Point groups in four dimensions

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File:Polychoral group tree.png
A hierarchy of 4D point groups and some subgroups. Vertical positioning is grouped by order. Blue, green, and pink colors show reflectional, hybrid, and rotational groups.
File:Polychoral group tree-conway.png
Some 4D point groups in Conway's notation

In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere.

History on four-dimensional groups

  • 1889 Édouard Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Annales Scientifiques de l'École Normale Supérieure, Sér. 3, 6, (pp. 9–102, pp. 80–81 tetrahedra), Goursat tetrahedron
  • 1951, A. C. Hurley, Finite rotation groups and crystal classes in four dimensions, Proceedings of the Cambridge Philosophical Society, vol. 47, issue 04, p. 650[1]
  • 1962 A. L. MacKay Bravais Lattices in Four-dimensional Space[2]
  • 1964 Patrick du Val, Homographies, quaternions and rotations, quaternion-based 4D point groups
  • 1975 Jan Mozrzymas, Andrzej Solecki, R4 point groups, Reports on Mathematical Physics, Volume 7, Issue 3, p. 363-394 [3]
  • 1978 H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space.[4]
  • 1982 N. P. Warner, The symmetry groups of the regular tessellations of S2 and S3 [5]
  • 1985 E. J. W. Whittaker, An atlas of hyperstereograms of the four-dimensional crystal classes
  • 1985 H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, Coxeter notation for 4D point groups
  • 2003 John Conway and Smith, On Quaternions and Octonions, Completed quaternion-based 4D point groups
  • 2018 N. W. Johnson Geometries and Transformations, Chapter 11,12,13, Full polychoric groups, p. 249, duoprismatic groups p. 269

Isometries of 4D point symmetry

Further information: Rotations in 4-dimensional Euclidean space

There are four basic isometries of 4-dimensional point symmetry: reflection symmetry, rotational symmetry, rotoreflection, and double rotation.

Notation for groups

Point groups in this article are given in Coxeter notation, which are based on Coxeter groups, with markups for extended groups and subgroups.[6] Coxeter notation has a direct correspondence the Coxeter diagram like [3,3,3], [4,3,3], [31,1,1], [3,4,3], [5,3,3], and [p,2,q]. These groups bound the 3-sphere into identical hyperspherical tetrahedral domains. The number of domains is the order of the group. The number of mirrors for an irreducible group is nh/2, where h is the Coxeter group's Coxeter number, n is the dimension (4).[7] For cross-referencing, also given here are quaternion based notations by Patrick du Val (1964)[8] and John Conway (2003).[9] Conway's notation allows the order of the group to be computed as a product of elements with chiral polyhedral group orders: (T=12, O=24, I=60). In Conway's notation, a (±) prefix implies central inversion, and a suffix (.2) implies mirror symmetry. Similarly Du Val's notation has an asterisk (*) superscript for mirror symmetry.

Involution groups

There are five involutional groups: no symmetry [ ]+, reflection symmetry [ ], 2-fold rotational symmetry [2]+, 2-fold rotoreflection [2+,2+], and central point symmetry [2+,2+,2+] as a 2-fold double rotation.

Rank 4 Coxeter groups

A polychoric group is one of five symmetry groups of the 4-dimensional regular polytopes. There are also three polyhedral prismatic groups, and an infinite set of duoprismatic groups. Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes. The dihedral angles between the mirrors determine order of dihedral symmetry. The Coxeter–Dynkin diagram is a graph where nodes represent mirror planes, and edges are called branches, and labeled by their dihedral angle order between the mirrors. The term polychoron (plural polychora, adjective polychoric), from the Greek roots poly ("many") and choros ("room" or "space") and was advocated[10] by Norman Johnson and George Olshevsky in the context of uniform polychora (4-polytopes), and their related 4-dimensional symmetry groups.[11]

Orthogonal subgroups

B4 can be decomposed into 2 orthogonal groups, 4A1 and D4:

