Architectonic and catoptric tessellation explained

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:

Label	Description	space group Intl symbol	Geometric notation^[2]	Coxeter notation	Fibrifold notation
bc	bicubic symmetry or extended cubic symmetry	(221) Imm	I43		8°:2
nc	normal cubic symmetry	(229) Pmm	P43	[4,3,4]	4^−:2
fc	half-cubic symmetry	(225) Fmm	F43	[4,3<sup>1,1</sup>] = [4,3,4,1<sup>+</sup>]	2^−:2
d	diamond symmetry or extended quarter-cubic symmetry	(227) Fdm	Fd4n3	3^[4] = 1^+,4,3,4,1^+	2^+:2

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

• Book: Conway . John H. . John Horton Conway . Burgiel . Heidi . Goodman-Strauss . Chaim . The Symmetries of Things . A K Peters, Ltd. . 2008 . 21. Naming Archimedean and Catalan Polyhedra and Tilings . 292–298 . 978-1-56881-220-5.
• Inchbald . Guy . The Archimedean honeycomb duals . The Mathematical Gazette . 81 . 491 . 213–219 . The Mathematical Association . Leicester . July 1997 . 3619198. 10.2307/3619198 . http://www.steelpillow.com/polyhedra/AHD/AHD.htm
• 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 https://web.archive.org/web/20140429195143/http://media.accademiaxl.it/memorie/Serie3_T14.pdf
• George Olshevsky, (2006) Uniform Panoploid Tetracombs, Manuscript PDF http://bendwavy.org/4HONEYS.pdf
• Book: Pearce, Peter . Structure in Nature is a Strategy for Design . The MIT Press . 1980 . 41–47 . 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318 https://books.google.com/books?id=fUm5Mwfx8rAC&dq=%22quarter+cubic+honeycomb%22+q%7B4%2C3%2C4%7D&pg=PA318

Notes and References

1. For cross-referencing of Architectonic solids, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28). Coxeters names are based on δ₄ as a cubic honeycomb, hδ₄ as an alternated cubic honeycomb, and qδ₄ as a quarter cubic honeycomb.
2. Hestenes . David . Holt . Jeremy . 2007-02-27 . Crystallographic space groups in geometric algebra . Journal of Mathematical Physics . 48 . 2 . 023514 . AIP Publishing LLC . 1089-7658 . 10.1063/1.2426416 .