Point groups in four dimensions

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,, 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

Notation for groups

Involution groups

Rank 4 Coxeter groups

Chiral subgroups

Pentachoric symmetry

• Pentachoric group – A₄, ,, order 120,, named for the 5-cell, given by ringed Coxeter diagram. It is also sometimes called the hyper-tetrahedral group for extending the tetrahedral group . There are 10 mirror hyperplanes in this group. It is isomorphic to the abstract symmetric group, S₅.
• *The extended pentachoric group, Aut,,, order 240,. It is isomorphic to the direct product of abstract groups: S₅×C₂.
• ** The chiral extended pentachoric group is ⁺,, order 120,. This group represents the construction of the omnisnub 5-cell,, although it can not be made uniform. It is isomorphic to the direct product of abstract groups: A₅×C₂.
• *The chiral pentachoric group is ⁺,, order 60,. It is isomorphic to the abstract alternating group, A₅.
• ** The extended chiral pentachoric group is 3,3,3]⁺], order 120,. Coxeter relates this group to the abstract group. It is also isomorphic to the abstract symmetric group, S₅.

  Hexadecachoric symmetry

• Hexadecachoric group – B₄, ,, order 384,, named for the 16-cell,. There are 16 mirror hyperplanes in this group, which can be identified in 2 orthogonal sets: 12 from a subgroup, and 4 from a subgroup. It is also called a hyper-octahedral group for extending the 3D octahedral group , and the tesseractic group for the tesseract,.
• * The chiral hexadecachoric group is ⁺,, order 192,. This group represents the construction of an omnisnub tesseract,, although it can not be made uniform.
• * The ionic diminished hexadecachoric group is ,, order 192,. This group leads to the snub 24-cell with construction.
• * The half hexadecachoric group is ,, order 192, and same as the #demitesseractic symmetry: . This group is expressed in the tesseract alternated construction of the 16-cell, =.
• ** The group ,, order 96, and same as the chiral demitesseractic group ⁺ and also is the commutator subgroup of .
• * A high-index reflective subgroup is the prismatic octahedral symmetry, , order 96, subgroup index 4,. The truncated cubic prism has this symmetry with Coxeter diagram and the cubic prism is a lower symmetry construction of the tesseract, as.
• ** Its chiral subgroup is ⁺,, order 48,. An example is the snub cubic antiprism,, although it can not be made uniform.
• ** The ionic subgroups are:
• *** ,, order 48,. The snub cubic prism has this symmetry with Coxeter diagram.
• **** ,, order 24,.
• *** ,, order 48,.
• **** = = ,, order 24,. This is the 3D pyritohedral group, .
• **** ,, order 24,.
• *** ,, order 48,.
• *** ,, order 48,.
• ** A half subgroup = = ,, order 48. It is called the octahedral pyramidal group and is 3D octahedral symmetry, . A cubic pyramid can have this symmetry, with Schläfli symbol: ∨.File:Sphere symmetry group oh.png|thumb|,, octahedral pyramidal group is isomorphic to 3d octahedral symmetry
• *** A chiral half subgroup = ⁺ = ⁺,, order 24. This is the 3D chiral octahedral group, ⁺. A snub cubic pyramid can have this symmetry, with Schläfli symbol: ∨ sr.
• * Another high-index reflective subgroup is the prismatic tetrahedral symmetry, ,, order 48, subgroup index 8,.
• ** The chiral subgroup is ⁺,, order 24,. An example is the snub tetrahedral antiprism,, although it can not be made uniform.
• ** The ionic subgroup is ,, order 24,. An example is the snub tetrahedral prism,.
• ** The half subgroup is = = ,, order 24,. It is called the tetrahedral pyramidal group and is the 3D tetrahedral group, . A regular tetrahedral pyramid can have this symmetry, with Schläfli symbol: ∨.File:Sphere symmetry group td.png|thumb|,, tetrahedral pyramidal group is isomorphic to 3d tetrahedral symmetry
• *** The chiral half subgroup = ⁺, order 12,. This is the 3D chiral tetrahedral group, ⁺. A snub tetrahedral pyramid can have this symmetry, with Schläfli symbol: ∨ sr.
• * Another high-index radial reflective subgroup is , index 24, removes mirrors with order-3 dihedral angles, creating , order 16. Others are , , with subgroup indices 6 and 12, order 64 and 32. These groups are lower symmetries of the tesseract:,, and. These groups are #duoprismatic symmetry.

  Icositetrachoric symmetry

• Icositetrachoric group – F₄, ,, order 1152,, named for the 24-cell,. There are 24 mirror planes in this symmetry, which can be decomposed into two orthogonal sets of 12 mirrors in demitesseractic symmetry subgroups, as and , as index 6 subgroups.
• *The extended icositetrachoric group, Aut,, has order 2304,.
• ** The chiral extended icositetrachoric group, ⁺, has order 1152,. This group represents the construction of the omnisnub 24-cell,, although it can not be made uniform.
• *The ionic diminished icositetrachoric groups, and ,, have order 576,. This group leads to the snub 24-cell with construction or.
• **The double diminished icositetrachoric group, , order 288, is the commutator subgroup of .
• *** It can be extended as 3⁺,4,3⁺, order 576,.
• * The chiral icositetrachoric group is ⁺,, order 576,.
• ** The extended chiral icositetrachoric group, 3,4,3]⁺] has order 1152,. Coxeter relates this group to the abstract group.