  1. File:CDel node c1.pngFile:CDel 4.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.pngFile:CDel 3g.pngFile:CDel node g.png = File:CDel node c1.pngFile:CDel 2.pngFile:CDel nodeab c1.pngFile:CDel 2.pngFile:CDel node c1.png (4 orthogonal mirrors)
  2. File:CDel node h0.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c4.png = File:CDel nodeab c2.pngFile:CDel split2.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c4.png (12 mirrors)

F4 can be decomposed into 2 orthogonal D4 groups:

  1. File:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.pngFile:CDel 4.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c4.png = File:CDel node c3.pngFile:CDel branch3 c3.pngFile:CDel splitsplit2.pngFile:CDel node c4.png (12 mirrors)
  2. File:CDel node c1.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.png = File:CDel node c1.pngFile:CDel splitsplit1.pngFile:CDel branch3 c2.pngFile:CDel node c2.png (12 mirrors)

B3×A1 can be decomposed into orthogonal groups, 4A1 and D3:

  1. File:CDel node c1.pngFile:CDel 4.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.pngFile:CDel 2.pngFile:CDel node c4.png = File:CDel node c1.pngFile:CDel 2.pngFile:CDel nodeab c1.pngFile:CDel 2.pngFile:CDel node c4.png (3+1 orthogonal mirrors)
  2. File:CDel node h0.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c3.pngFile:CDel 2.pngFile:CDel node h0.png = File:CDel nodeab c2.pngFile:CDel split2.pngFile:CDel node c3.png (6 mirrors)

Rank 4 Coxeter groups allow a set of 4 mirrors to span 4-space, and divides the 3-sphere into tetrahedral fundamental domains. Lower rank Coxeter groups can only bound hosohedron or hosotope fundamental domains on the 3-sphere. Like the 3D polyhedral groups, the names of the 4D polychoric groups given are constructed by the Greek prefixes of the cell counts of the corresponding triangle-faced regular polytopes.[12] Extended symmetries exist in uniform polychora with symmetric ring-patterns within the Coxeter diagram construct. Chiral symmetries exist in alternated uniform polychora. Only irreducible groups have Coxeter numbers, but duoprismatic groups [p,2,p] can be doubled to p,2,p by adding a 2-fold gyration to the fundamental domain, and this gives an effective Coxeter number of 2p, for example the [4,2,4] and its full symmetry B4, [4,3,3] group with Coxeter number 8.

