Point groups in four dimensions

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

Isometries of 4D point symmetry

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

Enumeration of groups

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

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

The 16-cell, with construction , 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. = (4 orthogonal mirrors)
  2. = (12 mirrors)

F4 can be decomposed into 2 orthogonal D4 groups:

  1. = (12 mirrors)
  2. = (12 mirrors)

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

  1. = (3+1 orthogonal mirrors)
  2. = (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].21S5 [3,3,3] 120 [3,3,3]+ 510
D4 ±1/3[T×T].21/2.2S4 [31,1,1] 192 [31,1,1]+612
B4 ±1/6[O×O].22S4 = S2≀S4 [4,3,3] 384 8412
F4 ±1/2[O×O].233.2S4 [3,4,3] 1152[3+,4,3+]121212
H4 ±[I×I].22.(A5×A5).2 [5,3,3] 14400 [5,3,3]+3060
Full polyhedral prismatic groups
A3A1 +1/24[O×O].23S4×D1 [3,3,2] = [3,3]×[ ] 48 [3,3]+ - 61
B3A1 ±1/24[O×O].2S4×D1 [4,3,2] = [4,3]×[ ] 96 - 361
H3A1 ±1/60[I×I].2A5×D1 [5,3,2] = [5,3]×[ ] 240[5,3]+ - 151
Full duoprismatic groups
4A1 = 2D2 ±1/2[D4×D4]D14 = D22 [2,2,2] = [ ]4 = [2]2 16[ ]+4 1111
D2B2 ±1/2[D4×D8]D2×D4 [2,2,4] = [2]×[4] 32[2]+ - 1122
D2A2 ±1/2[D4×D6]D2×D3 [2,2,3] = [2]×[3] 24[3]+ - 113
D2G2 ±1/2[D4×D12]D2×D6 [2,2,6] = [2]×[6] 48 - 1133
D2H2 ±1/2[D4×D10]D2×D5 [2,2,5] = [2]×[5] 40[5]+ - 115
2B2 ±1/2[D8×D8]D42 [4,2,4] = [4]2 64[2+,2,2+]8 2222
B2A2 ±1/2[D8×D6]D4×D3 [4,2,3] = [4]×[3] 48[2+,2,3+] - 223
B2G2 ±1/2[D8×D12]D4×D6 [4,2,6] = [4]×[6] 96 - 2233
B2H2 ±1/2[D8×D10]D4×D5 [4,2,5] = [4]×[5] 80[2+,2,5+] - 225
2A2 ±1/2[D6×D6]D32 [3,2,3] = [3]2 36[3+,2,3+]6 33
A2G2 ±1/2[D6×D12]D3×D6 [3,2,6] = [3]×[6] 72 - 333
2G2 ±1/2[D12×D12]D62 [6,2,6] = [6]2 14412 3333
A2H2 ±1/2[D6×D10]D3×D5 [3,2,5] = [3]×[5] 60[3+,2,5+] - 35
G2H2 ±1/2[D12×D10]D6×D5 [6,2,5] = [6]×[5] 120 - 335
2H2 ±1/2[D10×D10]D52 [5,2,5] = [5]2 100[5+,2,5+]10 55
In general, p,q=2,3,4...
2I2(2p) ±1/2[D4p×D4p]D2p2 [2p,2,2p] = [2p]216p2[p+,2,p+]2p pppp
2I2(p) ±1/2[D2p×D2p]Dp2 [p,2,p] = [p]24p22p pp
I2(p)I2(q) ±1/2[D4p×D4q]D2p×D2q [2p,2,2q] = [2p]×[2q] 16pq[p+,2,q+] - ppqq
I2(p)I2(q) ±1/2[D2p×D2q]Dp×Dq [p,2,q] = [p]×[q] 4pq - pq

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 A4D4B4F4H4
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 =
4-polytope
net
Omnitruncation omni. 5-cell omni. demitesseract omni. tesseract omni. 24-cell omni. 120-cell
Omnitruncation
dual
net
Coxeter diagram
Cells 5×24 = 120 (16/2)×24 = 192 8×48 = 384 24×48 = 1152 120×120 = 14400

Chiral subgroups

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

Coxeter
notation
Conway
Quaterion
Structure Order Gyration axes
Polychoric groups
[3,3,3]+ +1/60[I] A5 60103102
[[3,3,3]]+ ±1/60[I]A5×C2 120103(10+?)2
[31,1,1]+ ±1/3[T×T]1/2.2A496163?2
[4,3,3]+ ±1/6[O×O]2A4 = A2≀A4 19264163?2
[3,4,3]+ ±1/2[O×O]3.2A4576184163163722
[3+,4,3+] ±[T×T] 288163163(72+18)2
[[3+,4,3+]] ±[O×T] 576 323(72+18+?)2
[[3,4,3]]+ ±[O×O]1152184 323(72+?)2
[5,3,3]+ ±[I×I]2.(A5×A5)720072520034502
Polyhedral prismatic groups
[3,3,2]+ +1/24[O]A4×C2 244343(6+6)2
[4,3,2]+ ±1/24[O×O]S4×C2 966483(3+6+12)2
[5,3,2]+ ±1/60[I×I]A5×C2 240125203(15+30)2
Duoprismatic groups
[2,2,2]+ +1/2[D4×D4] 8 121242
[3,2,3]+ +1/2[D6×D6] 18 131392
[4,2,4]+ +1/2[D8×D8] 32 1414162
(p,q=2,3,4...)
[p,2,p]+ +1/2[D2p×D2p]2p2 1p1p(pp)2
[p,2,q]+ +1/2[D2p×D2q]2pq1p1q(pq)2
[p+,2,q+] +[Cp×Cq]Cp×Cqpq1p1q

Pentachoric symmetry

Hexadecachoric symmetry

Icositetrachoric symmetry

Demitesseractic symmetry

Hexacosichoric symmetry

Duoprismatic symmetry

Summary

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

References

  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. http://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. 1 2 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)

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