-26- Higher Still Four-Dimensional Point Groups There are only a few types of higher dimensional group for which we have complete enumerations. Next in sequence are the four- dimensional point groups, which of course can also be regarded as three-dimensional spherical groups, since they act on the unit sphere x2 +y2+z2 +t2 = 1. Their enumeration was started by Goursat [16], continued by Seifert and Threlfall [28], and also by Du Val [29]—we shall comment on their roles later. The groups are usually described in terms of quaternions, because the general orthogonal operation has the form [q,r]:x → q−1xr or ∗[q,r]:x → q−1xr¯ accordingly as its determinant is plus or minus 1. We should only briefly explain Tables 26.1, 26.2, and 26.3, which are quoted from On Quaternions and Octonians [26]. The name ± 1 × − of the typical group has a form f [L R]ifitcontains 1 (the negative of the identity matrix), so that its elements come in pairs ± 1 × g, and otherwise + f [L R]. These names are appropriate because, 1 up to sign, the group contains the proportion f of the elements of a direct product of the groups L and R that consist of left and right multiplications, respectively. A generator [l, r] denotes the map x → lxr¯ while the names of individual quaternions are given below Table 26.1. When only a + or − is given, it is to be applied to a generator written explicitly (opposite page) A portion of the polydodecahedron, stereographically projected into 3- space. 383 © 2016 by Taylor & Francis Group, LLC 384 26. Higher Still Group Generators ±[I × O][iI , 1], [ω,1], [1,iO], [1,ω]; ±[I × T ][iI , 1], [ω,1], [1,i], [1,ω]; ±[I × D2n][iI , 1], [ω,1], [1,en], [1,j]; ±[I × Cn][iI , 1], [ω,1], [1,en]; ±[O × T ][iO, 1], [ω,1], [1,i], [1,ω]; ±[O × D2n][iO, 1], [ω,1], [1,en], [1,j]; 1 ± 2 [O × D2n][i, 1], [ω,1], [1,en];[iO,j] 1 ± 2 [O × D4n][i, 1], [ω,1], [1,en], [1,j];[iO,e2n] 1 ± 6 [O × D6n][i, 1], [j, 1], [1,en];[iO,j], [ω,e3n] ±[O × Cn][iO, 1], [ω,1], [1,en]; 1 ± 2 [O × C2n][i, 1], [ω,1], [1,en];[iO,e2n] ±[T × D2n][i, 1], [ω,1], [1,en], [1,j]; ±[T × Cn][i, 1], [ω,1], [1,en]; 1 ± 3 [T × C3n][i, 1], [1,en];[ω,e3n] 1 ± 2 [D2m × D4n][em, 1], [1,en], [1,j];[j, e2n] ±[D2m × Cn][em, 1], [j, 1], [1,en]; 1 ± 2 [D2m × C2n][em, 1], [1,en];[j, e2n] 1 + 2 [D2m × C2n] − , − ; + 1 ± 2 [D4m × C2n][em, 1], [j, 1], [1,en];[e2m,e2n] √ √ Key: i + σj + τk 5 − 1 5+1 iI = (σ = ,τ = ), 2 2 2 − j + k 1+i + j + k πi/n iO = √ ,ω = ,iT = i, en = e 2 2 Ta b l e 2 6 . 