Cubic Polyhedra
Abstract
A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal surfaces (under an appropriate smoothing flow, keeping their symmetries). Here we give a complete classification of the cubic polyhedra. Among these are five new infinite uniform polyhedra and an uncountable collection of new infinite semiregular polyhedra. We also consider the somewhat larger class of all discrete minimal surfaces in the cubic lattice.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2002
 arXiv:
 arXiv:math/0205145
 Bibcode:
 2002math......5145G
 Keywords:

 Metric Geometry;
 Primary: 52B10;
 Secondary: 52B70;
 53A10
 EPrint:
 18 pages, many figures