Point groups in four dimensions

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A hierarchy of 4D polychoric point groups and some subgroups. Vertical positioning is grouped by order. Blue, green, and pink colors show reflectional, hybrid, and rotational groups.
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[edit]

  • 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
  • 2015 N. W. Johnson Geometries and Transformations, Extended Coxeter notation for 4D point groups

Isometries of 4D point symmetry[edit]

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

Enumeration of groups[edit]

For cross-referencing, also given here are quaternion based notations by Patrick du Val (1964)[6] and John Conway (2003).[7] 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[edit]

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[edit]

The 16-cell, with construction CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png, projected onto a 3-sphere show the symmetry of [2,2,2]. The curved edges can be seen as six great circles, each circle represents the intersection pairs of mirrors on the 3-sphere.

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 is advocated[8] by Norman Johnson and George Olshevsky in the context of uniform polychora (4-polytopes), and their related 4-dimensional symmetry groups.[9]

Orthogonal subgroups

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

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

F4 can be decomposed into 2 orthogonal D4 groups:

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

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

  1. CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 2.pngCDel node c4.png = CDel node c1.pngCDel 2.pngCDel nodeab c1.pngCDel 2.pngCDel node c4.png (3+1 orthogonal mirrors)
  2. CDel node h0.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node h0.png = CDel nodeab c2.pngCDel split2.pngCDel 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.[10] Extended symmetries exist in uniform polychora with symmetric ring-patterns within the Coxeter diagram construct. Chiral symmetries exist in alternated uniform polychora. The groups are named in this article in Coxeter's Bracket notation (1985).[11] 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).[12]

