Open AccessThis article is
- freely available
Symmetry 2012, 4(3), 545-565; doi:10.3390/sym4030545
Received: 29 June 2012; in revised form: 13 August 2012 / Accepted: 19 August 2012 / Published: 30 August 2012
(This article belongs to the Special Issue Polyhedra)
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract: This paper describes 4-valent tiling-like structures, called pseudotilings, composed of barrel tiles and apeirogonal pseudotiles in Euclidean 3-space. These (frequently face-to-face) pseudotilings naturally rise in columns above 3-valent plane tilings by convex polygons, such that each column is occupied by stacked congruent barrel tiles or congruent apeirogonal pseudotiles. No physical space is occupied by the apeirogonal pseudotiles. Many interesting pseudotilings arise from plane tilings with high symmetry. As combinatorial structures, these are abstract polytopes of rank 4 with both finite and infinite 2-faces and facets.
Keywords: tiling; pseudotiling; abstract polytope; realization