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In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.

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HistoryEdit

According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to finite regular skew polyhedra in 4-dimensions, and infinite regular skew apeirohedra in 3-dimensions (described here).

Coxeter identified 3 forms, with planar faces and skew vertex figures, two are complements of each other. They are all named with a modified Schläfli symbol {l,m|n}, where there are l-gonal faces, m faces around each vertex, with holes identified as n-gonal missing faces.

Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.

The regular skew polyhedra, represented by {l,m|n}, follow this equation:

  • 2 sin(π/l) · sin(π/m) = cos(π/n)

Regular skew apeirohedra of Euclidean 3-spaceEdit

The three Euclidean solutions in 3-space are {4,6|4}, {6,4|4}, and {6,6|3}. John Conway named them mucube, muoctahedron, and mutetrahedron respectively for multiple cube, octahedron, and tetrahedron.[1]

  1. Mucube: {4,6|4}: 6 squares on a vertex (related to cubic honeycomb, constructed by cubic cells, removing two opposite faces from each, and linking sets of six together around a faceless cube.)
  2. Muoctahedron: {6,4|4}: 4 hexagons on a vertex (related to bitruncated cubic honeycomb, constructed by truncated octahedron with their square faces removed and linking hole pairs of holes together.)
  3. Mutetrahedron: {6,6|3}: 6 hexagons on a vertex (related to quarter cubic honeycomb, constructed by truncated tetrahedron cells, removing triangle faces, and linking sets of four around a faceless tetrahedron.)

Coxeter gives these regular skew apeirohedra {2q,2r|p} with extended chiral symmetry [[(p,q,p,r)]+] which he says is isomorphic to his abstract group (2q,2r|2,p). The related honeycomb has the extended symmetry [[(p,q,p,r)]].[2]

14 Compact regular skew apeirohedra
Coxeter group
Symmetry
Apeirohedron
{p,q|l}
Image Face
{p}
Hole
{l}
Vertex
figure
Related
honeycomb
   
[[4,3,4]]
[[4,3,4]+]
{4,6|4}
Mucube
 
animation
         
t0,3{4,3,4}
 
{6,4|4}
Muoctahedron
 
animation
       
2t{4,3,4}
 
   
[[3[4]]]
[[3[4]]+]
{6,6|3}
Mutetrahedron
 
animation
         
q{4,3,4}
 

Regular skew apeirohedra in hyperbolic 3-spaceEdit

In 1967, C. W. L. Garner identified 31 hyperbolic skew apeirohedra with regular skew polygon vertex figures, found in a similar search to the 3 above from Euclidean space.[3]

These represent 14 compact and 17 paracompact regular skew polyhedra in hyperbolic space, constructed from the symmetry of a subset of linear and cyclic Coxeter groups graphs of the form [[(p,q,p,r)]], These define regular skew polyhedra {2q,2r|p} and dual {2r,2q|p}. For the special case of linear graph groups r = 2, this represents the Coxeter group [p,q,p]. It generates regular skews {2q,4|p} and {4,2q|p}. All of these exist as a subset of faces of the convex uniform honeycombs in hyperbolic space.

The skew apeirohedron shares the same antiprism vertex figure with the honeycomb, but only the zig-zag edge faces of the vertex figure are realized, while the other faces make "holes".

14 Compact regular skew apeirohedra
Coxeter
group
Apeirohedron
{p,q|l}
Face
{p}
Hole
{l}
Honeycomb Vertex
figure
Apeirohedron
{p,q|l}
Face
{p}
Hole
{l}
Honeycomb Vertex
figure
    
[3,5,3]
{10,4|3}         
2t{3,5,3}
  {4,10|3}         
t0,3{3,5,3}
 
   
[5,3,5]
{6,4|5}        
2t{5,3,5}
  {4,6|5}        
t0,3{5,3,5}
 
    
[(4,3,3,3)]
{8,6|3}         
ct{(4,3,3,3)}
  {6,8|3}         
ct{(3,3,4,3)}
 
    
[(5,3,3,3)]
{10,6|3}         
ct{(5,3,3,3)}
  {6,10|3}         
ct{(3,3,5,3)}
 
     
[(4,3,4,3)]
{8,8|3}          
ct{(4,3,4,3)}
  {6,6|4}          
ct{(3,4,3,4)}
 
     
[(5,3,4,3)]
{8,10|3}          
ct{(4,3,5,3)}
  {10,8|3}          
ct{(5,3,4,3)}
 
     
[(5,3,5,3)]
{10,10|3}          
ct{(5,3,5,3)}
  {6,6|5}          
ct{(3,5,3,5)}
 
17 Paracompact regular skew apeirohedra
Coxeter
group
Apeirohedron
{p,q|l}
Face
{p}
Hole
{l}
Honeycomb Vertex
figure
Apeirohedron
{p,q|l}
Face
{p}
Hole
{l}
Honeycomb Vertex
figure
    
[4,4,4]
{8,4|4}         
2t{4,4,4}
  {4,8|4}         
t0,3{4,4,4}
 
    
[3,6,3]
{12,4|3}         
2t{3,6,3}
  {4,12|3}         
t0,3{3,6,3}
 
   
[6,3,6]
{6,4|6}        
2t{6,3,6}
  {4,6|6}        
t0,3{6,3,6}
 
    
[(4,4,4,3)]
{8,6|4}         
ct{(4,4,3,4)}
  {6,8|4}         
ct{(3,4,4,4)}
 
     
[(4,4,4,4)]
{8,8|4}          
q{4,4,4}
 
     
[(6,3,3,3)]
{12,6|3}         
ct{(6,3,3,3)}
  {6,12|3}         
ct{(3,3,6,3)}
 
     
[(6,3,4,3)]
{12,8|3}          
ct{(6,3,4,3)}
  {8,12|3}          
ct{(4,3,6,3)}
 
     
[(6,3,5,3)]
{12,10|3}          
ct{(6,3,5,3)}
  {10,12|3}          
ct{(5,3,6,3)}
 
     
[(6,3,6,3)]
{12,12|3}          
ct{(6,3,6,3)}
  {6,6|6}          
ct{(3,6,3,6)}
 

See alsoEdit

ReferencesEdit

  1. ^ The Symmetry of Things, 2008, Chapter 23 Objects with Primary Symmetry, Infinite Platonic Polyhedra, pp. 333–335
  2. ^ Coxeter, Regular and Semi-Regular Polytopes II 2.34)
  3. ^ Garner, C. W. L. Regular Skew Polyhedra in Hyperbolic Three-Space. Can. J. Math. 19, 1179–1186, 1967. [1] Note: His paper says there are 32, but one is self-dual, leaving 31.
  • Petrie–Coxeter Maps Revisited PDF, Isabel Hubard, Egon Schulte, Asia Ivic Weiss, 2005
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5,
  • Peter McMullen, Four-Dimensional Regular Polyhedra, Discrete & Computational Geometry September 2007, Volume 38, Issue 2, pp 355–387
  • Coxeter, Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8
  • 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 [2]
    • (Paper 2) H.S.M. Coxeter, "The Regular Sponges, or Skew Polyhedra", Scripta Mathematica 6 (1939) 240–244.
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)
    • Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33–62, 1937.