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 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png [3,3,3] 120 [3,3,3]+ 5 10CDel node c1.png
D4 ±1/3[T×T].2 1/2.2S4 CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png CDel nodeab c1.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c1.png [31,1,1] 192 [31,1,1]+ 6 12CDel node c1.png
B4 ±1/6[O×O].2 2S4 = S2≀S4 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png [4,3,3] 384 8 4CDel node c2.png 12CDel node c1.png
F4 ±1/2[O×O].23 3.2S4 CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node c2.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png [3,4,3] 1152 [3+,4,3+] 12 12CDel node c2.png 12CDel node c1.png
H4 ±[I×I].2 2.(A5×A5).2 CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png [5,3,3] 14400 [5,3,3]+ 30 60CDel node c1.png
Full polyhedral prismatic groups
A3A1 +1/24[O×O].23 S4×D1 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png [3,3,2] = [3,3]×[ ] 48 [3,3]+ - 6CDel node c1.png 1CDel node c3.png
B3A1 ±1/24[O×O].2 S4×D1 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png [4,3,2] = [4,3]×[ ] 96 - 3CDel node c2.png 6CDel node c1.png 1CDel node c3.png
H3A1 ±1/60[I×I].2 A5×D1 CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png [5,3,2] = [5,3]×[ ] 240 [5,3]+ - 15CDel node c1.png 1CDel node c3.png
Full duoprismatic groups
4A1 = 2D2 ±1/2[D4×D4] D14 = D22 CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png [2,2,2] = [ ]4 = [2]2 16 [ ]+ 4 1CDel node c1.png 1CDel node c2.png 1CDel node c3.png 1CDel node c4.png
D2B2 ±1/2[D4×D8] D2×D4 CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 4.pngCDel node c4.png [2,2,4] = [2]×[4] 32 [2]+ - 1CDel node c1.png 1CDel node c2.png 2CDel node c3.png 2CDel node c4.png
D2A2 ±1/2[D4×D6] D2×D3 CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 3.pngCDel node c3.png [2,2,3] = [2]×[3] 24 [3]+ - 1CDel node c1.png 1CDel node c2.png 3CDel node c3.png
D2G2 ±1/2[D4×D12] D2×D6 CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png [2,2,6] = [2]×[6] 48 - 1CDel node c1.png 1CDel node c2.png 3CDel node c3.png 3CDel node c4.png
D2H2 ±1/2[D4×D10] D2×D5 CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c3.png [2,2,5] = [2]×[5] 40 [5]+ - 1CDel node c1.png 1CDel node c2.png 5CDel node c3.png
2B2 ±1/2[D8×D8] D42 CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 4.pngCDel node c4.png [4,2,4] = [4]2 64 [2+,2,2+] 8 2CDel node c1.png 2CDel node c2.png 2CDel node c3.png 2CDel node c4.png
B2A2 ±1/2[D8×D6] D4×D3 CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 3.pngCDel node c3.png [4,2,3] = [4]×[3] 48 [2+,2,3+] - 2CDel node c1.png 2CDel node c2.png 3CDel node c3.png
B2G2 ±1/2[D8×D12] D4×D6 CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png [4,2,6] = [4]×[6] 96 - 2CDel node c1.png 2CDel node c2.png 3CDel node c3.png 3CDel node c4.png
B2H2 ±1/2[D8×D10] D4×D5 CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c3.png [4,2,5] = [4]×[5] 80 [2+,2,5+] - 2CDel node c1.png 2CDel node c2.png 5CDel node c3.png
2A2 ±1/2[D6×D6] D32 CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel 3.pngCDel node c3.png [3,2,3] = [3]2 36 [3+,2,3+] 6 3CDel node c1.png 3CDel node c3.png
A2G2 ±1/2[D6×D12] D3×D6 CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png [3,2,6] = [3]×[6] 72 - 3CDel node c1.png 3CDel node c3.png 3CDel node c4.png
2G2 ±1/2[D12×D12] D62 CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png [6,2,6] = [6]2 144 12 3CDel node c1.png 3CDel node c2.png 3CDel node c3.png 3CDel node c4.png
A2H2 ±1/2[D6×D10] D3×D5 CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c3.png [3,2,5] = [3]×[5] 60 [3+,2,5+] - 3CDel node c1.png 5CDel node c3.png
G2H2 ±1/2[D12×D10] D6×D5 CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c3.png [6,2,5] = [6]×[5] 120 - 3CDel node c1.png 3CDel node c2.png 5CDel node c3.png
2H2 ±1/2[D10×D10] D52 CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png CDel node c1.pngCDel 5.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c3.png [5,2,5] = [5]2 100 [5+,2,5+] 10 5CDel node c1.png 5CDel node c3.png
In general, p,q=2,3,4...
2I2(2p) ±1/2[D4p×D4p] D2p2 CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel p.pngCDel node.png CDel node c1.pngCDel 2x.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2x.pngCDel p.pngCDel node c4.png [2p,2,2p] = [2p]2 16p2 [p+,2,p+] 2p pCDel node c1.png pCDel node c2.png pCDel node c3.png pCDel node c4.png
2I2(p) ±1/2[D2p×D2p] Dp2 CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel p.pngCDel node c3.png [p,2,p] = [p]2 4p2 2p pCDel node c1.png pCDel node c3.png
I2(p)I2(q) ±1/2[D4p×D4q] D2p×D2q CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node.png CDel node c1.pngCDel 2x.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2x.pngCDel q.pngCDel node c4.png [2p,2,2q] = [2p]×[2q] 16pq [p+,2,q+] - pCDel node c1.png pCDel node c2.png qCDel node c3.png qCDel node c4.png
I2(p)I2(q) ±1/2[D2p×D2q] Dp×Dq CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel q.pngCDel node c3.png [p,2,q] = [p]×[q] 4pq - pCDel node c1.png qCDel 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 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4-polytope
net
5-cell net.png 16-cell nets.png 8-cell net.png 24-cell net.png 120-cell net.png
Omnitruncation omni. 5-cell omni. demitesseract omni. tesseract omni. 24-cell omni. 120-cell
Omnitruncation
dual
net
Dual gippid net.png Dual tico net.png Dual gidpith net.png Dual gippic net.png Dual gidpixhi net.png
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel split1.pngCDel nodes f11.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Cells 5×24 = 120 (16/2)×24 = 192 8×48 = 384 24×48 = 1152 120×120 = 14400

Chiral subgroups[edit]

