- Source: Architectonic and catoptric tessellation
In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations (or honeycombs) of Euclidean 3-space with prime space groups and their duals, as three-dimensional analogue of the Platonic, Archimedean, and Catalan tiling of the plane. The singular vertex figure of an architectonic tessellation is the dual of the cell of the corresponding catoptric tessellation, and vice versa. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual. There are other uniform honeycombs constructed as gyrations or prismatic stacks (and their duals) which are excluded from these categories.
Enumeration
The pairs of architectonic and catoptric tessellations are listed below with their symmetry group. These tessellations only represent four symmetry space groups, and also all within the cubic crystal system. Many of these tessellations can be defined in multiple symmetry groups, so in each case the highest symmetry is expressed.
Vertex Figures
The vertex figures of all architectonic honeycombs, and the dual cells of all catoptric honeycombs are shown below, at the same scale and the same orientation:
Symmetry
These four symmetry groups are labeled as:
References
Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi (2009), p. 54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry
Further reading
Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "21. Naming Archimedean and Catalan Polyhedra and Tilings". The Symmetries of Things. A K Peters, Ltd. pp. 292–298. ISBN 978-1-56881-220-5.
Inchbald, Guy (July 1997). "The Archimedean honeycomb duals". The Mathematical Gazette. 81 (491). Leicester: The Mathematical Association: 213–219. doi:10.2307/3619198. JSTOR 3619198. [1]
Branko Grünbaum, (1994) Uniform tilings of 3-space. Geombinatorics 4, 49 - 56.
Norman Johnson (1991) Uniform Polytopes, Manuscript
A. Andreini, (1905) Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 75–129. PDF [2]
George Olshevsky, (2006) Uniform Panoploid Tetracombs, Manuscript PDF [3]
Pearce, Peter (1980). Structure in Nature is a Strategy for Design. The MIT Press. pp. 41–47. ISBN 9780262660457.
Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [4]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318 [5]