  Demitesseractic symmetry

• Demitesseractic group – D₄, , or ,, order 192,, named for the 4-demicube construction of the 16-cell, or. There are 12 mirrors in this symmetry group.
• *There are two types of extended symmetries by adding mirrors: <> which becomes by bisecting the fundamental domain by a mirror, with 3 orientations possible; and the full extended group .
• * The chiral demitesseractic group is ⁺ or ,, order 96,. This group leads to the snub 24-cell with construction =.

  Hexacosichoric symmetry

• Hexacosichoric group – H₄, ,, order 14400,, named for the 600-cell,. It is also sometimes called the hyper-icosahedral group for extending the 3D icosahedral group , and hecatonicosachoric group or dodecacontachoric group from the 120-cell,.
• * The chiral hexacosichoric group is ⁺,, order 7200,. This group represents the construction of the snub 120-cell,, although it can not be made uniform.
• * A high-index reflective subgroup is the prismatic icosahedral symmetry, ,, order 240, subgroup index 60,.
• ** Its chiral subgroup is ⁺,, order 120,. This group represents the construction of the snub dodecahedral antiprism,, although it can't be made uniform.
• ** An ionic subgroup is ,, order 120,. This group represents the construction of the snub dodecahedral prism,.
• ** A half subgroup is = = ,, order 120,. It is called the icosahedral pyramidal group and is the 3D icosahedral group, . A regular dodecahedral pyramid can have this symmetry, with Schläfli symbol: ∨.
• *** A chiral half subgroup is = ⁺ = ⁺,, order 60,. This is the 3D chiral icosahedral group, ⁺. A snub dodecahedral pyramid can have this symmetry, with Schläfli symbol: ∨ sr.

  Duoprismatic symmetry

• Duoprismatic groups – ,, 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: and .
• * The chiral subgroup is ⁺,, order 2pq. It can be doubled as 2p,2,2p]⁺].
• * If p and q are equal, ,, the symmetry can be doubled as,.
• ** Doublings: p⁺,2,p⁺,, 2p,2⁺,2p, 2p⁺,2⁺,2p⁺.
• * ,, it represents a line groups in 3-space,
• * , it represents the Euclidean plane symmetry with two sets of parallel mirrors and a rectangular domain.
• * Subgroups include: ,, ,, ,.
• * And for even values: ,, ,, ,, ,, ,, ,, ,, and communtator subgroup, index 16, ⁺,.
• Digonal duoprismatic group – ,, order 16.
• * The chiral subgroup is ⁺,, order 8.
• * Extended,, order 32. The 4-4 duoprism has this extended symmetry,.
• ** The chiral extended group is ⁺, order 16.
• ** Extended chiral subgroup is 2,2,2]⁺], order 16, with rotoreflection generators. It is isomorphic to the abstract group.
• * Other extended 2,2,2=, order 384, #Hexadecachoric symmetry. The tesseract has this symmetry, as or.
• * Ionic diminished subgroups is , order 8.
• ** The double diminished subgroup is , order 4.
• *** Extended as 2⁺,2,2⁺, order 8.
• ** The rotoreflection subgroups are , , , order 4.
• ** The triple diminished subgroup is ,, order 2. It is a 2-fold double rotation and a 4D central inversion.
• * Half subgroup is =, order 8.
• Triangular duoprismatic group – ,, order 36.
• * The chiral subgroup is ⁺, order 18.
• * Extended, order 72. The 3-3 duoprism has this extended symmetry,.
• ** The chiral extended group is ⁺, order 36.
• ** Extended chiral subgroup is 3,2,3]⁺], order 36, with rotoreflection generators. It is isomorphic to the abstract group.
• * Other extended 3],2,3], and .
• * And 3],2,.
• * And ,2,, order 288, isomorphic to. The 6–6 duoprism has this symmetry, as or.
• * Ionic diminished subgroups are , , order 18.
• ** The double diminished subgroup is , order 9.
• *** Extended as 3⁺,2,3⁺, order 18.
• * A high index subgroup is , order 12, index 3, which is isomorphic to the dihedral symmetry in three dimensions group, , D_3h.
• ** ⁺, order 6
• Square duoprismatic group – ,, order 64.
• * The chiral subgroup is ⁺, order 32.
• * Extended, order 128. The 4–4 duoprism has this extended symmetry,.
• ** The chiral extended group is ⁺, order 64.
• ** Extended chiral subgroup is 4,2,4]⁺], order 64, with rotoreflection generators. It is isomorphic to the abstract group.
• * Other extended 4],2,4], and . The 4–8 duoprism has this symmetry, as or.
• * And 4],2,.
• * And ,2,, order 512, isomorphic to. The 8–8 duoprism has this symmetry, as or.
• * Ionic diminished subgroups are , , order 32.
• ** The double diminished subgroup is , order 16.
• *** Extended as 4⁺,2,4⁺, order 32.
• ** The rotoreflection subgroups are , , , , order 16.
• ** The triple diminished subgroup is ,, order 8.
• * Half subgroups are =,, =,, order 32.
• ** ⁺=⁺,, ⁺=⁺,, order 16.
• * Half again subgroup is =,, order 16.
• ** ⁺ = = ⁺, order 8

  Summary