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

Coxeter
notation
Conway
Quaterion
Structure Order Gyration axes
Polychoric groups
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png [3,3,3]+ +1/60[I×I] A5 60 103Armed forces red triangle.svg 102Rhomb.svg
CDel branch h2h2.pngCDel 3ab.pngCDel nodes h2h2.png [[3,3,3]]+ ±1/60[I×I] A5×C2 120 103Armed forces red triangle.svg (10+?)2Rhomb.svg
CDel nodes h2h2.pngCDel split2.pngCDel node h2.pngCDel 3.pngCDel node h2.png [31,1,1]+ ±1/3[T×T] 1/2.2A4 96 163Armed forces red triangle.svg ?2Rhomb.svg
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png [4,3,3]+ ±1/6[O×O] 2A4 = A2≀A4 192 64Monomino.png 163Armed forces red triangle.svg ?2Rhomb.svg
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png [3,4,3]+ ±1/2[O×O] 3.2A4 576 184Monomino.png 163Armed forces red triangle.svg 163Purple Fire.svg 722Rhomb.svg
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel 4.pngCDel 2.pngCDel node h2.pngCDel 3.pngCDel node h2.png [3+,4,3+] ±[T×T] 288 163Armed forces red triangle.svg 163Purple Fire.svg (72+18)2Rhomb.svg
CDel label4.pngCDel branchgap h2h2.pngCDel 3ab.pngCDel nodes h2h2.png [[3+,4,3+]] ±[O×T] 576 323Purple Fire.svg (72+18+?)2Rhomb.svg
CDel label4.pngCDel branch h2h2.pngCDel 3ab.pngCDel nodes h2h2.png [[3,4,3]]+ ±[O×O] 1152 184Monomino.png 323Purple Fire.svg (72+?)2Rhomb.svg
CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png [5,3,3]+ ±[I×I] 2.(A5×A5) 7200 725Patka piechota.png 2003Armed forces red triangle.svg 4502Rhomb.svg
Polyhedral prismatic groups
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png [3,3,2]+ +1/24[O×O] A4×C2 24 43Purple Fire.svg 43Armed forces red triangle.svg (6+6)2Rhomb.svg
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png [4,3,2]+ ±1/24[O×O] S4×C2 96 64Monomino.png 83Armed forces red triangle.svg (3+6+12)2Rhomb.svg
CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png [5,3,2]+ ±1/60[I×I] A5×C2 240 125Patka piechota.png 203Armed forces red triangle.svg (15+30)2Rhomb.svg
Duoprismatic groups
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png [2,2,2]+ +1/2[D4×D4] 8 12Rhomb.svg 12Rhomb.svg 42Rhomb.svg
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 3.pngCDel node h2.png [3,2,3]+ +1/2[D6×D6] 18 13Purple Fire.svg 13Armed forces red triangle.svg 92Rhomb.svg
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 4.pngCDel node h2.png [4,2,4]+ +1/2[D8×D8] 32 14Blue square.png 14Monomino.png 162Rhomb.svg
(p,q=2,3,4...)
CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel p.pngCDel node h2.png [p,2,p]+ +1/2[D2p×D2p] 2p2 1pDisc Plain blue.svg 1pDisc Plain cyan.svg (pp)2Rhomb.svg
CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel q.pngCDel node h2.png [p,2,q]+ +1/2[D2p×D2q] 2pq 1pDisc Plain blue.svg 1qDisc Plain cyan.svg (pq)2Rhomb.svg
CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2.pngCDel node h2.pngCDel q.pngCDel node h2.png [p+,2,q+] +[Cp×Cq] Cp×Cq pq 1pDisc Plain blue.svg 1qDisc Plain cyan.svg

Pentachoric symmetry[edit]

