User:Tomruen/complex polytopes

Regular and uniform complex polytopes

Complex Polygons (C2)

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5{}+{}
 
    =    
 
   
 
    =    
 
   
 
   
 
    =    

The complex reflection group is p[q]r, order  [1] has, configuration matrix:[2]  

    =     (Order 2p2 and p2) - Related to p-p duoprisms

Regular
G(p,1,2)     k-face fk f0 f1 k-fig Notes
A1     ( ) f0 p2 2 { } G(p,1,2)/A1 = 2p2/2 = p2
p[ ]     p{ } f1 p 2p ( ) G(p,1,2)/p[ ] = 2p2/p = 2p
Quasiregular
p[]p[]     k-face fk f0 f1 k-fig Notes
    ( ) f0 p2 1 1 { } p[]p[] = p2
p[]     p{ } f1 p p * ( ) p[]p[]/p[] = p
p[]     p{ } p * p p[]p[]/p[] = p

    (order pq) - related to p-q duoprism

Quasiregular
p[]q[]     k-face fk f0 f1 k-fig Notes
    ( ) f0 pq 1 1 { } p[]q[] = pq
p[]     p{ } f1 p q * ( ) p[]q[]/p[] = q
q[]     q{ } q * p p[]q[]/q[] = p

    =     (order 18 and 9) - related to 3-3 duoprism

Regular
M2     k-face fk f0 f1 k-fig Notes
A1     ( ) f0 9 2 { } M2/A1 = 18/2 = 9
L1     3{ } f1 3 6 ( ) M2/L1 = 18/3 = 6
Quasiregular
L2
1
    k-face fk f0 f1 k-fig Notes
    ( ) f0 9 1 1 { } L2
1
= 9
L1     3{ } f1 3 3 * ( ) L2
1
/L1 = 9/3 = 3
    3 * 3

    (order 6) - related to triangular prism

Quasiregular
L1A1     k-face fk f0 f1 k-fig Notes
    ( ) f0 6 1 1 { } L1A1 = 6
L1     3{ } f1 3 2 * ( ) L1A1/L1 = 6/3 = 2
A1     { } 2 * 3 L1A1/A1 = 6/2 = 3

    (Order 18) - related 3-3 duopyramid

Regular
M2     k-face fk f0 f1 k-fig Notes
L1     ( ) f0 6 3 3{ } M2/L1 = 18/3 = 6
A1     { } f1 2 9 ( ) M2/A1 = 18/2 = 9

    (Order 18)

Quasiregular
M2     k-face fk f0 f1 k-fig Notes
    ( ) f0 18 1 1 { } M2 = 18
A1     { } f1 2 9 * ( ) M2/A1 = 18/2 = 9
L1     3{ } 3 * 6 M2/L1 = 18/3 = 6

Möbius–Kantor polygon     =    , (order 24)

Regular
L2     k-face fk f0 f1 k-fig Notes
L1     ( ) f0 8 3 3{ } L2/L1 = 4!/3 = 8
    3{ } f1 3 8 ( )

    =     (order 48 and 24)

Regular
G6     k-face fk f0 f1 k-fig Notes
A1     ( ) f0 24 2 { } G6/A1 = 48/2 = 24
L1     3{ } f1 3 16 ( ) G6/L1 = 48/3 = 16
Quasiregular
L2     k-face fk f0 f1 k-fig Notes
    ( ) f0 24 1 1 { } L2 = 24
L1     3{ } f1 3 8 * ( ) L2/L1 = 24/3 = 8
    3 * 8

Complex polyhedra (C3)

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There are 9 unique regular and uniform complex polyhedra from 14 Wythoff constructions (ringed patterns) in the L3 and M3 Shephard groups. These polyhedra can be seen a complex analogues of tetrahedral symmetry and octahedral symmetry of the regular tetrahedron, cube, and octahedron.