Weyl
group 		Conway
Quaternion 		Abstract
structure 		Coxeter
diagram 		Coxeter
notation 		Order Commutator
subgroup 		Coxeter
number
(h) 		Mirrors
(m)
Full polychoric groups
A4 +1/60[I×I].21 S5 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.png [3,3,3] 120 [3,3,3]+ 5 10File:CDel node c1.png
D4 ±1/3[T×T].2 1/2.2S4 File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.png [31,1,1] 192 [31,1,1]+ 6 12File:CDel node c1.png
B4 ±1/6[O×O].2 2S4 = S2≀S4 File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node c2.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.png [4,3,3] 384 8 4File:CDel node c2.png 12File:CDel node c1.png
F4 ±1/2[O×O].23 3.2S4 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node c2.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.png [3,4,3] 1152 [3+,4,3+] 12 12File:CDel node c2.png 12File:CDel node c1.png
H4 ±[I×I].2 2.(A5×A5).2 File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 5.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.png [5,3,3] 14400 [5,3,3]+ 30 60File:CDel node c1.png
Full polyhedral prismatic groups
A3A1 +1/24[O×O].23 S4×D1 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 2.pngFile:CDel node c3.png [3,3,2] = [3,3]×[ ] 48 [3,3]+ - 6File:CDel node c1.png 1File:CDel node c3.png
B3A1 ±1/24[O×O].2 S4×D1 File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png File:CDel node c2.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 2.pngFile:CDel node c3.png [4,3,2] = [4,3]×[ ] 96 - 3File:CDel node c2.png 6File:CDel node c1.png 1File:CDel node c3.png
H3A1 ±1/60[I×I].2 A5×D1 File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 5.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 2.pngFile:CDel node c3.png [5,3,2] = [5,3]×[ ] 240 [5,3]+ - 15File:CDel node c1.png 1File:CDel node c3.png
Full duoprismatic groups
4A1 = 2D2 ±1/2[D4×D4] D14 = D22 File:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 2.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 2.pngFile:CDel node c4.png [2,2,2] = [ ]4 = [2]2 16 [ ]+ 4 1File:CDel node c1.png 1File:CDel node c2.png 1File:CDel node c3.png 1File:CDel node c4.png
D2B2 ±1/2[D4×D8] D2×D4 File:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 2.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 4.pngFile:CDel node c4.png [2,2,4] = [2]×[4] 32 [2]+ - 1File:CDel node c1.png 1File:CDel node c2.png 2File:CDel node c3.png 2File:CDel node c4.png
D2A2 ±1/2[D4×D6] D2×D3 File:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 2.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c3.png [2,2,3] = [2]×[3] 24 [3]+ - 1File:CDel node c1.png 1File:CDel node c2.png 3File:CDel node c3.png
D2G2 ±1/2[D4×D12] D2×D6 File:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 2.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 6.pngFile:CDel node c4.png [2,2,6] = [2]×[6] 48 - 1File:CDel node c1.png 1File:CDel node c2.png 3File:CDel node c3.png 3File:CDel node c4.png
D2H2 ±1/2[D4×D10] D2×D5 File:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 2.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 5.pngFile:CDel node c3.png [2,2,5] = [2]×[5] 40 [5]+ - 1File:CDel node c1.png 1File:CDel node c2.png 5File:CDel node c3.png
2B2 ±1/2[D8×D8] D42 File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 4.pngFile:CDel node c4.png [4,2,4] = [4]2 64 [2+,2,2+] 8 2File:CDel node c1.png 2File:CDel node c2.png 2File:CDel node c3.png 2File:CDel node c4.png
B2A2 ±1/2[D8×D6] D4×D3 File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c3.png [4,2,3] = [4]×[3] 48 [2+,2,3+] - 2File:CDel node c1.png 2File:CDel node c2.png 3File:CDel node c3.png
B2G2 ±1/2[D8×D12] D4×D6 File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 6.pngFile:CDel node c4.png [4,2,6] = [4]×[6] 96 - 2File:CDel node c1.png 2File:CDel node c2.png 3File:CDel node c3.png 3File:CDel node c4.png
B2H2 ±1/2[D8×D10] D4×D5 File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 5.pngFile:CDel node c3.png [4,2,5] = [4]×[5] 80 [2+,2,5+] - 2File:CDel node c1.png 2File:CDel node c2.png 5File:CDel node c3.png
2A2 ±1/2[D6×D6] D32 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c3.png [3,2,3] = [3]2 36 [3+,2,3+] 6 3File:CDel node c1.png 3File:CDel node c3.png
A2G2 ±1/2[D6×D12] D3×D6 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 6.pngFile:CDel node c4.png [3,2,6] = [3]×[6] 72 - 3File:CDel node c1.png 3File:CDel node c3.png 3File:CDel node c4.png
2G2 ±1/2[D12×D12] D62 File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 6.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 6.pngFile:CDel node c4.png [6,2,6] = [6]2 144 12 3File:CDel node c1.png 3File:CDel node c2.png 3File:CDel node c3.png 3File:CDel node c4.png
A2H2 ±1/2[D6×D10] D3×D5 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 5.pngFile:CDel node c3.png [3,2,5] = [3]×[5] 60 [3+,2,5+] - 3File:CDel node c1.png 5File:CDel node c3.png
G2H2 ±1/2[D12×D10] D6×D5 File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 6.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 5.pngFile:CDel node c3.png [6,2,5] = [6]×[5] 120 - 3File:CDel node c1.png 3File:CDel node c2.png 5File:CDel node c3.png
2H2 ±1/2[D10×D10] D52 File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 5.pngFile:CDel node c1.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 5.pngFile:CDel node c3.png [5,2,5] = [5]2 100 [5+,2,5+] 10 5File:CDel node c1.png 5File:CDel node c3.png
In general, p,q=2,3,4...
2I2(2p) ±1/2[D4p×D4p] D2p2 File:CDel node.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node c4.png [2p,2,2p] = [2p]2 16p2 [p+,2,p+] 2p pFile:CDel node c1.png pFile:CDel node c2.png pFile:CDel node c3.png pFile:CDel node c4.png
2I2(p) ±1/2[D2p×D2p] Dp2 File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel p.pngFile:CDel node.png File:CDel node c1.pngFile:CDel p.pngFile:CDel node c1.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel p.pngFile:CDel node c3.png [p,2,p] = [p]2 4p2 2p pFile:CDel node c1.png pFile:CDel node c3.png
I2(p)I2(q) ±1/2[D4p×D4q] D2p×D2q File:CDel node.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2x.pngFile:CDel q.pngFile:CDel node.png File:CDel node c1.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node c2.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel 2x.pngFile:CDel q.pngFile:CDel node c4.png [2p,2,2q] = [2p]×[2q] 16pq [p+,2,q+] - pFile:CDel node c1.png pFile:CDel node c2.png qFile:CDel node c3.png qFile:CDel node c4.png
I2(p)I2(q) ±1/2[D2p×D2q] Dp×Dq File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node.png File:CDel node c1.pngFile:CDel p.pngFile:CDel node c1.pngFile:CDel 2.pngFile:CDel node c3.pngFile:CDel q.pngFile:CDel node c3.png [p,2,q] = [p]×[q] 4pq - pFile:CDel node c1.png qFile:CDel node c3.png

