 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 ...
3
votes
0answers
128 views

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 ...
2
votes
1answer
156 views

Finding the vertices of a convex polyhedron from a set of planes

I'm new to computational geometry and advanced mathematics in general here so bear with me. I've spent a decent amount of time attempting to figure out this problem and I just can't find a solution. ...
8
votes
3answers
491 views

On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1. For example, it's easy to prove that ...
9
votes
2answers
203 views

Do maximal polyhedra have algebraic volume?

Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number? Update: What ...
2
votes
1answer
133 views

Equiprojective polyhedra

Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open, and which some might find intriguing. Define an ...
5
votes
1answer
205 views

Standard (special) spines and hyperbolic structure on 3-manifolds

My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...
2
votes
0answers
70 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 ...
8
votes
1answer
257 views

Complexity of the union of randomly rotated unit cubes

It is a remarkable fact that the union of congrent cubes has only at most near-quadratic combinatorial complexity, $O^*(n^2)$ for $n$ cubes, known to be almost tight. This contrasts with the union of ...
6
votes
2answers
166 views

Volumes of convex vs non-convex polyhedra with prescribed facets areas

It is known that given a set of Areas $A_f$ and normals $\vec{n}_f$ if $\sum_f A_f \vec{n}_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov ...
1
vote
1answer
182 views

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| \, ...
15
votes
1answer
309 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas? Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...
2
votes
2answers
281 views

Polyhedra Classification

The following is inspired by this question. From time to time I search the web for tables of polyhedra, but without much success. Part of the problem is that there are many non-equivalent questions ...
7
votes
3answers
432 views

Solid angles of a tetrahedron

This is a problem I have had for a while. For a triangle, the side opposite the largest angle has the largest length (and similarly for smallest angle). For a tetrahedron, the question is whether the ...
0
votes
0answers
78 views

Convex Polyhedra with Largest Constant Dihedral Angle for Given Number of Faces

Which $20$ faced convex polyhedron has the largest constant dihedral angle? Which $24$ faced convex polyhedron has the largest constant dihedral angle? Also, what about $30$ or $32$ faced polyhedra? ...
4
votes
3answers
427 views

Visualizing polyhedra from their 1-skeletons

Except for a few simple cases (typically pyramids and prisms) I find it hard to visualize a polyhedron from its 1-skeleton embedded in the plane, e.g. the hexahedral graph 5, as can be seen here. ...
8
votes
1answer
346 views

Polyhedra that combinatorially shadow a sequence

Let $P$ be a polyhedron in $\mathbb{R}^3$. Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if there is a continuous rotation of $P$ such that its orthogonal-projection shadows ...
9
votes
2answers
696 views

Is the tensor product of polyhedra a polyhedron?

Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
2
votes
2answers
359 views

Dilogarithm, tetrahedrons, and hyperbolic space

The Bloch-Wigner function $D(z)$, gives the volume of a ideal tetrahedron in hyperbolic space $\mathbb H3$, where z is the cross-ratio (z1,z2,z3,z4) parametrizing the tetrahedron in $\mathbb CP1$. ...
2
votes
2answers
227 views

Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb{H}^3$, has a boundary $\mathbb{CP}^1$. A ideal tetrahedron in $\mathbb{H}^3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb{CP}^1$. The four ...
1
vote
2answers
389 views

Does Euler's formula imply bounds on the degree of vertices in a 3-polytopal graph?

A corollary of Euler's formula tells us that the edge-vertex graph of every convex 3-polyhedron must have a face with either 3, 4, or 5 edges, using an argument about the average degree of vertices. ...
1
vote
1answer
174 views

Polar duality and -1 [closed]

We define the polar dual of a polytope $P$ as the set $$\{x\in \mathbb{R}^n: x \cdot a\geq -1 \text{ for all } a\in P\}$$ Why do we require $-1$ instead of $-2$ or any other constant?
2
votes
1answer
125 views

coarser than triangulations “almost partitions” into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an ...
6
votes
1answer
338 views

Wrapping a convex polyhedron with string

This is a meta-question, rather than a specific mathematical question. I am seeking a mathematical definition that captures the following physical idea. Suppose you have a convex polyhedron $P ...
2
votes
1answer
140 views

How can pushing a vertex in a polytope lead to merging facets?

I'm trying to understand the Lemma 2.2 and Corollary 2.3 of Francisco Santos paper "A counterexample to the Hirsch conjecture": http://arxiv.org/abs/1006.2814 Corollary 2.3 is a proof of a result of ...
8
votes
0answers
547 views

Building an $n$-toroid from cubes

I wonder if this has been studied: What is the fewest number of unit cubes from which one can build an $n$-toroid? The cubes must be glued face-to-face, and the boundary of the resulting ...
8
votes
2answers
522 views

Fractal Tiling of Rhombic Dodecahedra

Hello, this is my first question on Math Overflow... Rhombic dodecahedra can be tiled in 3-space, leaving no gaps. This tiling corresponds to the close-packing of spheres. Consider a "nucleus" ...
3
votes
1answer
254 views

Grading a non-graded poset as squeezed as possible

Here is a curiosity question (motivated by the recent revamp of ranked-poset routines in Sage). Let $P$ be a finite poset. We look for a family $\left(a_p\right)_{p\in P}$ of real numbers summing up ...
9
votes
1answer
893 views

What nets fold to polyhedra?

There is a classic (and open) problem asking whether every polyhedron can be unfolded to give a non-overlapping net. The converse problem has been studied asking which polygons can be folded in some ...
10
votes
2answers
257 views

Acute triangulation

Assume that $S$ is a finite 2-dimensional simplicial complex equipped with a metric $d$ such that each triangle is isometric to a plane triangle (so $(S,d)$ is a polyhedral space). Is it possible ...
5
votes
0answers
271 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 ...
8
votes
3answers
610 views

Efficient topological triangulations of non-convex polyhedra

Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a topological triangulation with complexity $O(n)$? Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ ...
10
votes
0answers
238 views

Update to Shephard's “Twenty Problems on Convex Polyhedra”

Forty-three years ago, Geoffrey Shephard published an influential list of open problems on convex polyhedra. Progress has been made on several of his problems, and perhaps some have been completely ...
13
votes
2answers
1k views

How many vertices/edges/faces at most for a convex polyhedron that tiles space?

I wonder if this problem has already been examined before: Consider a convex polyhedron that tiles $\mathbb R^3$. What is the maximum of vertices/edges/faces that such a polyhedron can have? ...
17
votes
2answers
760 views

Placing points on a sphere so that no 3 lie close to the same plane

Motivation I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...