A zonohedron is a polyhedron in which every
face is centrally symmetric.
Zonohedra can be defined in various ways, for instance as the Minkowski
sums of line segments. The combinatorics
of their faces are equivalent to those of line arrangements in the plane.
(See my Mathematica Zonotope
notebook for details of this equivalence as well as more examples of
zonohedra.)
Polyhedra Blender.
Mathematica software and Java-based interactive web gallery for what look like
Minkowski sums of polyhedra. If the inputs to the Minkowski
sums were line segments, cubes, or zonohedra, the results would be again
zonohedra, but the ability to supply other inputs allows more general
polyhedra to be formed.
Sylvester's theorem.
This states that any finite non-colinear point set has
a line containing only two points (equivalently, every zonohedron has a
quadrilateral face). Michael Larsen, Tim Chow, and Noam
Elkies discuss two proofs and a complex-number generalization.
(They omit the very simple generalization from Euler's
formula: every convex polyhedron has a face of degree at most five.)
Voronoi diagrams of lattices.
Greg Kuperberg
discusses an algorithm for constructing the Voronoi cells in a planar
lattice of points. This problem is closely related to some important
number theory: Euclid's algorithm for integer GCD's, continued
fractions, and good approximations of real numbers by rationals.
Higher-dimensional generalizations (in which the Voronoi cells form
zonotopes) are much harder -- one can find a basis of
short vectors using the well-known LLL algorithm, but this doesn't
necessarily find
the vectors corresponding to Voronoi adjacencies. (In fact, according
to Senechal's Quasicrystals and Geometry, although the set
of Voronoi adjacencies of any lattice generates the lattice, it's not
known whether this set always contains a basis.)
Zonohedra
and cubic partial cubes. Connecting the geometric problem of
classifying simplicial line arrangements to the graph-theoretic one of
finding regular graphs that can be isometrically embedded on a cube.
Zonotiles.
Russell Towle investigates tilings of zonogons (centrally symmetric
polygons) by smaller zonogons, and their relation to line arrangements,
with an implementation in Mathematica.
Zonotopes.
Helena Verrill wonders in how many ways one can decompose a polygon into
parallelograms. The answer turns out to be equivalent to certain problems
of counting pseudo-line arrangements.