The symmetry order is equal to the number of cells of the regular polychoron times the symmetry of its cells. The omnitruncated dual polychora have cells that match the fundamental domains of the symmetry group.

Nets for convex regular 4-polytopes and omnitruncated duals
Symmetry A4 D4 B4 F4 H4
4-polytope 5-cell demitesseract tesseract 24-cell 120-cell
Cells 5 {3,3} 16 {3,3} 8 {4,3} 24 {3,4} 120 {5,3}
Cell symmetry [3,3], order 24 [4,3], order 48 [5,3], order 120
Coxeter diagram File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
4-polytope
net 		File:5-cell net.png File:16-cell nets.png File:8-cell net.png File:24-cell net.png File:120-cell net.png
Omnitruncation omni. 5-cell omni. demitesseract omni. tesseract omni. 24-cell omni. 120-cell
Omnitruncation
dual
net 		File:Dual gippid net.png File:Dual tico net.png File:Dual gidpith net.png File:Dual gippic net.png File:Dual gidpixhi net.png
Coxeter diagram File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel split1.pngFile:CDel nodes f11.png File:CDel node f1.pngFile:CDel 4.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 4.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node f1.pngFile:CDel 5.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png
Cells 5×24 = 120 (16/2)×24 = 192 8×48 = 384 24×48 = 1152 120×120 = 14400

Chiral subgroups

File:Stereographic polytope 16cell colour.png
The 16-cell edges projected onto a 3-sphere represent 6 great circles of B4 symmetry. 3 circles meet at each vertex. Each circle represents axes of 4-fold symmetry.
File:Stereographic polytope 24cell faces.png
The 24-cell edges projected onto a 3-sphere represent the 16 great circles of F4 symmetry. Four circles meet at each vertex. Each circle represents axes of 3-fold symmetry.
File:Stereographic polytope 600cell.png
The 600-cell edges projected onto a 3-sphere represent 72 great circles of H4 symmetry. Six circles meet at each vertex. Each circle represent axes of 5-fold symmetry.

