Architectonic and catoptric tessellation

In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations (or honeycombs) of Euclidean 3-space 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 catoptric tessellation. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual.

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.

Ref. indices	Symmetry	Architectonic tessellation			Catoptric tessellation
		Name Coxeter diagram Image	Vertex figure Image	Cells	Name	Cell	Vertex figures
J11,15 A1 W1 G22 δ4	nc [4,3,4]	Cubille,	Octahedron,		Cubille,	Cube,
J12,32 A15 W14 G7 t1δ4	nc [4,3,4]	Cuboctahedrille, 40px40px	Cuboid, 60px		Oblate octahedrille 60px	Isosceles square bipyramid	30px ,
J13 A14 W15 G8 t0,1δ4	nc [4,3,4]	Truncated cubille, 40px40px	Isosceles square pyramid		Pyramidille, 60px	Isosceles square pyramid	,
J14 A17 W12 G9 t0,2δ4	nc [4,3,4]	2-RCO-trille, 40px40px	Wedge 60px		Quarter oblate octahedrille	irr. Triangular bipyramid	30px , ,
J16 A3 W2 G28 t1,2δ4	bc [[4,3,4]]	Truncated octahedrille, 40px40px	Tetragonal disphenoid 60px		Oblate tetrahedrille, 40px	Tetragonal disphenoid
J17 A18 W13 G25 t0,1,2δ4	nc [4,3,4]	n-tCO-trille, 40px	Mirrored sphenoid 60px		Triangular pyramidille,	Mirrored sphenoid	, ,
J18 A19 W19 G20 t0,1,3δ4	nc [4,3,4]	1-RCO-trille, 40px	Trapezoidal pyramid 60px		Square quarter pyramidille,	Irr. pyramid	, , ,
J19 A22 W18 G27 t0,1,2,3δ4	bc [[4,3,4]]	b-tCO-trille, 40px40px	Phyllic disphenoid 60px		Eighth pyramidille,	Phyllic disphenoid	,
J21,31,51 A2 W9 G1 hδ4	fc [4,31,1]	Tetroctahedrille, 40px	Cuboctahedron, 60px		Dodecahedrille,	30px Rhombic dodecahedron,	,
J22,34 A21 W17 G10 h2δ4	fc [4,31,1]	truncated tetraoctahedrille, 40px	Rectangular pyramid 60px		Half oblate octahedrille,	rhombic pyramid	30px , ,
J23 A16 W11 G5 h3δ4	fc [4,31,1]	3-RCO-trille, 40px40px	Truncated triangular pyramid 60px		Quarter cubille	irr. triangular bipyramid
J24 A20 W16 G21 h2,3δ4	fc [4,31,1]	f-tCO-trille, 40px40px	Mirrored sphenoid 60px		Half pyramidille	Mirrored sphenoid
J25,33 A13 W10 G6 qδ4	d [[3[4]]]	Truncated tetrahedrille, 40px40px	Isosceles triangular prism 60px		Oblate cubille	Trigonal trapezohedron

Symmetry

These are four of the 35 cubic space groups

These four symmetry groups are labeled as:

Label	Description	space group Intl symbol	Geometric notation	Coxeter notation	Fibrifold notation
bc	bicubic symmetry or extended cubic symmetry	(221) Im3m	I43	[[4,3,4]]	8°:2
nc	normal cubic symmetry	(229) Pm3m	P43	[4,3,4]	4−:2
fc	half-cubic symmetry	(225) Fm3m	F43	[4,31,1] = [4,3,4,1+]	2−:2
d	diamond symmetry or extended quarter-cubic symmetry	(227) Fd3m	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

• [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]
• 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]