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19
votes
3answers
788 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 ...
15
votes
1answer
214 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., ...
15
votes
1answer
341 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 ...
25
votes
3answers
957 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 $\frac{S^\prime}{S}$. ...
7
votes
0answers
104 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 ...
16
votes
2answers
528 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 ...
8
votes
3answers
656 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 ...
5
votes
1answer
125 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 ...
8
votes
1answer
413 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
1answer
132 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 ...