1 . Chiral groups, I. These are most of the “metachiral” groups—some others appear in the last few lines of Table 26.2. Downloaded by [University of Bergen Library] at 04:56 26 October 2016 in a previous line. Generators before a “;” are pure left or right multiplications; those after involve both. We now comment on the history of the enumerations. What Goursat enumerated, strictly speaking, were the groups of motions in elliptic three-space. This is equivalent to finding the subgroups − ± 1 × of SO4 that contain 1. He missed the group 4 (D4m D4n), as was noted by Seifert and Threlfall and independently by J. Sun- day. Seifert and Threlfall extended this to the general case and © 2016 by Taylor & Francis Group, LLC Four-Dimensional Point Groups 385 Group Generators Coxeter Name + ±[I × I][iI , 1], [ω,1], [1,iI], [1,ω]; [3, 3, 5] 1 + ± 60 [I × I] ;[ω,ω], [iI ,iI ]2.[3, 5] 1 + + 60 [I × I] ; + , +[3, 5] 1 + ± 60 [I × I] ;[ω,ω], [iI ,iI ]2.[3, 3, 3] 1 + + 60 [I × I] ; + , +[3, 3, 3] + ±[O × O][iO, 1], [ω,1], [1,iO], [1,ω]; [3, 4, 3] :2 1 + ± 2 [O × O][i, 1], [ω,1], [1,i], [1,ω];[iO,iO][3, 4, 3] 1 + ± 6 [O × O][i, 1], [j, 1], [1,i], [1,j];[ω,ω], [iO,iO][3, 3, 4] 1 + ± 24 [O × O] ;[ω,ω], [iO,iO]2.[3, 4] 1 + + 24 [O × O] ; + , +[3, 4] 1 + + 24 [O × O] ; + , − [2, 3, 3] ±[T × T ][i, 1], [ω,1], [1,i], [1,ω]; [+3, 4, 3+] 1 + + ± 3 [T × T ][i, 1], [j, 1], [1,i], [1,j];[ω,ω][3, 3, 4 ] ∼ 1 = ± 3 [T × T ][i, 1], [j, 1], [1,i], [1,j];[ω,ω]” 1 + ± 12 [T × T ] ;[ω,ω], [i, i]2.[3, 3] ∼ 1 = ± 12 [T × T ] ;[ω,ω], [i, −i]” 1 + + 12 [T × T ] ; + , +[3, 3] ∼ 1 = + 12 [T × T ] ; + , +” ±[D2m × D2n][em, 1], [j, 1], [1,en], [1,j]; 1 ± 2 [D4m × D4n][em, 1], [j, 1], [1,en], [1,j];[e2m,e2n] 1 ± 4 [D4m × D4n][em, 1], [1,en];[e2m,j], [j, e2n] Conditions 1 + 4 [D4m × D4n] − , − ; + , + m, n odd 1 (s) s Downloaded by [University of Bergen Library] at 04:56 26 October 2016 ± 2f [D2mf × D2nf ][em, 1], [1,en];[emf ,enf ], [j, j](s, f)=1 1 (s) + 2f [D2mf × D2nf ] − , − ; + , + m, n odd, (s, 2f)=1 1 (s) s ± f [Cmf × Cnf ][em, 1], [1,en];[emf ,enf ](s, f)=1 1 (s) + f [Cmf × Cnf ] − , − ; + m, n odd, (s, 2f)=1 Ta b l e 2 6 . 2 . Chiral groups, II. These groups are mostly “orthochiral,” with a few “parachiral” groups in the last few lines. The generators should be taken with both signs except in the haploid cases, for which we just indicate the proper choice of sign. © 2016 by Taylor & Francis Group, LLC 386 26. Higher Still Group Extending element Coxeter Name ±[I × I] · 2 ∗ [3, 3, 5] 1 ± 60 [I × I] · 2 ∗ 2.[3, 5] 1 ◦ + 60 [I × I] · 23 or 123 or 21 ∗ or −∗ [3, 5] or [3, 5] 1 ± 60 [I × I] · 2 ∗ 2.[3, 3, 3] 1 ◦ + 60 [I × I] · 23 or 21 ∗ or −∗ [3, 3, 3] or [3, 3, 3] ±[O × O] · 2 ∗ [3, 4, 3] : 2 1 +· ± 2 [O × O] · 2or2 ∗ or ∗ [1,iO][3, 4, 3] or [3, 4, 3] 2 1 ± 6 [O × O] · 2 ∗ [3, 3, 4] 1 ± 24 [O × O] · 2 ∗ 2.