Open AccessTwo-dimensionally infinite polyhedra with vertices related by symmetry operations.
Author(s) -
David Harker
Publication year - 1991
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.88.2.585
Subject(s) - polyhedron , equilateral triangle , combinatorics , symmetry (geometry) , mathematics , dual polyhedron , tetrahedron , plane (geometry) , geometry
It is shown that an infinitely extended polyhedron of one sheet composed of plane polygons of finite size, and in which all the vertices are related to one another by symmetry operations, can always be constructed by folding a stiff, unstretchable sheet (such as a piece of paper). It is also shown that only 11 such polyhedra have faces that are regular polygons; these polygons are equilateral triangles and squares. The 4 of these 11 polyhedra that are not known to have been published previously are presented here.