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Architectonic and catoptric tessellation

{{Short description|Uniform Euclidean 3D tessellations and their duals}}

{{use mdy dates|date=September 2021}}

{{Use American English|date = March 2019}}

File:Catoptric tessellation cells.png

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.

class="wikitable sortable"

!rowspan=2|Ref.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.

indices

!rowspan=2|Symmetry

!colspan=3|Architectonic tessellation

!colspan=3|Catoptric tessellation

Name
Coxeter diagram
Image

!Vertex figure
Image

!Cells

!Name

!Cell

!Vertex figures

align=center

!J11,15
A1
W1
G22
δ4

!nc
[4,3,4]
{{CDD|node|4|node|3|node|4|node}}

| Cubille
(Cubic honeycomb)
{{CDD|node_1|4|node|3|node|4|node}}
40px40px

| Octahedron, {{CDD|node_1|3|node|4|node}}
60px

| 30px

| Cubille
{{CDD|node_f1|4|node|3|node|4|node}}
40px

| 80px
Cube, {{CDD|node_f1|3|node|4|node}}

| 30px
{{CDD|node_f1|4|node|3|node}}

align=center

!J12,32
A15
W14
G7
t1δ4

!nc
[4,3,4]
{{CDD|node|4|node|3|node|4|node}}

| Cuboctahedrille
(Rectified cubic honeycomb)
{{CDD|node|4|node_1|3|node|4|node}}
40px40px

| Cuboid, {{CDD|node_1|2|node_1|4|node}}
60px

| 30px30px

| Oblate octahedrille
{{CDD|node|4|node_f1|3|node|4|node}}
60px

| 80px
Isosceles square bipyramid
{{CDD|node_f1|2x|node_f1|4|node}}

| 30px30px
{{CDD|node_f1|3|node|4|node}}, {{CDD|node|4|node_f1|3|node}}

align=center

!J13
A14
W15
G8
t0,1δ4

!nc
[4,3,4]
{{CDD|node|4|node|3|node|4|node}}

| Truncated cubille
(Truncated cubic honeycomb)
{{CDD|node_1|4|node_1|3|node|4|node}}
40px40px

| Isosceles square pyramid
60px

| 30px30px

| Pyramidille
{{CDD|node_f1|4|node_f1|3|node|4|node}}
60px

| 80px
Isosceles square pyramid

| 30px30px
{{CDD|node_f1|3|node|4|node}}, {{CDD|node_f1|4|node_f1|3|node}}

align=center

!J14
A17
W12
G9
t0,2δ4

!nc
[4,3,4]
{{CDD|node|4|node|3|node|4|node}}

| 2-RCO-trille
(Cantellated cubic honeycomb)
{{CDD|node_1|4|node|3|node_1|4|node}}
40px40px

| Wedge
60px

| 30px30px30px

| Quarter oblate octahedrille
{{CDD|node_f1|4|node|3|node_f1|4|node}}

| 80px
irr. Triangular bipyramid

| 30px30px30px
{{CDD|node_f1|4|node|3|node_f1}}, {{CDD|node|3|node_f1|4|node}}, {{CDD|node_f1|2x|node_f1|4|node}}

align=center

!J16
A3
W2
G28
t1,2δ4

!bc
{{brackets|4,3,4}}
{{CDD|branch_c1|4a4b|nodeab_c2}}

| Truncated octahedrille
(Bitruncated cubic honeycomb)
{{CDD|branch_11|4a4b|nodes}}
40px40px

| Tetragonal disphenoid
60px

| 30px

| Oblate tetrahedrille
{{CDD|node|4|node_f1|3|node_f1|4|node}}
40px

| 80px
Tetragonal disphenoid

| 30px
{{CDD|node_f1|3|node_f1|4|node}}

align=center

!J17
A18
W13
G25
t0,1,2δ4

!nc
[4,3,4]
{{CDD|node|4|node|3|node|4|node}}

| n-tCO-trille
(Cantitruncated cubic honeycomb)
{{CDD|node_1|4|node_1|3|node_1|4|node}}
40px40px

