Architectonic and catoptric tessellation

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The 13 architectonic or catoptric tessellations, shown as uniform cell centers, and catoptric cells, arranged as multiples of the smallest cell on top.

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]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cubille
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Partial cubic honeycomb.pngCubic honeycomb.png
Octahedron, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cubic honeycomb verf.png
Hexahedron.png Cubille
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Partial cubic honeycomb.png
Cubic full domain.png
Cube, CDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Octahedron.png
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
J12,32
A15
W14
G7
t1δ4
nc
[4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cuboctahedrille
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Rectified cubic honeycomb.pngRectified cubic tiling.png
Cuboid, CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Rectified cubic honeycomb verf.png
Octahedron.pngCuboctahedron.png Oblate octahedrille
CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Hexakis cubic honeycomb.png
Cubic square bipyramid.png
Isosceles square bipyramid
CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node.png
Hexahedron.pngRhombic dodecahedron.jpg
CDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
J13
A14
W15
G8
t0,1δ4
nc
[4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Truncated cubille
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Truncated cubic honeycomb.pngTruncated cubic tiling.png
Isosceles square pyramid
Truncated cubic honeycomb verf.png
Octahedron.pngTruncated hexahedron.png Pyramidille
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Hexakis cubic honeycomb.png
Cubic square pyramid.png
Isosceles square pyramid
Hexahedron.pngTriakis octahedron.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
J14
A17
W12
G9
t0,2δ4
nc
[4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
2-RCO-trille
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cantellated cubic honeycomb.pngCantellated cubic tiling.png
Wedge
Cantellated cubic honeycomb verf.png
Small rhombicuboctahedron.pngCuboctahedron.pngHexahedron.png Quarter oblate octahedrille
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Quarter oblate octahedrille cell.png
irr. Triangular bipyramid
Strombic icositetrahedron.pngRhombic dodecahedron.jpgOctahedron.png
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png, CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png, CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node.png
J16
A3
W2
G28
t1,2δ4
bc
[[4,3,4]]
CDel branch c1.pngCDel 4a4b.pngCDel nodeab c2.png
Truncated octahedrille
CDel branch 11.pngCDel 4a4b.pngCDel nodes.png
Bitruncated Cubic Honeycomb1.svgBitruncated cubic tiling.png
Tetragonal disphenoid
Bitruncated cubic honeycomb verf.png
Truncated octahedron.png Oblate tetrahedrille
CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Disphenoid tetrah hc.png
Oblate tetrahedrille cell.png
Tetragonal disphenoid
Tetrakis cube.png
CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
J17
A18
W13
G25
t0,1,2δ4
nc
[4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
n-tCO-trille
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cantitruncated Cubic Honeycomb.svgCantitruncated cubic tiling.png
Mirrored sphenoid
Cantitruncated cubic honeycomb verf.png
Great rhombicuboctahedron.pngTruncated octahedron.pngHexahedron.png Triangular pyramidille
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Triangular pyramidille cell1.png
Mirrored sphenoid
Disdyakis dodecahedron.pngTetrakis cube.pngOctahedron.png
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png, CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png, CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node.png
J18
A19
W19
G20
t0,1,3δ4
nc
[4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
1-RCO-trille
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Runcitruncated cubic honeycomb.jpgRuncitruncated cubic tiling.png
Trapezoidal pyramid
Runcitruncated cubic honeycomb verf.png
Small rhombicuboctahedron.pngTruncated hexahedron.pngOctagonal prism.pngHexahedron.png Square quarter pyramidille
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node f1.png
Square quarter pyramidille cell.png
Irr. pyramid
Strombic icositetrahedron.pngTriakis octahedron.pngOctagonal bipyramid.pngOctahedron.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node f1.png, CDel node f1.pngCDel 2x.pngCDel node.pngCDel 4.pngCDel node f1.png, CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 2x.pngCDel node f1.png, CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
J19
A22
W18
G27
t0,1,2,3δ4
bc
[[4,3,4]]
CDel branch c1.pngCDel 4a4b.pngCDel nodeab c2.png
