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 1962 A. L. MacKay Bravais Lattices in Four-dimensional Space 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 1978 H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. 1982 N. P. Warner, The symmetry groups of the regular tessellations of S2 and S3 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
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. 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).
For cross-referencing, also given here are quaternion based notations by Patrick du Val (1964) and John Conway (2003). 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 by Norman Johnson and George Olshevsky in the context of uniform polychora (4-polytopes), and their related 4-dimensional symmetry groups.
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. 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.