| Mirrored sphenoid
60px

| 30px30px30px

| Triangular pyramidille
{{CDD|node_f1|4|node_f1|3|node_f1|4|node}}

| 80px
Mirrored sphenoid

| 30px30px30px
{{CDD|node_f1|4|node_f1|3|node_f1}}, {{CDD|node_f1|3|node_f1|4|node}}, {{CDD|node_f1|2x|node_f1|4|node}}

align=center

!J18
A19
W19
G20
t0,1,3δ4

!nc
[4,3,4]
{{CDD|node|4|node|3|node|4|node}}

| 1-RCO-trille
(Runcitruncated cubic honeycomb)
{{CDD|node_1|4|node_1|3|node|4|node_1}}
40px40px

| Trapezoidal pyramid
60px

| 30px30px30px30px

| Square quarter pyramidille
{{CDD|node_f1|4|node_f1|3|node|4|node_f1}}

| 80px
Irr. pyramid

| 30px30px30px30px
{{CDD|node_f1|3|node|4|node_f1}}, {{CDD|node_f1|2x|node|4|node_f1}}, {{CDD|node_f1|4|node_f1|2x|node_f1}}, {{CDD|node_f1|4|node_f1|3|node}}

align=center

!J19
A22
W18
G27
t0,1,2,3δ4

!bc
{{brackets|4,3,4}}
{{CDD|branch_c1|4a4b|nodeab_c2}}

| b-tCO-trille
(Omnitruncated cubic honeycomb)
{{CDD|branch_11|4a4b|nodes_11}}
40px40px

| Phyllic disphenoid
60px

| 30px30px

| Eighth pyramidille
{{CDD|node_f1|4|node_f1|3|node_f1|4|node_f1}}

| 80px
Phyllic disphenoid

| 30px30px
{{CDD|node_f1|3|node_f1|4|node_f1}}, {{CDD|node_f1|2x|node_f1|4|node_f1}}

align=center

!J21,31,51
A2
W9
G1
hδ4

!fc
[4,31,1]
{{CDD|node|4|node|split1|nodes}}

| Tetroctahedrille
(Tetrahedral-octahedral honeycomb)
{{CDD|node_1|3|node|split1-43|nodes}} or {{CDD|node_h|4|node|3|node|4|node}}
40px40px

| Cuboctahedron, {{CDD|node|3|node_1|4|node}}
60px

| 30px30px

| Dodecahedrille
{{CDD|node_f1|3|node|split1-43|nodes}} or {{CDD|node_fh|4|node|3|node|4|node}}
40px

| 80px
Rhombic dodecahedron, {{CDD|node|3|node_f1|4|node}}

| 30px30px
{{CDD|node_f1|3|node|3|node}}, {{CDD|node_f1|4|node|3|node}}

align=center

!J22,34
A21
W17
G10
h2δ4

!fc
[4,31,1]
{{CDD|node|4|node|split1|nodes}}

| truncated tetraoctahedrille
(Truncated tetrahedral-octahedral honeycomb)
{{CDD|node_1|3|node_1|split1-43|nodes}} or {{CDD|node_h|4|node|3|node_1|4|node}}
40px40px

| Rectangular pyramid
60px

| 30px30px30px

| Half oblate octahedrille
{{CDD|node_f1|3|node_f1|split1-43|nodes}} or {{CDD|node_fh|4|node|3|node_f1|4|node}}

| 80px
rhombic pyramid

| 30px30px30px
{{CDD|node_f1|3|node_f1|4|node}}, {{CDD|node|3|node_f1|4|node}}, {{CDD|node_f1|3|node_f1|3|node}}

align=center

!J23
A16
W11
G5
h3δ4

!fc
[4,31,1]
{{CDD|node|4|node|split1|nodes}}

| 3-RCO-trille
(Cantellated tetrahedral-octahedral honeycomb)
{{CDD|nodes_10ru|split2|node|4|node_1}} or {{CDD|node_h|4|node|3|node|4|node_1}}
40px40px

| Truncated triangular pyramid
60px

| 30px30px30px

| Quarter cubille
{{CDD|node_fh|4|node|3|node|4|node_f1}}

| 80px 80px
irr. triangular bipyramid

| 30px30px30px

align=center

!J24
A20
W16
G21
h2,3δ4

!fc
[4,31,1]
{{CDD|node|4|node|split1|nodes}}

| f-tCO-trille
(Cantitruncated tetrahedral-octahedral honeycomb)
{{CDD|nodes_10ru|split2|node_1|4|node_1}} or {{CDD|node_h|4|node|3|node_1|4|node_1}}
40px40px