[3, 4] 1 ◦ + 24 [O × O] · 23 or 21 ∗ or −∗ [3, 4] or [3, 4] 1 ◦ + 24 [O × O] · 23 or 21 ∗ or −∗ [2, 3, 3] or [2, 3, 3] ±[T × T ] · 2 ∗ [3, 4, 3+] 1 + ± 3 [T × T ] · 2 ∗ [ 3, 3, 4] 1 + ± 3 [T × T ] · 2 ∗ [3, 3, 4 ] 1 + ± 12 [T × T ] · 2 ∗ 2.[ 3, 4] 1 ± 12 [T × T ] · 2 ∗ 2.[3, 3] 1 + + ◦ + 12 [T × T ] · 23 or 21 ∗ or −∗ [ 3, 4] or [ 3, 4] 1 ◦ + 12 [T × T ] · 23 or 21 ∗ or −∗ [3, 3] or [3, 3] ±[D2n × D2n] · 2 ∗ 1 ± 2 [D4n × D4n] · 2or2 ∗ or ∗ [1,e2n] 1 ± 4 [D4n × D4n] · 2 ∗ Conditions 1 + [D4n × D4n] · 23 or 21 ∗ or −∗ ⎫ n odd 4 ⎪ ± 1 × (s) · (α,β) ∗ α αs+βf ∗ ⎪ 2f [D2nf D2nf ] 2 or 2 [e2nf ,e2nf ]or [1,j] ⎪ 1 (s) (α,β) α αs+βf ⎬⎪ + 2f [D2nf × D2nf ] · 2 or 2 ∗[e2nf ,e2nf ]or ∗ [1,j] See [23, 1 (s) (γ) γ(f,s+1) ± [Cnf × C ] · 2 ∗[1,e ] ⎪ pages 50–53] f nf 2nf ⎪ ⎪ 1 × (s) · (γ) ∗ γ(f,s+1) ⎭ Downloaded by [University of Bergen Library] at 04:56 26 October 2016 + f [Cnf Cnf ] 2 [1,e2nf ] Ta b l e 2 6 . 3 . Achiral groups. studied the orbifolds (as they are now known) of some of the groups. Du Val’s elegant book on Quaternions and Homographies also con- tains an enumeration, but sadly omits the relatives of the group ± 1 4 (D4m,D4n) that Goursat missed. There is a minor respect in which all enumerations before that of [26] were incomplete—namely © 2016 by Taylor & Francis Group, LLC Regular Polytopes 387 some of the groups depend on several parameters and the exact con- ditions on the parameters for two groups to be isomorphic were first given in [26]. The four-dimensional space groups have also been enumerated. However, their enumeration takes up an entire book [2] and so we cannot summarize it here. Higher-dimensional point groups have also been listed, but in more and more abbreviated ways as the dimension increases. Regular Polytopes The term regular polytope is usually understood to mean “convex regular polytope.” These are • in one dimension, the line segment with Sch¨afli symbol {}; • in two dimensions, the regular polygons with symbols {3}, {4}, {5}, ...; • in three dimensions, {3, 3}, {3, 4}, {3, 5}, {4, 3}, {5, 3}; • in four dimensions, – {3, 3, 3} the simplex, or 5-cell, – {4, 3, 3} the tesseract, or 8-cell, – {3, 3, 4} the orthoplex, or 16-cell, – {3, 4, 3} the polyoctahedron, or 24-cell, Downloaded by [University of Bergen Library] at 04:56 26 October 2016 – {5, 3, 3} the polydodecahedron, or 120-cell, – {3, 3, 5} the polytetrahedron, or 600-cell; • in n ≥ 5 dimensions, – {3, 3,...,3, 3} the n-simplex, – {4, 3,...,3, 3} the n-hypercube, – {3, 3,...3, 4} the n-orthoplex. © 2016 by Taylor & Francis Group, LLC 388 26. Higher Still A gratuitous but pretty figure of a 5-coloring of the polydodecahedron.