b-tCO-trille
CDel branch 11.pngCDel 4a4b.pngCDel nodes 11.png
HC A6-Pr8.pngOmnitruncated cubic tiling.png
Phyllic disphenoid
Omnitruncated cubic honeycomb verf2.png
Great rhombicuboctahedron.pngOctagonal prism.png Eighth pyramidille
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Eighth pyramidille cell.png
Phyllic disphenoid
Disdyakis dodecahedron.pngOctagonal bipyramid.png
CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png, CDel node f1.pngCDel 2x.pngCDel node f1.pngCDel 4.pngCDel node f1.png
J21,31,51
A2
W9
G1
4
fc
[4,31,1]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
Tetroctahedrille
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes.png or CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Tetrahedral-octahedral honeycomb.pngAlternated cubic tiling.png
Cuboctahedron, CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Alternated cubic honeycomb verf.svg
Tetrahedron.pngOctahedron.png Dodecahedrille
CDel node f1.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes.png or CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Rhombic dodecahedra.png
Dodecahedrille cell.png
Rhombic dodecahedron, CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Tetrahedron.pngHexahedron.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
J22,34
A21
W17
G10
h2δ4
fc
[4,31,1]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
truncated tetraoctahedrille
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1-43.pngCDel nodes.png or CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Truncated Alternated Cubic Honeycomb.svgTruncated alternated cubic tiling.png
Rectangular pyramid
Truncated alternated cubic honeycomb verf.png
Truncated octahedron.pngCuboctahedron.pngTruncated tetrahedron.png Half oblate octahedrille
CDel node f1.pngCDel 3.pngCDel node f1.pngCDel split1-43.pngCDel nodes.png or CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Half oblate octahedrille cell.png
rhombic pyramid
Tetrakis cube.pngRhombic dodecahedron.jpgTriakis tetrahedron.png
CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png, CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png, CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png
J23
A16
W11
G5
h3δ4
fc
[4,31,1]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
3-RCO-trille
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.png or CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Runcinated alternated cubic honeycomb.jpgRuncinated alternated cubic tiling.png
Truncated triangular pyramid
Runcinated alternated cubic honeycomb verf.png
Small rhombicuboctahedron.pngHexahedron.pngTetrahedron.png Quarter cubille
CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node f1.png
Quarter cubille cell.png Quarter cubille cell-dodeca.png
irr. triangular bipyramid
Strombic icositetrahedron.pngOctahedron.pngTetrahedron.png
J24
A20
W16
G21
h2,3δ4
fc
[4,31,1]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
f-tCO-trille
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node 1.png or CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Cantitruncated alternated cubic honeycomb.jpgCantitruncated alternated cubic tiling.png
Mirrored sphenoid
Runcitruncated alternate cubic honeycomb verf.png
Great rhombicuboctahedron.pngTruncated hexahedron.pngTruncated tetrahedron.png Half pyramidille
CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Half pyramidille cell.png Half pyramidille cell-dodeca.png
Mirrored sphenoid
Disdyakis dodecahedron.pngTriakis octahedron.pngTriakis tetrahedron.png
J25,33
A13
W10
G6
4
d
[[3[4]]]
CDel branch c1.pngCDel 3ab.pngCDel branch c2.png
Truncated tetrahedrille
CDel branch 11.pngCDel 3ab.pngCDel branch.png or CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
Quarter cubic honeycomb2.pngBitruncated alternated cubic tiling.png
Isosceles triangular prism
T01 quarter cubic honeycomb verf2.png
Tetrahedron.pngTruncated tetrahedron.png Oblate cubille
CDel labelh.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel labelh.png
Oblate cubille cell.png
Trigonal trapezohedron
Tetrahedron.pngTriakis tetrahedron.png

Symmetry[edit]

These are four of the 35 cubic space groups

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) Im3m I43 [[4,3,4]]
CDel branch c1.pngCDel 4a4b.pngCDel nodeab c2.png
8°:2
nc normal cubic symmetry (229) Pm3m P43 [4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
4:2
fc half-cubic symmetry (225) Fm3m F43 [4,31,1] = [4,3,4,1+]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
2:2
d diamond symmetry
or extended quarter-cubic symmetry
(227) Fd3m Fd4n3 [[3[4]]] = [[1+,4,3,4,1+]]
CDel branch c1.pngCDel 3ab.pngCDel branch c2.png
2+:2

References[edit]

  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[edit]

  • 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. [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]