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Point groups in four dimensions

Ingeometry, 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

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• 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
• 2018 N. W. Johnson Geometries and Transformations, Chapter 11,12,13, Full polychoric groups, p. 249, duoprismatic groups p. 269

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

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Point groups in this article are given in Coxeter notation, which are based on Coxeter groups, with markups for extended groups and subgroups.^[6] Coxeter notation has a direct correspondence the Coxeter diagram like [3,3,3], [4,3,3], [3^1,1,1], [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).^[7]

For cross-referencing, also given here are quaternion based notations by Patrick du Val (1964)^[8] and John Conway (2003).^[9] 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.

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.

Apolychoric 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^[10]byNorman Johnson and George Olshevsky in the context of uniform polychora (4-polytopes), and their related 4-dimensional symmetry groups.^[11]

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 hosohedronorhosotope 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.^[12] 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.

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			[3,3,3]	120	[3,3,3]+	5		10
D4	±1/3[T×T].2	1/2.2S4			[31,1,1]	192	[31,1,1]+	6		12
B4	±1/6[O×O].2	2S4 = S2≀S4			[4,3,3]	384		8	4	12
F4	±1/2[O×O].23	3.2S4			[3,4,3]	1152	[3+,4,3+]	12	12	12
H4	±[I×I].2	2.(A5×A5).2			[5,3,3]	14400	[5,3,3]+	30		60
Full polyhedral prismatic groups
A3A1	+1/24[O×O].23	S4×D1			[3,3,2] = [3,3]×[ ]	48	[3,3]+	-		6	1
B3A1	±1/24[O×O].2	S4×D1			[4,3,2] = [4,3]×[ ]	96		-	3	6	1
H3A1	±1/60[I×I].2	A5×D1			[5,3,2] = [5,3]×[ ]	240	[5,3]+	-		15	1
Full duoprismatic groups
4A1 = 2D2	±1/2[D4×D4]	D14 = D22			[2,2,2] = [ ]4 = [2]2	16	[ ]+	4	1	1	1	1
D2B2	±1/2[D4×D8]	D2×D4			[2,2,4] = [2]×[4]	32	[2]+	-	1	1	2	2
D2A2	±1/2[D4×D6]	D2×D3			[2,2,3] = [2]×[3]	24	[3]+	-	1	1	3
D2G2	±1/2[D4×D12]	D2×D6			[2,2,6] = [2]×[6]	48		-	1	1	3	3
D2H2	±1/2[D4×D10]	D2×D5			[2,2,5] = [2]×[5]	40	[5]+	-	1	1	5
2B2	±1/2[D8×D8]	D42			[4,2,4] = [4]2	64	[2+,2,2+]	8	2	2	2	2
B2A2	±1/2[D8×D6]	D4×D3			[4,2,3] = [4]×[3]	48	[2+,2,3+]	-	2	2	3
B2G2	±1/2[D8×D12]	D4×D6			[4,2,6] = [4]×[6]	96		-	2	2	3	3
B2H2	±1/2[D8×D10]	D4×D5			[4,2,5] = [4]×[5]	80	[2+,2,5+]	-	2	2	5
2A2	±1/2[D6×D6]	D32			[3,2,3] = [3]2	36	[3+,2,3+]	6	3		3
A2G2	±1/2[D6×D12]	D3×D6			[3,2,6] = [3]×[6]	72		-	3		3	3
2G2	±1/2[D12×D12]	D62			[6,2,6] = [6]2	144		12	3	3	3	3
A2H2	±1/2[D6×D10]	D3×D5			[3,2,5] = [3]×[5]	60	[3+,2,5+]	-	3		5
G2H2	±1/2[D12×D10]	D6×D5			[6,2,5] = [6]×[5]	120		-	3	3	5
2H2	±1/2[D10×D10]	D52			[5,2,5] = [5]2	100	[5+,2,5+]	10	5		5
In general, p,q=2,3,4...
2I2(2p)	±1/2[D4p×D4p]	D2p2			[2p,2,2p] = [2p]2	16p2	[p+,2,p+]	2p	p	p	p	p
2I2(p)	±1/2[D2p×D2p]	Dp2			[p,2,p] = [p]2	4p2		2p	p		p
I2(p)I2(q)	±1/2[D4p×D4q]	D2p×D2q			[2p,2,2q] = [2p]×[2q]	16pq	[p+,2,q+]	-	p	p	q	q
I2(p)I2(q)	±1/2[D2p×D2q]	Dp×Dq			[p,2,q] = [p]×[q]	4pq		-	p		q

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		=
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

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

Coxeter notation		Conway Quaternion	Structure	Order	Gyration axes
Polychoric groups
	[3,3,3]+	+1/60[I×I]	A5	60			103	102
	3,3,3+	±1/60[I×I]	A5×Z2	120			103	(10+?)2
	[31,1,1]+	±1/3[T×T]	1/2.2A4	96			163	182
	[4,3,3]+	±1/6[O×O]	2A4 = A2≀A4	192	64		163	362
	[3,4,3]+	±1/2[O×O]	3.2A4	576	184	163	163	722
	[3+,4,3+]	±[T×T]		288		163	163	(72+18)2
	[[3+,4,3+]]	±[O×T]		576			323	(72+18+?)2
	3,4,3+	±[O×O]		1152	184		323	(72+?)2
	[5,3,3]+	±[I×I]	2.(A5×A5)	7200		725	2003	4502
Polyhedral prismatic groups
	[3,3,2]+	+1/24[O×O]	A4×Z2	24		43	43	(6+6)2
	[4,3,2]+	±1/24[O×O]	S4×Z2	48		64	83	(3+6+12)2
	[5,3,2]+	±1/60[I×I]	A5×Z2	120		125	203	(15+30)2
Duoprismatic groups
	[2,2,2]+	+1/2[D4×D4]		8		12	12	42
	[3,2,3]+	+1/2[D6×D6]		18		13	13	92
	[4,2,4]+	+1/2[D8×D8]		32		14	14	162
(p,q=2,3,4...), gcd(p,q)=1
	[p,2,p]+	+1/2[D2p×D2p]		2p2		1p	1p	(pp)2
	[p,2,q]+	+1/2[D2p×D2q]		2pq		1p	1q	(pq)2
	[p+,2,q+]	+[Cp×Cq]	Zp×Zq	pq		1p	1q

