5
votes
2answers
221 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 ...
10
votes
2answers
448 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 ...
4
votes
1answer
173 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 $X$ ...
19
votes
3answers
754 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 ...
7
votes
0answers
100 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 ...
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 ...
15
votes
1answer
329 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 ...
3
votes
1answer
256 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 ...
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? ...
7
votes
3answers
625 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 ...