  • Pentachoric groupA4, [3,3,3], (CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png), order 120, (Du Val #51' (I/C1;I/C1)†*, Conway +1/60[I×I].21), named for the 5-cell (pentachoron), given by ringed Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png. It is also sometimes called the hyper-tetrahedral group for extending the tetrahedral group [3,3]. There are 10 mirror hyperplanes in this group. It is isomorphic to the abstract symmetric group, S5.
    • The extended pentachoric group, Aut(A4), [[3,3,3]], (The doubling can be hinted by a folded diagram, CDel branch.pngCDel 3ab.pngCDel nodes.png), order 240, (Du Val #51 (I†*/C2;I/C2)†*, Conway ±1/60[I×I].2). It is isomorphic to the direct product of abstract groups: S5×C2.
      • The chiral extended pentachoric group is [[3,3,3]]+, (CDel branch h2h2.pngCDel 3ab.pngCDel nodes h2h2.png), order 120, (Du Val #32 (I/C2;I/C2), Conway ±1/60[IxI]). This group represents the construction of the omnisnub 5-cell, CDel branch hh.pngCDel 3ab.pngCDel nodes hh.png, although it can not be made uniform. It is isomorphic to the direct product of abstract groups: A5×C2.
    • The chiral pentachoric group is [3,3,3]+, (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 60, (Du Val #32' (I/C1;I/C1), Conway +1/60[I×I]). It is isomorphic to the abstract alternating group, A5.
      • The extended chiral pentachoric group is [[3,3,3]+], order 120, (Du Val #51" (I/C1;I/C1)†*, Conway +1/60[IxI].23). Coxeter relates this group to the abstract group (4,6|2,3).[13] It is also isomorphic to the abstract symmetric group, S5.

Hexadecachoric symmetry[edit]

