# Tagged Questions

Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.

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### Can the sphere be partitioned into small congruent cells?

On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
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### Bodies of constant width?

In two-dimensional case one can generalize figures of constant width as figures which can rotate in a covex polygon. Here is one example which can be used to drill triangular holes: I would like to ...
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### 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 ...
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### Shortest grid-graph paths with random diagonal shortcuts

Suppose you have a network of edges connecting each integer lattice point in the 2D square grid $[0,n]^2$ to each of its (at most) four neighbors, {N,S,E,W}. Within each of the $n^2$ unit cells of ...
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### Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:       Starting with any triangle $T$, one forms $T'$ by connecting ...
We have a circle and two families of $n$ red arcs and $n$ blue arcs, positioned on the circle so that every two arcs of different colors intersect. Can one show that there is a point in the perimeter ...