Type L3 =      , order 648 M3 =      , order 1296
Regular       =       (27,72,27)       (54,216,72)       =       (72,216,54)
Truncation       =       (27,72+216,27+27)       (648,216+432,72+72)       =       (648,216+432,72+72)
Quasiregular       =       (27,216,54+54)       =       (216,432,54+72)
Cantellation       =       (216,216+216,27+27+72)       (216,216+432,54+72)
Cantitruncation       =       (648,216+216+216,27+27+72)       (1296,432+432+648,54+54+216)

      =       - analogous to real tetrahedron

Regular
L3       k-face fk f0 f1 f2 k-fig Notes
L2       ( ) f0 27 8 8 3{3}3 L3/L2 = 27*4!/4! = 27
L1L1       3{ } f1 3 72 3 3{ } L3/L1L1 = 27*4!/9 = 72
L2       3{3}3 f2 8 8 27 ( ) L3/L2 = 27*4!/4! = 27

Rectified Hessian polyhedron

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      =       - analogous to real octahedron

Regular
M3       k-face fk f0 f1 f2 k-fig Notes
M2       ( ) f0 72 9 6 3{4}2 M3/M2 = 1296/18 = 72
L1A1       3{ } f1 3 216 2 { } M3/L1A1 = 1296/3/2 = 216
L2       3{3}3 f2 8 8 54 ( ) M3/L2 = 1296/24 = 54
Quasiregular
L3       k-face fk f0 f1 f2 k-fig Notes
L1L1       ( ) f0 72 9 3 3 3{ }×3{ } L3/L1L1 = 648/9 = 72
L1       3{ } f1 3 216 1 1 { } L3/L1 = 648/3 = 216
L2       3{3}3 f2 8 8 27 * ( ) L3/L2 = 648/24 = 27
      8 8 * 27

Truncated Hessian polyhedron

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      =       - analogous to real truncated tetrahedron

Truncated
L3       k-face fk f0 f1 f2 k-fig Notes
L1       ( ) f0 27 1 3 3 3 L3/L1 = 648/24 = 27
L1L1       3{ } f1 3 72 * 3 0 L3/L1L1 = 648/3/3 = 72
L1       3 * 216 1 2 L3/L1 = 648/3 = 216
L2       t(3{3}3) f2 24 8 8 27 * ( ) L3/L2 = 648/24 = 27
      3{3}3 8 0 8 * 27

Cantellated Hessian polyhedron

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      =       - analogous to real cuboctahedron

Cantellated
L3       k-face fk f0 f1 f2 k-fig Notes
L1       ( ) f0 216 1 3 3 3 0 3{ }×{ } L3/L1 = 648/3 = 216
      3{ } f1 3 216 * 2 0 0 { }
      3 * 216 1 1 0
L2       3{3}3 f2 8 8 0 27 * * ( ) L3/L2 = 648/24 = 27
L1L1       3{ }×3{ } 9 3 3 * 72 * L3/L1L1 = 648/9 = 72
L2       3{3}3 8 0 8 * * 27 L3/L2 = 648/24 = 27
Rectified
M3       k-face fk f0 f1 f2 k-fig Notes
L1A1       ( ) f0 216 6 3 2 3{ }×{ } M3/L1A1 = 1296/6 = 216
L1       3{ } f1 3 432 1 1 { } M3/L1 = 1296/3 = 432
L2       3{3}3 f2 8 8 54 * ( ) M3/L2 = 1296/24 = 54
M2       3{4}2 9 6 * 72 M3/M2 = 1296/18 = 72

Cantitruncated Hessian polyhedron

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      =       - analogous to real truncated octahedron

Truncated
M3       k-face fk f0 f1 f2 k-fig Notes
A1       ( ) f0 648 ? ? ? ? M3/L1 = 1296/2 = 648
L1A1       3{ } f1 3 216 * ? ? M3/L1A1 = 1296/3/2 = 216
L1       3 * 432 ? ? M3/L1 = 1296/3 = 432
L2       t(3{3}3) f2 24 8 8 54 * ( ) M3/L2 = 1296/24 = 54
M2       3{4}2 9 0 6 * 72 M3/M2 = 1296/48 = 27
Cantitruncated
L3       k-face fk f0 f1 f2 k-fig Notes
      ( ) f0 648 ? ? ? ? ? ? L3 = 648
L1       3{ } f1 3 216 * * ? ? ? L3/L1 = 648/3 = 216
      3 * 216 * ? ? ?
      3 * * 216 ? ? ?
L2       3{3}3 f2 24 8 8 0 27 * * ( ) L3/L2 = 648/24 = 27
L1L1       3{ }×3{ } 9 3 0 3 * 72 * L3/L1/L1 = 648/3/3 = 72
L2       3{3}3 24 0 8 8 * * 27 L3/L2 = 648/24 = 27