Direct subgroups of the reflective 4-dimensional point groups are:

Coxeter
notation 		Conway
Quaternion 		Structure Order Gyration axes
Polychoric groups
File:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png [3,3,3]+ +1/60[I×I] A5 60 103File:Armed forces red triangle.svg 102File:Rhomb.svg
File:CDel branch h2h2.pngFile:CDel 3ab.pngFile:CDel nodes h2h2.png 3,3,3+ ±1/60[I×I] A5×Z2 120 103File:Armed forces red triangle.svg (10+?)2File:Rhomb.svg
File:CDel nodes h2h2.pngFile:CDel split2.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png [31,1,1]+ ±1/3[T×T] 1/2.2A4 96 163File:Armed forces red triangle.svg 182File:Rhomb.svg
File:CDel node h2.pngFile:CDel 4.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png [4,3,3]+ ±1/6[O×O] 2A4 = A2≀A4 192 64File:Monomino.png 163File:Armed forces red triangle.svg 362File:Rhomb.svg
File:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 4.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png [3,4,3]+ ±1/2[O×O] 3.2A4 576 184File:Monomino.png 163File:Purple Fire.svg 163File:Armed forces red triangle.svg 722File:Rhomb.svg
File:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 2.pngFile:CDel 4.pngFile:CDel 2.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png [3+,4,3+] ±[T×T] 288 163File:Purple Fire.svg 163File:Armed forces red triangle.svg (72+18)2File:Rhomb.svg
File:CDel label4.pngFile:CDel branchgap h2h2.pngFile:CDel 3ab.pngFile:CDel nodes h2h2.png [[3+,4,3+]] ±[O×T] 576 323File:Armed forces red triangle.svg (72+18+?)2File:Rhomb.svg
File:CDel label4.pngFile:CDel branch h2h2.pngFile:CDel 3ab.pngFile:CDel nodes h2h2.png 3,4,3+ ±[O×O] 1152 184File:Monomino.png 323File:Armed forces red triangle.svg (72+?)2File:Rhomb.svg
File:CDel node h2.pngFile:CDel 5.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png [5,3,3]+ ±[I×I] 2.(A5×A5) 7200 725File:Patka piechota.png 2003File:Armed forces red triangle.svg 4502File:Rhomb.svg
Polyhedral prismatic groups
File:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.png [3,3,2]+ +1/24[O×O] A4×Z2 24 43File:Purple Fire.svg 43File:Armed forces red triangle.svg (6+6)2File:Rhomb.svg
File:CDel node h2.pngFile:CDel 4.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.png [4,3,2]+ ±1/24[O×O] S4×Z2 48 64File:Monomino.png 83File:Armed forces red triangle.svg (3+6+12)2File:Rhomb.svg
File:CDel node h2.pngFile:CDel 5.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.png [5,3,2]+ ±1/60[I×I] A5×Z2 120 125File:Patka piechota.png 203File:Armed forces red triangle.svg (15+30)2File:Rhomb.svg
Duoprismatic groups
File:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.png [2,2,2]+ +1/2[D4×D4] 8 12File:Rhomb.svg 12File:Rhomb.svg 42File:Rhomb.svg
File:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png [3,2,3]+ +1/2[D6×D6] 18 13File:Purple Fire.svg 13File:Armed forces red triangle.svg 92File:Rhomb.svg
File:CDel node h2.pngFile:CDel 4.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.pngFile:CDel 4.pngFile:CDel node h2.png [4,2,4]+ +1/2[D8×D8] 32 14File:Blue square.png 14File:Monomino.png 162File:Rhomb.svg
(p,q=2,3,4...), gcd(p,q)=1
File:CDel node h2.pngFile:CDel p.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.pngFile:CDel p.pngFile:CDel node h2.png [p,2,p]+ +1/2[D2p×D2p] 2p2 1pFile:Disc Plain blue.svg 1pFile:Disc Plain cyan.svg (pp)2File:Rhomb.svg
File:CDel node h2.pngFile:CDel p.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.pngFile:CDel q.pngFile:CDel node h2.png [p,2,q]+ +1/2[D2p×D2q] 2pq 1pFile:Disc Plain blue.svg 1qFile:Disc Plain cyan.svg (pq)2File:Rhomb.svg
File:CDel node h2.pngFile:CDel p.pngFile:CDel node h2.pngFile:CDel 2.pngFile:CDel node h2.pngFile:CDel q.pngFile:CDel node h2.png [p+,2,q+] +[Cp×Cq] Zp×Zq pq 1pFile:Disc Plain blue.svg 1qFile:Disc Plain cyan.svg