  • Hexadecachoric groupB4, [4,3,3], (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png), order 384, (Du Val #47 (O/V;O/V)*, Conway ±1/6[O×O].2), named for the 16-cell (hexadecachoron), CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png. There are 16 mirror hyperplanes in this group, which can be identified in 2 orthogonal sets: 12 from a [31,1,1] subgroup, and 4 from a [2,2,2] subgroup. It is also called a hyper-octahedral group for extending the 3D octahedral group [4,3], and the tesseractic group for the tesseract, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png.
    • The chiral hexadecachoric group is [4,3,3]+, (CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 192, (Du Val #27 (O/V;O/V), Conway ±1/6[O×O]). This group represents the construction of an omnisnub tesseract, CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png, although it can not be made uniform.
    • The ionic diminished hexadecachoric group is [4,(3,3)+], (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 192, (Du Val #41 (T/V;T/V)*, Conway ±1/3[T×T].2). This group leads to the omnisnub 24-cell with construction CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png.
    • The half hexadecachoric group is [1+,4,3,3], (CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png), order 192, and same as the #demitesseractic symmetry: [31,1,1]. This group is expressed in the tesseract alternated construction of the 16-cell, CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png.
      • The group [1+,4,(3,3)+], (CDel node h0.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel nodes h2h2.pngCDel split2.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 96, and same as the chiral demitesseractic group [31,1,1]+ and also is the commutator subgroup of [4,3,3].
    • A high-index reflective subgroup is the prismatic octahedral symmetry, [4,3,2] (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png), order 96, subgroup index 4, (Du Val #44 (O/C2;O/C2)*, Conway ±1/24[O×O].2). The truncated cubic prism has this symmetry with Coxeter diagram CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png and the cubic prism is a lower symmetry construction of the tesseract, as CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png.
      • Its chiral subgroup is [4,3,2]+, (CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), order 48, (Du Val #26 (O/C2;O/C2), Conway ±1/24[O×O]). An example is the snub cubic antiprism, CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png, although it can not be made uniform.
      • The ionic subgroups are:
        • [(3,4)+,2], (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node h2.pngCDel 2.pngCDel node.png), order 48, (Du Val #44b' (O/C1;O/C1)*, Conway +1/24[O×O].21). The snub cubic prism has this symmetry with Coxeter diagram CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png.
          • [(3,4)+,2+], (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node h4.pngCDel 3.pngCDel node h2.png), order 24, (Du Val #44' (T/C2;T/C2)*, Conway +1/12[T×T].21).
        • [4,3+,2], (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node.png), order 48, (Du Val #39 (T/C2;T/C2)c*, Conway ±1/12[T×T].2).
          • [4,3+,2,1+] = [4,3+,1] = [4,3+], (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node h0.png = CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 24, (Du Val #44" (T/C2;T/C2)*, Conway +1/12[T×T].23). This is the 3D pyritohedral group, [4,3+].
          • [3+,4,2+], (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel 2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), order 24, (Du Val #21 (T/C2;T/C2), Conway ±1/12[T×T]).
        • [3,4,2+], (CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h2.pngCDel 2.pngCDel node h2.png), order 48, (Du Val #39' (T/C2;T/C2)*, Conway ±1/12[T×T].2).
        • [4,(3,2)+], (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), order 48, (Du Val #40b' (O/C1;O/C1)*, Conway +1/24[O×O].21).
      • A half subgroup [4,3,2,1+] = [4,3,1] = [4,3], (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node h0.png = CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png), order 48 (Du Val #44b" (O/C1;O/C1)c*, Conway +1/24[O×O].23). It is called the octahedral pyramidal group and is 3D octahedral symmetry, [4,3]. A cubic pyramid can have this symmetry, with Schläfli symbol: { } ∨ {4,3}.
        [4,3], CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png, octahedral pyramidal group is isomorphic to 3d octahedral symmetry
        • A chiral half subgroup [(4,3)+,2,1+] = [4,3,1]+ = [4,3]+, (CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node h0.png = CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 24 (Du Val #26b' (O/C1;O/C1), Conway +1/24[O×O]). This is the 3D chiral octahedral group, [4,3]+. A snub cubic pyramid can have this symmetry, with Schläfli symbol: { } ∨ sr{4,3}.
    • Another high-index reflective subgroup is the prismatic tetrahedral symmetry, [3,3,2], (CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png), order 48, subgroup index 8, (Du Val #40b" (O/C1;O/C1)*, Conway +1/24[O×O].23).
      • The chiral subgroup is [3,3,2]+, (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), order 24, (Du Val #26b" (O/C1;O/C1), Conway +1/24[O×O]). An example is the snub tetrahedral antiprism, CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png, although it can not be made uniform.
      • The ionic subgroup is [(3,3)+,2], (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node.png), order 24, (Du Val #39b' (T/C1;T/C1)c*, Conway +1/12[T×T].23). An example is the snub tetrahedral prism, CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png.
      • The half subgroup is [3,3,2,1+] = [3,3,1] = [3,3], (CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node h0.png = CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png), order 24, (Du Val #39b" (T/C1;T/C1)*, Conway +1/12[T×T].21). It is called the tetrahedral pyramidal group and is the 3D tetrahedral group, [3,3]. A regular tetrahedral pyramid can have this symmetry, with Schläfli symbol: { } ∨ {3,3}.
        [3,3], CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png, tetrahedral pyramidal group is isomorphic to 3d tetrahedral symmetry
        • The chiral half subgroup [(3,3)+,2,1+] = [3,3]+(CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node h0.png = CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 12, (Du Val #21b' (T/C1;T/C1), Conway +1/12[T×T]). This is the 3D chiral tetrahedral group, [3,3]+. A snub tetrahedral pyramid can have this symmetry, with Schläfli symbol: { } ∨ sr{3,3}.
    • Another high-index radial reflective subgroup is [4,(3,3)*], index 24, removes mirrors with order-3 dihedral angles, creating [2,2,2] (CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png), order 16. Others are [4,2,4] (CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png), [4,2,2] (CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png), with subgroup indices 6 and 12, order 64 and 32. These groups are lower symmetries of the tesseract: (CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png), (CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png), and (CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png). These groups are #duoprismatic symmetry.

Icositetrachoric symmetry[edit]