| Mirrored sphenoid
60px

| 30px30px30px

| Half pyramidille
{{CDD|node_fh|4|node|3|node_f1|4|node_f1}}

| 80px 80px
Mirrored sphenoid

| 30px30px30px

align=center

!J25,33
A13
W10
G6
qδ4

!d
{{brackets|3{{bracket|4}}}}
{{CDD|branch_c1|3ab|branch_c2}}

| Truncated tetrahedrille
(Cyclotruncated tetrahedral-octahedral honeycomb)
{{CDD|branch_11|3ab|branch}} or {{CDD|node_h1|4|node|3|node|4|node_h1}}
40px40px

| Isosceles triangular prism
60px

| 30px30px

| Oblate cubille
{{CDD|labelh|node_fh|4|node|3|node|4|node_fh|labelh}}

| 80px
Trigonal trapezohedron

| 30px30px

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:

1136px

Symmetry

File:35 cubic fibrifold groups.png

These four symmetry groups are labeled as:

class=wikitable

!Label

!Description

!space group

Intl symbol

!Geometric

notation{{cite journal | last1 = Hestenes | first1 = David | last2 = Holt | first2 = Jeremy | date = 2007-02-27 | title = Crystallographic space groups in geometric algebra | journal = Journal of Mathematical Physics | volume = 48 | issue = 2 | pages = 023514 | publisher = AIP Publishing LLC | issn = 1089-7658 | doi = 10.1063/1.2426416 | url = https://davidhestenes.net/geocalc/pdf/CrystalGA.pdf }}

![[Coxeter notation|Coxeter

notation]]

![[Fibrifold notation|Fibrifold

notation]]

align=center

!bc

| bicubic symmetry
or extended cubic symmetry

| (221) Im{{overline|3}}m

I43{{brackets|4,3,4}}
{{CDD|branch_c1|4a4b|nodeab_c2}}		8°:2
align=center

!nc

| normal cubic symmetry

| (229) Pm{{overline|3}}m

P43[4,3,4]
{{CDD|node|4|node|3|node|4|node}}		4−:2
align=center

!fc

| half-cubic symmetry

| (225) Fm{{overline|3}}m

F43[4,31,1] = [4,3,4,1+]
{{CDD|node|4|node|split1|nodes}}		2−:2
align=center

!d

| diamond symmetry
or extended quarter-cubic symmetry

| (227) Fd{{overline|3}}m

Fd4n3{{brackets|3{{bracket|4}}}} = 1+,4,3,4,1+
{{CDD|branch_c1|3ab|branch_c2}}		2+:2

References

{{reflist}}

Further reading

  • [https://books.google.com/books?id=nVx-tu596twC&q=space-filling+packings&pg=PA54 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
  • {{cite book | last1 = Conway | first1 = John H. | authorlink1 = John Horton Conway | last2 = Burgiel | first2 = Heidi | last3 = Goodman-Strauss | first3 = Chaim | title = The Symmetries of Things | publisher = A K Peters, Ltd. | date = 2008 | chapter = 21. Naming Archimedean and Catalan Polyhedra and Tilings | pages = 292–298 | isbn = 978-1-56881-220-5}}
  • {{cite journal | last = Inchbald | first = Guy | title = The Archimedean honeycomb duals | journal = The Mathematical Gazette | volume = 81 | issue = 491 | pages = 213–219 | publisher = The Mathematical Association | location = Leicester | date = July 1997 | jstor = 3619198| doi = 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]
  • {{cite book | last = Pearce | first = Peter | title = Structure in Nature is a Strategy for Design | publisher = The MIT Press | date = 1980 | pages = 41–47 | isbn = 978-0-262-66045-7 | url = https://books.google.com/books?id=sfc2OEuE8oQC&q=%22the+tetrahedron+and+octahedron+space+filling%22}}
  • 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}} [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]

{{Tessellation}}

Category:Honeycombs (geometry)

Category:John Horton Conway