Pentachoric symmetry

Hexadecachoric symmetry

Further information: Hyperoctahedral group

Icositetrachoric symmetry

Demitesseractic symmetry

Further information: Hyperoctahedral_group § Subgroups

Hexacosichoric symmetry

File:Coxeter 533 order-5 gyration axes.png
[5,3,3]+ 72 order-5 gyrations 		File:Coxeter 533 order-3 gyration axes.png
[5,3,3]+ 200 order-3 gyrations
File:Coxeter 533 order-2 gyration axes.png
[5,3,3]+ 450 order-2 gyrations 		File:Coxeter 533 all gyration axes.png
[5,3,3]+ all gyrations
File:Sphere symmetry group ih.png
[5,3], File:CDel node c2.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c2.png, icosahedral pyramidal group is isomorphic to 3d icosahedral symmetry

Duoprismatic symmetry

Summary of some 4-dimensional point groups

This is a summary of 4-dimensional point groups in Coxeter notation. 227 of them are crystallographic point groups (for particular values of p and q).[14][which?] (nc) is given for non-crystallographic groups. Some crystallographic group[which?] have their orders indexed (order.index) by their abstract group structure.[15]

Finite groups
[ ]: File:CDel node.png
Symbol Order
[1]+ 1.1
[1] = [ ] 2.1
[2]: File:CDel node.pngFile:CDel 2.pngFile:CDel node.png
Symbol Order
[1+,2]+ 1.1
[2]+ 2.1
[2] 4.1
[2,2]: File:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png
Symbol Order
[2+,2+]+
= [(2+,2+,2+)]		 1.1
[2+,2+] 2.1
[2,2]+ 4.1
[2+,2] 4.1
[2,2] 8.1
[2,2,2]: File:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png
Symbol Order
[(2+,2+,2+,2+)]
= [2+,2+,2+]+		 1.1
[2+,2+,2+] 2.1
[2+,2,2+] 4.1
[(2,2)+,2+] 4
[[2+,2+,2+]] 4
[2,2,2]+ 8
[2+,2,2] 8.1
[(2,2)+,2] 8
[[2+,2,2+]] 8.1
[2,2,2] 16.1
[[2,2,2]]+ 16
[[2,2+,2]] 16
[[2,2,2]] 32
[p]: File:CDel node.pngFile:CDel p.pngFile:CDel node.png
Symbol Order
[p]+ p
[p] 2p
[p,2]: File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png
Symbol Order
[p,2]+ 2p
[p,2] 4p
[2p,2+]: File:CDel node.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2x.pngFile:CDel node.png
Symbol Order
[2p,2+] 4p
[2p+,2+] 2p
[p,2,2]: File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png
Symbol Order
[p+,2,2+] 2p
[(p,2)+,2+] 2p
[p,2,2]+ 4p
[p,2,2+] 4p
[p+,2,2] 4p
[(p,2)+,2] 4p
[p,2,2] 8p
[2p,2+,2]: File:CDel node.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png
Symbol Order
[2p+,2+,2+]+ p
[2p+,2+,2+] 2p
[2p+,2+,2] 4p
[2p+,(2,2)+] 4p
[2p,(2,2)+] 8p
[2p,2+,2] 8p
[p,2,q]: File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node.png
Symbol Order
[p+,2,q+] pq
[p,2,q]+ 2pq
[p+,2,q] 2pq
[p,2,q] 4pq
[(p,2)+,2q]: File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2x.pngFile:CDel q.pngFile:CDel node.png
