Discrete geometry includes the study of covering, illumination, packing, convex bodies, convex polytopes, and other metric geometry.

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3
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2answers
353 views

Voronoi decomposition implementation in four dimensions?

I'm a software engineer and have been asked to research a Voronoi implementation in four dimensions. I'm not asking for "teh codez" but am interested in approachable tutorials on Voronoi decomposition ...
3
votes
1answer
124 views

Lower bound for a set of distances between pairs of points in a plane

There are $N>1$ points in a plane. Consider the set of all distances between pairs of the points. Let $n$ be the number of elements of this set. We know that $$n \leq {N \choose 2}.$$ What is the ...
1
vote
1answer
263 views

Black and White points on a grid

Some points on (integer, rectangular) grid in a plane are colored white, some black and some are not colored. In each step, one vertical or horizontal line can be selected, and all colored points on ...
1
vote
2answers
631 views

Distinct Hamiltonian cycles of the icosahedron and dodecahedron

I am seeking a listing of the distinct Hamiltonian cycles following the edges of the icosahedron and the dodecahedron. By distinct I mean they are not congruent by some symmetry of the icosahedron or ...
16
votes
5answers
997 views

Pick's Theorem on a triangular (or hex) grid

Pick's theorem says that given a square grid (that is, all points in the plane with integer coordinates) and a polygon without holes and non selt-intersecting whose vertices are grid points, its area ...
4
votes
3answers
316 views

Euclidean Tilings that are Uniform but not Vertex-Transitive

Basic definitions: a tiling of d-dimensional Euclidean space is a decomposition of that space into polyhedra such that there is no overlap between their interiors, and every point in the space is ...