  • Icositetrachoric groupF4, [3,4,3], (CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png), order 1152, (Du Val #45 (O/T;O/T)*, Conway [O×O].23), named for the 24-cell (icositetrachoron), CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png. There are 24 mirror planes in this symmetry, which can be decomposed into two orthogonal sets of 12 mirrors in demitesseractic symmetry [31,1,1] subgroups, as [3*,4,3] and [3,4,3*], as index 6 subgroups.
    • The extended icositetrachoric group, Aut(F4), [[3,4,3]], (CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.png) has order 2304, (Du Val #48 (O/O;O/O)*, Conway ±[O×O].2).
      • The chiral extended icositetrachoric group, [[3,4,3]]+, (CDel label4.pngCDel branch h2h2.pngCDel 3ab.pngCDel nodes h2h2.png) has order 1152, (Du Val #25 (O/O;O/O), Conway ±[OxO]). This group represents the construction of the omnisnub 24-cell, CDel label4.pngCDel branch hh.pngCDel 3ab.pngCDel nodes hh.png, although it can not be made uniform.
    • The ionic diminished icositetrachoric groups, [3+,4,3] and [3,4,3+], (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png or CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png), have order 576, (Du Val #43 (T/T;T/T)*, Conway ±[T×T].2). This group leads to the snub 24-cell with construction CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png or CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png.
      • The double diminished icositetrachoric group, [3+,4,3+] (the double diminishing can be shown by a gap in the diagram 4-branch: CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel 4.pngCDel 2.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 288, (Du Val #20 (T/T;T/T), Conway ±[T×T]) is the commutator subgroup of [3,4,3].
        • It can be extended as [[3+,4,3+]], (CDel label4.pngCDel branchgap h2h2.pngCDel 3ab.pngCDel nodes h2h2.png) order 576, (Du Val #23 (T/T;O/O), Conway ±[OxT]).
    • The chiral icositetrachoric group is [3,4,3]+, (CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 576, (Du Val #28 (O/T;O/T), Conway ±1/2[O×O]).
      • The extended chiral icositetrachoric group, [[3,4,3]+] has order 1152, (Du Val #46 (O/T;O/T)*, Conway ±1/2[OxO].2). Coxeter relates this group to the abstract group (4,8|2,3).[13]

Demitesseractic symmetry[edit]

  • Demitesseractic groupD4, [31,1,1], [31,1,3] or [1+,4,3,3], (CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png), order 192, (Du Val #42 (T/V;T/V)*, Conway ±1/3[T×T].2), named for the (demitesseract) 4-demicube construction of the 16-cell, CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png or CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png. There are 12 mirrors in this symmetry group.
    • There are two types of extended symmetries by adding mirrors: <[3,31,1]> which becomes [4,3,3] by bisecting the fundamental domain by a mirror, with 3 orientations possible; and the full extended group [3[31,1,1]] becomes [3,4,3].
    • The chiral demitesseractic group is [31,1,1]+ or [1+,4,(3,3)+], (CDel nodes h2h2.pngCDel split2.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel node h0.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 96, (Du Val #22 (T/V;T/V), Conway ±1/3[T×T]). This group leads to the snub 24-cell with construction CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 3.pngCDel node h.png = CDel node h0.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png.

Hexacosichoric symmetry[edit]

  • Hexacosichoric groupH4, [5,3,3], (CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png), order 14400, (Du Val #50 (I/I;I/I)*, Conway ±[I×I].2), named for the 600-cell (hexacosichoron), CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png. It is also sometimes called the hyper-icosahedral group for extending the 3D icosahedral group [5,3], and hecatonicosachoric group or dodecacontachoric group from the 120-cell, CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png.
    • The chiral hexacosichoric group is [5,3,3]+, (CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 7200, (Du Val #30 (I/I;I/I), Conway ±[I×I]). This group represents the construction of the snub 120-cell, CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png, although it can not be made uniform.
    • A high-index reflective subgroup is the prismatic icosahedral symmetry, [5,3,2], (CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png), order 240, subgroup index 60, (Du Val #49 (I/C2;I/C2)*, Conway ±1/60[IxI].2).
      • Its chiral subgroup is [5,3,2]+, (CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), order 120, (Du Val #31 (I/C2;I/C2), Conway ±1/60[IxI]). This group represents the construction of the snub dodecahedral antiprism, CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png, although it can't be made uniform.
      • An ionic subgroup is [(5,3)+,2], (CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node.png), order 120, (Du Val #49' (I/C1;I/C1)*, Conway +1/60[IxI].21). This group represents the construction of the snub dodecahedral prism, CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png.
      • A half subgroup is [5,3,2,1+] = [5,3,1] = [5,3], (CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node h0.png = CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png), order 120, (Du Val #49" (I/C1;I/C1)*, Conway +1/60[IxI].23). It is called the icosahedral pyramidal group and is the 3D icosahedral group, [5,3]. A regular dodecahedral pyramid can have this symmetry, with Schläfli symbol: { } ∨ {5,3}.
        [5,3], CDel node c2.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c2.png, icosahedral pyramidal group is isomorphic to 3d icosahedral symmetry
        • A chiral half subgroup is [(5,3)+,2,1+] = [5,3,1]+ = [5,3]+, (CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node h0.png = CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.png), order 60, (Du Val #31' (I/C1;I/C1), Conway +1/60[IxI]). This is the 3D chiral icosahedral group, [5,3]+. A snub dodecahedral pyramid can have this symmetry, with Schläfli symbol: { } ∨ sr{5,3}.

