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. There are other uniform honeycombs constructed as prismatic stacks (and their duals) which are excluded from these categories.

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.[1]
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

Cuboid,
Oblate octahedrille


Isosceles square bipyramid

,
J13
A14
W15
G8
t0,1δ4
nc
[4,3,4]
Truncated cubille

Isosceles square pyramid
Pyramidille


Isosceles square pyramid

,
J14
A17
W12
G9
t0,2δ4
nc
[4,3,4]
2-RCO-trille

Wedge
Quarter oblate octahedrille

irr. Triangular bipyramid

, ,
J16
A3
W2
G28
t1,2δ4
bc
[[4,3,4]]
Truncated octahedrille

Tetragonal disphenoid
Oblate tetrahedrille


Tetragonal disphenoid

J17
A18
W13
G25
t0,1,2δ4
nc
[4,3,4]
n-tCO-trille

Mirrored sphenoid
Triangular pyramidille

Mirrored sphenoid

, ,
J18
A19
W19
G20
t0,1,3δ4
nc
[4,3,4]
1-RCO-trille

Trapezoidal pyramid
Square quarter pyramidille

Irr. pyramid

, , ,
J19
A22
W18
G27
t0,1,2,3δ4
bc
[[4,3,4]]
b-tCO-trille

Phyllic disphenoid
Eighth pyramidille

Phyllic disphenoid

,
J21,31,51
A2
W9
G1
4
fc
[4,31,1]
Tetroctahedrille
or
Cuboctahedron,
Dodecahedrille
or

Rhombic dodecahedron,

,
J22,34
A21
W17
G10
h2δ4
fc
[4,31,1]
truncated tetraoctahedrille
or
Rectangular pyramid
Half oblate octahedrille
or

rhombic pyramid

, ,
J23
A16
W11
G5
h3δ4
fc
[4,31,1]
3-RCO-trille
or
Truncated triangular pyramid
Quarter cubille

irr. triangular bipyramid
J24
A20
W16
G21
h2,3δ4
fc
[4,31,1]
f-tCO-trille
or
Mirrored sphenoid
Half pyramidille

Mirrored sphenoid
J25,33
A13
W10
G6
4
d
[[3[4]]]
Truncated tetrahedrille
or
Isosceles triangular prism
Oblate cubille

Trigonal trapezohedron

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) Im3mI43[[4,3,4]]
8°:2
nc normal cubic symmetry (229) Pm3mP43[4,3,4]
4:2
fc half-cubic symmetry (225) Fm3mF43[4,31,1] = [4,3,4,1+]
2:2
d diamond symmetry
or extended quarter-cubic symmetry
(227) Fd3mFd4n3[[3[4]]] = [[1<sup>+</sup>,4,3,4,1<sup>+</sup>]]
2+:2

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" (PDF). Journal of Mathematical Physics. AIP Publishing LLC. 48 (2). doi:10.1063/1.2426416. ISSN 1089-7658.

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. Leicester: The Mathematical Association. 81 (491): 213–219. JSTOR 3619198.
  • 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
  • George Olshevsky, (2006) Uniform Panoploid Tetracombs, Manuscript PDF
  • 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
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
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