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30
votes
1answer
1k views

Pach's “Animals”: What if genus $> 0$ ?

Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today: Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...
29
votes
3answers
3k 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 ...
25
votes
3answers
953 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 $\frac{S^\prime}{S}$. ...
21
votes
1answer
987 views

Building a genus-$n$ torus 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 ...
19
votes
3answers
769 views

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 ...
19
votes
2answers
886 views

Four Dimensional Origami Axioms

What are the axioms of four dimensional Origami. If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded ...
17
votes
2answers
788 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 ...
15
votes
2answers
500 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 ...
15
votes
1answer
336 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 ...
15
votes
1answer
164 views

Higher dimensional generalization of: Any quadrilateral tiles the plane?

Any (non-self-intersecting) quadrilateral tiles the plane.     (MathWorld image.) Q. What is the strongest known generalization of this statement to higher dimensions? I.e., ...
13
votes
2answers
556 views

What was the Question that led Euler to his Investigations on Polyhedra?

The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and that story is a must in every introduction to graph theory. ...
13
votes
2answers
2k 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? ...
13
votes
0answers
345 views

Surface area of convex hull [duplicate]

Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?
10
votes
2answers
275 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 ...
10
votes
2answers
469 views

Cut and Fold Polyhedron!

I have two convex polyhedra such that their sums of side areas are equal. It is true that I can cut one of them and flatten it on the plane, then fold the flattened polygon to reach the other ...
10
votes
2answers
759 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 ...
10
votes
0answers
651 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 ...
10
votes
0answers
268 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 ...
9
votes
2answers
209 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 ...
9
votes
1answer
1k 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 ...
8
votes
3answers
639 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 ...
8
votes
3answers
498 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 ...
8
votes
1answer
403 views

Polyhedron not circumscribed about a sphere

Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere. My teacher ...
8
votes
3answers
630 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 $n$ ...
8
votes
1answer
129 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 ...
8
votes
1answer
271 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 ...
8
votes
2answers
584 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" ...
8
votes
1answer
373 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 ...
7
votes
3answers
631 views

Not quite regular polyhedra

Take a naive interpretation of regular polyhedra: All vertices (including epsilon ball) congruent All edges congruent All faces congruent We can now find interesting families by removing one ...
7
votes
5answers
117 views

The limit of edge-midpoint convex polyhedra

    Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$, replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$. Continuing this process, we obtain a ...
7
votes
3answers
1k views

polyhedra with equilateral pentagons faces.

In page http://loki3.com/poly/isohedra.html around six polyhedra with equilateral pentagons as faces are shown: a pyritohedron, icositetrahedrons... Is there a complete list of this kind of polyhedra? ...
7
votes
0answers
101 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 ...
7
votes
0answers
195 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 ...
6
votes
1answer
374 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 ...
6
votes
2answers
189 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 ...
6
votes
0answers
73 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 ...
5
votes
2answers
231 views

Put P inside Q! polygons/polyhedra

We have two Polygons/Polyhedra P and Q. Does there exist a polynomial time algorithm to decide if we can put P (using translation and rotation) inside Q or not? First think about the case of which P ...
5
votes
1answer
124 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 ...
5
votes
1answer
225 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 ...
5
votes
0answers
287 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 ...
4
votes
2answers
204 views

What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?

The question is mostly in the title. Imagine I have some non-convex polyhedron $P$, and I would like to find the smallest diameter ring that it can pass through in 3-space, undergoing any necessary ...
4
votes
1answer
372 views

Are there non-tiling polyhedra that pack arbitrarily well?

The fact that an upper bound on the packing density $< 1$ has only recently been exhibited for regular tetrahedra in $\mathbb{R}^3$ (see this question) suggests that proving concrete bounds of ...
4
votes
3answers
477 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. ...
4
votes
3answers
202 views

Can anyone suggest a text on polyhedral theory?

Can anyone suggest a text on polyhedral theory? Particularly on increasing the number of faces under projections. 0,1 polytopes
4
votes
2answers
148 views

Build a topological polytope with a specified CW-structure

I am a topologist and not quite familiar with the tools for building a polytope. I would like to build some topological polytope which is an somewhere in between permutohedron and associahedron which ...
4
votes
1answer
1k views

intersection of convex and non-convex polyhedra

Hi everyone, I am trying to find the best appropriate way to intersect polyhedra which may be non-convex. The number of vertices that build the polyhedron is hence always small (up to 20 or so). ...
4
votes
1answer
74 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 ...
4
votes
1answer
177 views

Light inside a polyhedron

I have two questions the same as Mostafa's Question: 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 $X$ ...
4
votes
0answers
126 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:             ...
3
votes
3answers
448 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 ...