Duoprismatic symmetry[edit]

  • Duoprismatic groups – [p,2,q], (CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png), order 4pq, exist for all 2 ≤ p,q < ∞. There are p+q mirrors in this symmetry, which are trivially decomposed into two orthogonal sets of p and q mirrors of dihedral symmetry: [p] and [q].
    • The chiral subgroup is [p,2,p]+,(CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel q.pngCDel node h2.png), order 2pq. It can be doubled as [[2p,2,2p]+].
    • If p and q are equal, [p,2,p], (CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png), the symmetry can be doubled as [[p,2,p]], (CDel labelp.pngCDel branch.pngCDel 2.pngCDel branch.pngCDel labelp.png).
      • Doublings: [[p+,2,p+]], (CDel labelp.pngCDel branch h2h2.pngCDel 2.pngCDel branch h2h2.pngCDel labelp.png), [[2p,2+,2p]], [[2p+,2+,2p+]].
    • [p,2,∞], (CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png), it represents a line groups in 3-space,
    • [∞,2,∞], (CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png) it represents the Euclidean plane symmetry with two sets of parallel mirrors and a rectangular domain (orbifold *2222).
    • Subgroups include: [p+,2,q], (CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png), [p,2,q+], (CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node h2.pngCDel q.pngCDel node h2.png), [p+,2,q+], (CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2.pngCDel node h2.pngCDel q.pngCDel node h2.png).
    • And for even values: [2p,2+,2q], (CDel node.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel q.pngCDel node.png), [2p,2+,2q+], (CDel node.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h2.png), [(p,2)+,2q], (CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel q.pngCDel node.png), [2p,(2,q)+], (CDel node.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel q.pngCDel node h2.png), [(p,2)+,2q+], (CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h2.png), [2p+,(2,q)+], (CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel q.pngCDel node h2.png), [2p+,2+,2q+], (CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h2.png).
  • Digonal duoprismatic group – [2,2,2], (CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png), order 16.
    • The chiral subgroup is [2,2,2]+, (CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), order 8.
    • Extended [[2,2,2]], (CDel nodes.pngCDel 2.pngCDel nodes.png), order 32. The 4-4 duoprism has this extended symmetry, CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png.
      • The chiral extended group is [[2,2,2]]+, order 16.
      • Extended chiral subgroup is [[2,2,2]+], order 16, with rotoreflection generators. It is isomorphic to the abstract group (4,4|2,2).
    • Other extended [(3,3)[2,2,2]]=[4,3,3], order 384, #Hexadecachoric symmetry. The tesseract has this symmetry, as CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png or CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png.
    • Ionic diminished subgroups is [2+,2,2], order 8.
      • The double diminished subgroup is [2+,2,2+], order 4.
        • Extended as [[2+,2,2+]], order 8.
      • The rotoreflection subgroups are [2+,2+,2], [2,2+,2+], [2+,(2,2)+], [(2,2)+,2+] order 4.
      • The triple diminished subgroup is [2+,2+,2+], (CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png), order 2. It is a 2-fold double rotation and a 4D central inversion.
    • Half subgroup is [1+,2,2,2]=[1,2,2], order 8.
  • Triangular duoprismatic group – [3,2,3], CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png, order 36.
    • The chiral subgroup is [3,2,3]+, order 18.
    • Extended [[3,2,3]], order 72. The 3-3 duoprism has this extended symmetry, CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png.
      • The chiral extended group is [[3,2,3]]+, order 36.
      • Extended chiral subgroup is [[3,2,3]+], order 36, with rotoreflection generators. It is isomorphic to the abstract group (4,4|2,3).
    • Other extended [[3],2,3], [3,2,[3]], order 72, and are isomorphic to [6,2,3] and [3,2,6].
    • And [[3],2,[3]], order 144, and is isomorphic to [6,2,6].
    • And [[[3],2,[3]]], order 288, isomorphic to [[6,2,6]]. The 6–6 duoprism has this symmetry, as CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png or CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png.
    • Ionic diminished subgroups are [3+,2,3], [3,2,3+], order 18.
      • The double diminished subgroup is [3+,2,3+], order 9.
        • Extended as [[3+,2,3+]], order 18.
    • A high index subgroup is [3,2] , order 12, index 3, which is isomorphic to the dihedral symmetry in three dimensions group, [3,2], D3h.
      • [3,2]+, order 6
  • Square duoprismatic group – [4,2,4], CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png, order 64.
    • The chiral subgroup is [4,2,4]+, order 32.
    • Extended [[4,2,4]], order 128. The 4–4 duoprism has this extended symmetry, CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png.
      • The chiral extended group is [[4,2,4]]+, order 64.
      • Extended chiral subgroup is [[4,2,4]+], order 64, with rotoreflection generators. It is isomorphic to the abstract group (4,4|2,4).
    • Other extended [[4],2,4], [4,2,[4]], order 128, and are isomorphic to [8,2,4] and [4,2,8]. The 4–8 duoprism has this symmetry, as CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png.
    • And [[4],2,[4]], order 256, and is isomorphic to [8,2,8].
    • And [[[4],2,[4]]], order 288, isomorphic to [[8,2,8]]. The 8–8 duoprism has this symmetry, as CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png or CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel node.png.
    • Ionic diminished subgroups are [4+,2,4], [4,2,4+], order 32.
      • The double diminished subgroup is [4+,2,4+], order 16.
        • Extended as [[4+,2,4+]], order 32.
      • The rotoreflection subgroups are [4+,2+,4], [4,2+,4+], [4+,(2,4)+], [(4,2)+,4+] order 16.
      • The triple diminished subgroup is [4+,2+,4+], order 8.
    • Half subgroups are [1+,4,2,4]=[2,2,4], [4,2,4,1+]=[4,2,2], order 16.
      • [1+,4,2,4]+=[2,2,4]+, [4,2,4,1+]+=[4,2,2]+, order 8.
    • Half again subgroup is [1+,4,2,4,1+]=[2,2,2], order 8.
      • [1+,4,2,4,1+]+ = [1+,4,2+,4,1+] = [2,2,2]+, order 4