• Pentachoric group – A₄, [3,3,3], (), 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 . 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, ), 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]]⁺, (), order 120, (Du Val #32 (I^†/C₂;I/C₂)^†, Conway ±¹/₆₀[IxI]). 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 [3,3,3]⁺, (), 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 group – B₄, [4,3,3], (), order 384, (Du Val #47 (O/V;O/V)^*, Conway ±¹/₆[O×O].2), named for the 16-cell (hexadecachoron), . There are 16 mirror hyperplanes in this group, which can be identified in 2 orthogonal sets: 12 from a [3^1,1,1] 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, .
  • The chiral hexadecachoric group is [4,3,3]⁺, (), order 192, (Du Val #27 (O/V;O/V), Conway ±¹/₆[O×O]). This group represents the construction of an omnisnub tesseract, , although it can not be made uniform.
  • The ionic diminished hexadecachoric group is [4,(3,3)⁺], (), order 192, (Du Val #41 (T/V;T/V)^*, Conway ±¹/₃[T×T].2). This group leads to the snub 24-cell with construction .
  • The half hexadecachoric group is [1⁺,4,3,3], ( = ), order 192, and same as the #demitesseractic symmetry: [3^1,1,1]. This group is expressed in the tesseract alternated construction of the 16-cell, = .
    • The group [1⁺,4,(3,3)⁺], ( = ), order 96, and same as the chiral demitesseractic group [3^1,1,1]⁺ and also is the commutator subgroup of [4,3,3].
  • A high-index reflective subgroup is the prismatic octahedral symmetry, [4,3,2] (), 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 and the cubic prism is a lower symmetry construction of the tesseract, as .
    • Its chiral subgroup is [4,3,2]⁺, (), order 48, (Du Val #26 (O/C₂;O/C₂), Conway ±¹/₂₄[O×O]). An example is the snub cubic antiprism, , although it can not be made uniform.
    • The ionic subgroups are:
      • [(3,4)⁺,2], (), 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 .
        • [(3,4)⁺,2⁺], (), order 24, (Du Val #44' (T/C₂;T/C₂)₋^*, Conway +¹/₁₂[T×T].2₁).
      • [4,3⁺,2], (), 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⁺], ( = ), 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⁺], (), order 24, (Du Val #21 (T/C₂;T/C₂), Conway ±¹/₁₂[T×T]).
      • [3,4,2⁺], (), order 48, (Du Val #39' (T/C₂;T/C₂)₋^*, Conway ±¹/₁₂[T×T].2).
      • [4,(3,2)⁺], (), 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], ( = ), 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}.

      

      • A chiral half subgroup [(4,3)⁺,2,1⁺] = [4,3,1]⁺ = [4,3]⁺, ( = ), 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], (), 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]⁺, (), order 24, (Du Val #26b" (O/C₁;O/C₁), Conway +¹/₂₄[O×O]). An example is the snub tetrahedral antiprism, , although it can not be made uniform.
    • The ionic subgroup is [(3,3)⁺,2], (), order 24, (Du Val #39b' (T/C₁;T/C₁)_c^*, Conway +¹/₁₂[T×T].2₃). An example is the snub tetrahedral prism, .
    • The half subgroup is [3,3,2,1⁺] = [3,3,1] = [3,3], ( = ), 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}.

      

      • The chiral half subgroup [(3,3)⁺,2,1⁺] = [3,3]⁺( = ), 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] (), order 16. Others are [4,2,4] (), [4,2,2] (), 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 group – F₄, [3,4,3], (), order 1152, (Du Val #45 (O/T;O/T)^*, Conway ±¹/₂[OxO].2), named for the 24-cell (icositetrachoron), . There are 24 mirror planes in this symmetry, which can be decomposed into two orthogonal sets of 12 mirrors in demitesseractic symmetry [3^1,1,1] subgroups, as [3^*,4,3] and [3,4,3^*], as index 6 subgroups.
  • The extended icositetrachoric group, Aut(F₄), [[3,4,3]], () has order 2304, (Du Val #48 (O/O;O/O)^*, Conway ±[O×O].2).
    • The chiral extended icositetrachoric group, [[3,4,3]]⁺, () has order 1152, (Du Val #25 (O/O;O/O), Conway ±[OxO]). This group represents the construction of the omnisnub 24-cell, , although it can not be made uniform.
  • The ionic diminished icositetrachoric groups, [3⁺,4,3] and [3,4,3⁺], (or), have order 576, (Du Val #43 (T/T;T/T)^*, Conway ±[T×T].2). This group leads to the snub 24-cell with construction or.
    • The double diminished icositetrachoric group, [3⁺,4,3⁺] (the double diminishing can be shown by a gap in the diagram 4-branch: ), 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⁺]], () order 576, (Du Val #23 (T/T;O/O), Conway ±[OxT]).
  • The chiral icositetrachoric group is [3,4,3]⁺, (), 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]