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. Catoptric tessellation cells.png
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 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.

Contents

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.

Ref. [1]

indices

Symmetry Architectonic tessellationCatoptric tessellation
Name
Coxeter diagram
Image
Vertex figure
Image
CellsNameCell 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
(Cubic honeycomb)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Partial cubic honeycomb.png Cubic honeycomb.png
Octahedron, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cubic honeycomb verf.svg
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
(Rectified cubic honeycomb)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Rectified cubic honeycomb.png Rectified cubic tiling.png
Cuboid, CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
Rectified cubic honeycomb verf.png
Octahedron.png Cuboctahedron.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.png Rhombic 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
(Truncated cubic honeycomb)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Truncated cubic honeycomb.png Truncated cubic tiling.png
Isosceles square pyramid
Truncated cubic honeycomb verf.png
Octahedron.png Truncated 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.png Triakis 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
(Cantellated cubic honeycomb)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cantellated cubic honeycomb.png Cantellated cubic tiling.png
Wedge
Cantellated cubic honeycomb verf.png
Small rhombicuboctahedron.png Cuboctahedron.png Hexahedron.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.png Rhombic dodecahedron.jpg Octahedron.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
(Bitruncated cubic honeycomb)
CDel branch 11.pngCDel 4a4b.pngCDel nodes.png
Bitruncated Cubic Honeycomb1.svg Bitruncated 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
(Cantitruncated cubic honeycomb)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cantitruncated Cubic Honeycomb.svg Cantitruncated cubic tiling.png
Mirrored sphenoid
Cantitruncated cubic honeycomb verf.png
Great rhombicuboctahedron.png Truncated octahedron.png Hexahedron.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.png Tetrakis cube.png Octahedron.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
(Runcitruncated cubic honeycomb)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Runcitruncated cubic honeycomb.jpg Runcitruncated cubic tiling.png
Trapezoidal pyramid
Runcitruncated cubic honeycomb verf.png
Small rhombicuboctahedron.png Truncated hexahedron.png Octagonal prism.png Hexahedron.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.png Triakis octahedron.png Octagonal bipyramid.png Octahedron.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
(Omnitruncated cubic honeycomb)
CDel branch 11.pngCDel 4a4b.pngCDel nodes 11.png
HC A6-Pr8.png Omnitruncated cubic tiling.png
Phyllic disphenoid
Omnitruncated cubic honeycomb verf2.png
Great rhombicuboctahedron.png Octagonal 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.png Octagonal 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
(Tetrahedral-octahedral honeycomb)
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.png Alternated cubic tiling.png
Cuboctahedron, CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Alternated cubic honeycomb verf.svg
Tetrahedron.png Octahedron.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.png Hexahedron.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
(Truncated tetrahedral-octahedral honeycomb)
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.svg Truncated alternated cubic tiling.png
Rectangular pyramid
Truncated alternated cubic honeycomb verf.png
Truncated octahedron.png Cuboctahedron.png Truncated 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.png Rhombic dodecahedron.jpg Triakis 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
(Cantellated tetrahedral-octahedral honeycomb)
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.jpg Runcinated alternated cubic tiling.png
Truncated triangular pyramid
Runcinated alternated cubic honeycomb verf.png
Small rhombicuboctahedron.png Hexahedron.png Tetrahedron.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.png Octahedron.png Tetrahedron.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
(Cantitruncated tetrahedral-octahedral honeycomb)
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.jpg Cantitruncated alternated cubic tiling.png
Mirrored sphenoid
Runcitruncated alternate cubic honeycomb verf.png
Great rhombicuboctahedron.png Truncated hexahedron.png Truncated 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.png Triakis octahedron.png Triakis tetrahedron.png
J25,33
A13
W10
G6
4
d
[[3[4]]]
CDel branch c1.pngCDel 3ab.pngCDel branch c2.png
Truncated tetrahedrille
(Cyclotruncated tetrahedral-octahedral honeycomb)
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.png Bitruncated alternated cubic tiling.png
Isosceles triangular prism
T01 quarter cubic honeycomb verf2.png
Tetrahedron.png Truncated 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.png Triakis tetrahedron.png

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:

Architectonic Vertex Figures and Dual Cells.svg

Symmetry

These are four of the 35 cubic space groups 35 cubic fibrifold groups.png
These are four of the 35 cubic space groups

These four symmetry groups are labeled as:

LabelDescription space group

Intl symbol

Geometric

notation [2]

Coxeter notation Fibrifold notation
bcbicubic symmetry
or extended cubic symmetry
(221) Im3mI43[[4,3,4]]
CDel branch c1.pngCDel 4a4b.pngCDel nodeab c2.png
8°:2
ncnormal cubic symmetry(229) Pm3mP43[4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
4:2
fchalf-cubic symmetry(225) Fm3mF43[4,31,1] = [4,3,4,1+]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
2:2
ddiamond symmetry
or extended quarter-cubic symmetry
(227) Fd3mFd4n3[[3[4]]] = [[1+,4,3,4,1+]]
CDel branch c1.pngCDel 3ab.pngCDel branch c2.png
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 (February 27, 2007). "Crystallographic space groups in geometric algebra" (PDF). Journal of Mathematical Physics. 48 (2). AIP Publishing LLC: 023514. doi:10.1063/1.2426416. ISSN   1089-7658.

Further reading