Symbol Order
[(p,2)+,2q+] 2pq
[(p,2)+,2q] 4pq
[2p,2,2q]: File:CDel node.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2x.pngFile:CDel q.pngFile:CDel node.png
Symbol Order
[2p+,2+,2q+]+=
[(2p+,2+,2q+,2+)]		 pq
[2p+,2+,2q+] 2pq
[2p,2+,2q+] 4pq
[((2p,2)+,(2q,2)+)] 4pq
[2p,2+,2q] 8pq
[[p,2,p]]: File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel p.pngFile:CDel node.png
Symbol Order
[[p+,2,p+]] 2p2
[[p,2,p]]+ 4p2
[[p,2,p]+] 4p2
[[p,2,p]] 8p2
[[2p,2,2p]]: File:CDel node.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel node.png
Symbol Order
[[(2p+,2+,2p+,2+)]] 2p2
[[2p+,2+,2p+]] 4p2
[[((2p,2)+,(2p,2)+)]] 8p2
[[2p,2+,2p]] 16p2
[3,3,2]: File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png
Symbol Order
[(3,3)Δ,2,1+]
≅ [2,2]+		 4
[(3,3)Δ,2]
≅ [2,(2,2)+]		 8
[(3,3)⅄,2,1+]
≅ [4,2+]		 8
[(3,3)+,2,1+]
= [3,3]+		 12.5
[(3,3)⅄,2]
≅ [2,4,2+]		 16
[3,3,2,1+]
= [3,3]		 24
[(3,3)+,2] 24.10
[3,3,2]+ 24.10
[3,3,2] 48.36
[4,3,2]: File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png
Symbol Order
[1+,4,3+,2,1+]
= [3,3]+		 12
[3+,4,2+] 24
[(3,4)+,2+] 24
[1+,4,3+,2]
= [(3,3)+,2]		 24.10
[3+,4,2,1+]
= [3+,4]		 24.10
[(4,3)+,2,1+]
= [4,3]+		 24.15
[1+,4,3,2,1+]
= [3,3]		 24
[1+,4,(3,2)+]
= [3,3,2]+		 24
[3,4,2+] 48
[4,3+,2] 48.22
[4,(3,2)+] 48
[(4,3)+,2] 48.36
[1+,4,3,2]
= [3,3,2]		 48.36
[4,3,2,1+]
= [4,3]		 48.36
[4,3,2]+ 48.36
[4,3,2] 96.5
[5,3,2]: File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png
Symbol Order
[(5,3)+,2,1+]
= [5,3]+		 60.13
[5,3,2,1+]
= [5,3]		 120.2
[(5,3)+,2] 120.2
[5,3,2]+ 120.2
[5,3,2] 240 (nc)
[31,1,1]: File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png
Symbol Order
[31,1,1]Δ
≅[[4,2+,4]]+		 32
[31,1,1]⅄ 64
[31,1,1]+ 96.1
[31,1,1] 192.2
<[3,31,1]>
= [4,3,3]		 384.1
[3[31,1,1]]
= [3,4,3]		 1152.1
[3,3,3]: File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Symbol Order
[3,3,3]+ 60.13
[3,3,3] 120.1
[[3,3,3]]+ 120.2
[[3,3,3]+] 120.1
[[3,3,3]] 240.1
[4,3,3]: File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Symbol Order
[1+,4,(3,3)Δ]
= [31,1,1]Δ
≅[[4,2+,4]]+		 32
[4,(3,3)Δ]
= [2+,4[2,2,2]+]
≅[[4,2+,4]]		 64
[1+,4,(3,3)⅄]
= [31,1,1]⅄		 64
[1+,4,(3,3)+]
= [31,1,1]+		 96.1
[4,(3,3)⅄]
≅ [[4,2,4]]		 128
[1+,4,3,3]
= [31,1,1]		 192.2
[4,(3,3)+] 192.1
[4,3,3]+ 192.3
[4,3,3] 384.1
[3,4,3]: File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Symbol Order
[3+,4,3+] 288.1
[3,4,3⅄]
= [4,3,3]		 384.1
[3,4,3]+ 576.2
[3+,4,3] 576.1
[[3+,4,3+]] 576 (nc)
[3,4,3] 1152.1
[[3,4,3]]+ 1152 (nc)
[[3,4,3]+] 1152 (nc)
[[3,4,3]] 2304 (nc)
[5,3,3]: File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Symbol Order
[5,3,3]+ 7200 (nc)
[5,3,3] 14400 (nc)