Summary[edit]

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] (nc) is given for non-crystallographic groups. Some crystallographic group have their orders indexed by their abstract group structure.[15]

See also[edit]

References[edit]

  1. ^ http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2039540
  2. ^ http://met.iisc.ernet.in/~lord/webfiles/Alan/CV25.pdf
  3. ^ "R4 point groups". Reports on Mathematical Physics. 7: 363–394. 1975. Bibcode:1975RpMP....7..363M. doi:10.1016/0034-4877(75)90040-3. 
  4. ^ http://journals.iucr.org/a/issues/2002/03/00/au0290/au0290.pdf
  5. ^ https://www.jstor.org/discover/10.2307/2397289?uid=3739736
  6. ^ Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
  7. ^ Conway and Smith, On Quaternions and Octonions, 2003 Chapter 4, section 4.4 Coxeter's Notations for the Polyhedral Groups
  8. ^ "Convex and abstract polytopes", Programme and abstracts, MIT, 2005
  9. ^ Johnson (2015), Chapter 11, Section 11.5 Spherical Coxeter groups
  10. ^ What Are Polyhedra?, with Greek Numerical Prefixes
  11. ^ Coxeter, Regular and Semi-Regular Polytopes II,1985, 2.2 Four-dimensional reflection groups, 2.3 Subgroups of small index
  12. ^ Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
  13. ^ a b 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 Cryst. A43: 294. 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, (2015)
  • 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-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)

External links[edit]