Double Hessian polyhedron

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Double Hessian polyhedron       - analogous to real cube

Regular
M3       k-face fk f0 f1 f2 k-fig Notes
L2       ( ) f0 54 8 8 3{3}3 M3/L2 = 1296/24 = 54
L1A1       { } f1 2 216 3 3{ } M3/L1A1 = 1296/3/2 = 216
M2       2{4}3 f2 6 9 72 ( ) M3/M2 = 1296/18 = 72

Truncated double Hessian polyhedron

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      - analogous to real truncated cube

Truncated
M3       k-face fk f0 f1 f2 k-fig Notes
L1       ( ) f0 648 ? ? ? ? M3/L1 = 1296/3 = 432
L1A1       { } f1 2 216 * ? ? M3/L1A1 = 1296/6 = 216
L1       3{ } 3 * 432 ? ? M3/L1 = 1296/3 = 432
M2       t(3{4}2) f2 24 8 8 72 * ( ) M3/M2 = 1296/18 = 72
L2       3{3}3 8 0 8 * 72 M3/L2 = 1296/24 = 54

Cantellated double Hessian polyhedron

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      - analogous to real rhombicuboctahedron

Cantellated
M3       k-face fk f0 f1 f2 k-fig Notes
L1       ( ) f0 216 1 3 3 3 0 M3/L1 = 1296/3 = 216
A1       { } f1 3 648 * 2 0 0 { } M3/A1 = 1296/2 = 648
L1       3{ } 3 * 216 1 1 0 M3/L1 = 1296/3 = 216
M2       3{4}2 f2 9 6 0 72 * * ( ) M3/M2 =1296/18 = 72
L1A1       3{ }×{ } 6 3 2 * 216 * M3/L1A1 = 1296/6 = 216
L2       3{3}3 8 0 8 * * 54 M3/L2 = 1296/24 = 54

Cantitruncated double Hessian polyhedron

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      - analogous to real truncated cuboctahedron

Cantitruncated
M3       k-face fk f0 f1 f2 k-fig Notes
      ( ) f0 1296 ? ? ? ? ? ? M3 = 1296
L1       3{ } f1 3 432 * * ? ? ? M3/L1 = 1296/3 = 432
      3 * 432 * ? ? ?
A1       { } 3 * * 648 ? ? ? M3/A1 = 1296/2 = 648
L2       t(3{3}3) f2 24 8 8 0 54 * * ( ) M3/L2 = 1296/24 = 54
L1A1       3{ }×{ } 6 3 0 2 * 216 * M3/L1/A1 = 1296/6 = 216
M2       t(3{4}2) 18 0 9 6 * * 27 M3/M2 = 1296/48 = 27

Witting polytope (C4)

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Witting polytope -         - Real representation 421 polytope

L4         k-face fk f0 f1 f2 f3 k-fig Notes
L3         ( ) f0 240 27 72 27 3{3}3{3}3 L4/L3 = 216*6!/27/4! = 240
L3L1         3{ } f1 3 2160 8 8 3{3}3 L4/L3L1 = 216*6!/4!/3 = 2160
        3{3}3 f2 8 8 2160 3 3{ }
L3         3{3}3{3}3 f3 27 72 27 240 ( ) L4/L3 = 216*6!/27/4! = 240

          - Honeycomb of Witting polytope: L5 is order 155520N - Real representation 521 honeycomb

L5           k-face fk f0 f1 f2 f3 f4 k-figure Notes
L4           ( ) f0 N 240 2160 2160 240 3{3}3{3}3{3}3 L5/L4 = N
L3L1           3{ } f1 3 80N 27 72 27 3{3}3{3}3 L5/L3L1 = NL4/L3L1 = 80N
L2L2           3{3}3 f2 8 8 270N 8 8 3{3}3 L5/L2L2 = NL4/L2L2 = 270N
L3L1           3{3}3{3}3 f3 27 72 27 80N 3 3{ } L5/L3L1 = NL4/L3L1 = 80N
L4           3{3}3{3}3{3}3 f4 240 2160 2160 240 N ( ) L5/L4 = NL4/L4 = N

Notes

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  1. ^ Lehrer & Taylor 2009, p.87
  2. ^ Complex Regular Polytopes, p. 117