See also

References

  1. ↑ Hurley, A. C.; Dirac, P. A. M. (1951). "Finite rotation groups and crystal classes in four dimensions". Mathematical Proceedings of the Cambridge Philosophical Society. 47 (4): 650–661. Bibcode:1951PCPS...47..650H. doi:10.1017/S0305004100027109. S2CID 122468489.
  2. ↑ http://met.iisc.ernet.in/~lord/webfiles/Alan/CV25.pdf [bare URL PDF]
  3. ↑ Mozrzymas, Jan; Solecki, Andrzej (1975). "R4 point groups". Reports on Mathematical Physics. 7 (3): 363–394. Bibcode:1975RpMP....7..363M. doi:10.1016/0034-4877(75)90040-3.
  4. ↑ Brown, H; Bülow, R; Neubüser, J; Wondratschek, H; Zassenhaus, H (1978). Crystallographic Groups of Four-Dimensional Space (PDF). Wiley.
  5. ↑ Warner, N. P. (1982). "The Symmetry Groups of the Regular Tessellations of S2 and S3". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 383 (1785): 379–398. Bibcode:1982RSPSA.383..379W. doi:10.1098/rspa.1982.0136. JSTOR 2397289. S2CID 119786906.
  6. ↑ Coxeter, Regular and Semi-Regular Polytopes II,1985, 2.2 Four-dimensional reflection groups, 2.3 Subgroups of small index
  7. ↑ Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
  8. ↑ Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
  9. ↑ Conway and Smith, On Quaternions and Octonions, 2003 Chapter 4, section 4.4 Coxeter's Notations for the Polyhedral Groups
  10. ↑ "Convex and abstract polytopes", Programme and abstracts, MIT, 2005
  11. ↑ Johnson (2015), Chapter 11, Section 11.5 Spherical Coxeter groups
  12. ↑ What Are Polyhedra?, with Greek Numerical Prefixes
  13. ↑ 13.0 13.1 Coxeter, The abstract groups Gm;n;p, (1939)
  14. ↑ Weigel, D.; Phan, T.; Veysseyre, R. (1987). "Crystallography, geometry and physics in higher dimensions. III. Geometrical symbols for the 227 crystallographic point groups in four-dimensional space". Acta Crystallogr. A43 (3): 294. Bibcode:1987AcCrA..43..294W. doi:10.1107/S0108767387099367.
  15. ↑ Coxeter, Regular and Semi-Regular Polytopes II (1985)
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • 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
    • (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]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
  • H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980 p92, p122.
  • John .H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p. 249
  • John H. Conway and Derek A. Smith, On Quaternions and Octonions, 2003, ISBN 978-1-56881-134-5
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)

External links

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