Architectonic and catoptric tessellation

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.

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

Enumeration

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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
(Cubic honeycomb)
       
  
Octahedron,      
 
  Cubille
       
 
 
Cube,      
 
     
J12,32
A15
W14
G7
t1δ4
nc
[4,3,4]
       
Cuboctahedrille
(Rectified cubic honeycomb)
       
  
Cuboid,      
 
   Oblate octahedrille
       
 
 
Isosceles square bipyramid
     
  
     ,      
J13
A14
W15
G8
t0,1δ4
nc
[4,3,4]
       
Truncated cubille
(Truncated cubic honeycomb)
       
  
Isosceles square pyramid
 
   Pyramidille
       
 
 
Isosceles square pyramid
  
     ,      
J14
A17
W12
G9
t0,2δ4
nc
[4,3,4]
       
2-RCO-trille
(Cantellated cubic honeycomb)
       
  
Wedge
 
    Quarter oblate octahedrille
       
 
irr. Triangular bipyramid
   
     ,      ,      
J16
A3
W2
G28
t1,2δ4
bc
[[4,3,4]]
   
Truncated octahedrille
(Bitruncated cubic honeycomb)
   
  
Tetragonal disphenoid
 
  Oblate tetrahedrille
       
 
 
Tetragonal disphenoid
 
     
J17
A18
W13
G25
t0,1,2δ4
nc
[4,3,4]
       
n-tCO-trille
(Cantitruncated cubic honeycomb)
       
  
Mirrored sphenoid
 
    Triangular pyramidille
       
 
Mirrored sphenoid
   
     ,      ,      
J18
A19
W19
G20
t0,1,3δ4
nc
[4,3,4]
       
1-RCO-trille
(Runcitruncated cubic honeycomb)
       
  
Trapezoidal pyramid
 
     Square quarter pyramidille
       
 
Irr. pyramid
    
     ,      ,      ,      
J19
A22
W18
G27
t0,1,2,3δ4
bc
[[4,3,4]]
   
b-tCO-trille
(Omnitruncated cubic honeycomb)
   
  
Phyllic disphenoid
 
   Eighth pyramidille
       
 
Phyllic disphenoid
  
     ,      
J21,31,51
A2
W9
G1
4
fc
[4,31,1]
     
Tetroctahedrille
(Tetrahedral-octahedral honeycomb)
      or        
  
Cuboctahedron,      
 
   Dodecahedrille
      or        
 
 
Rhombic dodecahedron,      
  
     ,      
J22,34
A21
W17
G10
h2δ4
fc
[4,31,1]
     
truncated tetraoctahedrille
(Truncated tetrahedral-octahedral honeycomb)
      or        
  
Rectangular pyramid
 
    Half oblate octahedrille
      or        
 
rhombic pyramid
   
     ,      ,      
J23
A16
W11
G5
h3δ4
fc
[4,31,1]
     
3-RCO-trille
(Cantellated tetrahedral-octahedral honeycomb)
      or        
  
Truncated triangular pyramid
 
    Quarter cubille
       
   
irr. triangular bipyramid
   
J24
A20
W16
G21
h2,3δ4
fc
[4,31,1]
     
f-tCO-trille
(Cantitruncated tetrahedral-octahedral honeycomb)
      or        
  
Mirrored sphenoid
 
    Half pyramidille
       
   
Mirrored sphenoid
   
J25,33
A13
W10
G6
4
d
[[3[4]]]
   
Truncated tetrahedrille
(Cyclotruncated tetrahedral-octahedral honeycomb)
    or        
  
Isosceles triangular prism
 
   Oblate cubille
         
 
Trigonal trapezohedron
  

Vertex Figures

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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:

 

Symmetry

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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]]
   
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

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  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

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  • 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. 81 (491). Leicester: The Mathematical Association: 213–219. doi:10.2307/3619198. 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]