Complex polytope

For nonstarry groups, the order of the group _p[q]_r can be computed as ${\displaystyle g=8/q\cdot (1/p+2/q+1/r-1)^{-2}}$.^[9]

The

${\displaystyle h=2/(1/p+2/q+1/r-1)}$

${\displaystyle g=2h^{2}/q}$

The rank 2 solutions that generate complex polygons are:

Group	G3=G(q,1,1)	G2=G(p,1,2)	G4	G6	G5	G8	G14	G9	G10	G20	G16	G21	G17	G18
	2[q]2, q=3,4...	p[4]2, p=2,3...	3[3]3	3[6]2	3[4]3	4[3]4	3[8]2	4[6]2	4[4]3	3[5]3	5[3]5	3[10]2	5[6]2	5[4]3
Order	2q	2p2	24	48	72	96	144	192	288	360	600	720	1200	1800
h	q	2p	6	12			24			30		60

Excluded solutions with odd q and unequal p and r are: ₆[3]₂, ₆[3]₃, ₉[3]₃, ₁₂[3]₃, ..., ₅[5]₂, ₆[5]₂, ₈[5]₂, ₉[5]₂, ₄[7]₂, ₉[5]₂, ₃[9]₂, and ₃[11]₂.

Other whole q with unequal p and r, create starry groups with overlapping fundamental domains: , , , , , and .

The dual polygon of _p{q}_r is _r{q}_p. A polygon of the form _p{q}_p is self-dual. Groups of the form _p[2q]₂ have a half symmetry _p[q]_p, so a regular polygon is the same as quasiregular . As well, regular polygon with the same node orders, , have an alternated construction , allowing adjacent edges to be two different colors.^[10]

The group order, g, is used to compute the total number of vertices and edges. It will have g/r vertices, and g/p edges. When p=r, the number of vertices and edges are equal. This condition is required when q is odd.

Matrix generators

The group p[q]r, , can be represented by two matrices:^[11]

Name	R1	R2
Order	p	r
Matrix	${\displaystyle \left[{\begin{smallmatrix}e^{2\pi i/p}&0\\(e^{2\pi i/p}-1)k&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&(e^{2\pi i/r}-1)k\\0&e^{2\pi i/r}\\\end{smallmatrix}}\right]}$

With

k=${\displaystyle {\sqrt {\frac {cos({\frac {\pi }{p}}-{\frac {\pi }{r}})+cos({\frac {2\pi }{q}})}{2\sin {\frac {\pi }{p}}\sin {\frac {\pi }{r}}}}}}$

Examples

Name R1 R2 Order p q Matrix ${\displaystyle \left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&e^{2\pi i/q}\\\end{smallmatrix}}\right]}$	Name R1 R2 Order p 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]}$	Name R1 R2 Order 3 3 Matrix ${\displaystyle \left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3}}i}{2}}&0\\{\frac {-3+{\sqrt {3}}i}{2}}&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&{\frac {-3+{\sqrt {3}}i}{2}}\\0&{\frac {-1+{\sqrt {3}}i}{2}}\\\end{smallmatrix}}\right]}$
Name R1 R2 Order 4 4 Matrix ${\displaystyle \left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&i\\\end{smallmatrix}}\right]}$	Name R1 R2 Order 4 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]}$	Name R1 R2 Order 3 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3}}i}{2}}&0\\{\frac {-3+{\sqrt {3}}i}{2}}&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&-2\\0&-1\\\end{smallmatrix}}\right]}$

Enumeration of regular complex polygons

Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes.^[12]

Group	Order	Coxeter number	Polygon		Vertices	Edges		Notes
G(q,q,2) 2[q]2 = [q] q=2,3,4,...	2q	q	2{q}2		q	q	{}	Real regular polygons Same as Same as if q even

Group	Order	Coxeter number	Polygon			Vertices	Edges		Notes
G(p,1,2) p[4]2 p=2,3,4,...	2p2	2p	p(2p2)2	p{4}2		p2	2p	p{}	same as p{}×p{} or ${\displaystyle \mathbb {R} ^{4}}$ representation as p-p duoprism
			2(2p2)p	2{4}p		2p	p2	{}	${\displaystyle \mathbb {R} ^{4}}$ representation as p-p duopyramid
G(2,1,2) 2[4]2 = [4]	8	4		2{4}2 = {4}		4	4	{}	same as {}×{} or Real square
G(3,1,2) 3[4]2	18	6	6(18)2	3{4}2		9	6	3{}	same as 3{}×3{} or ${\displaystyle \mathbb {R} ^{4}}$ representation as 3-3 duoprism
			2(18)3	2{4}3		6	9	{}	${\displaystyle \mathbb {R} ^{4}}$ representation as 3-3 duopyramid
G(4,1,2) 4[4]2	32	8	8(32)2	4{4}2		16	8	4{}	same as 4{}×4{} or ${\displaystyle \mathbb {R} ^{4}}$ representation as 4-4 duoprism or {4,3,3}
			2(32)4	2{4}4		8	16	{}	${\displaystyle \mathbb {R} ^{4}}$ representation as 4-4 duopyramid or {3,3,4}
G(5,1,2) 5[4]2	50	25	5(50)2	5{4}2		25	10	5{}	same as 5{}×5{} or ${\displaystyle \mathbb {R} ^{4}}$ representation as 5-5 duoprism
			2(50)5	2{4}5		10	25	{}	${\displaystyle \mathbb {R} ^{4}}$ representation as 5-5 duopyramid
G(6,1,2) 6[4]2	72	36	6(72)2	6{4}2		36	12	6{}	same as 6{}×6{} or ${\displaystyle \mathbb {R} ^{4}}$ representation as 6-6 duoprism
			2(72)6	2{4}6		12	36	{}	${\displaystyle \mathbb {R} ^{4}}$ representation as 6-6 duopyramid
G4=G(1,1,2) 3[3]3 <2,3,3>	24	6	3(24)3	3{3}3		8	8	3{}	Möbius–Kantor configuration self-dual, same as ${\displaystyle \mathbb {R} ^{4}}$ representation as {3,3,4}
G6 3[6]2	48	12	3(48)2	3{6}2		24	16	3{}	same as
				3{3}2					starry polygon
			2(48)3	2{6}3		16	24	{}
				2{3}3					starry polygon
G5 3[4]3	72	12	3(72)3	3{4}3		24	24	3{}	self-dual, same as ${\displaystyle \mathbb {R} ^{4}}$ representation as {3,4,3}
G8 4[3]4	96	12	4(96)4	4{3}4		24	24	4{}	self-dual, same as ${\displaystyle \mathbb {R} ^{4}}$ representation as {3,4,3}
G14 3[8]2	144	24	3(144)2	3{8}2		72	48	3{}	same as
				3{8/3}2					starry polygon, same as
			2(144)3	2{8}3		48	72	{}
				2{8/3}3					starry polygon
G9 4[6]2	192	24	4(192)2	4{6}2		96	48	4{}	same as
			2(192)4	2{6}4		48	96	{}
				4{3}2		96	48	{}	starry polygon
				2{3}4		48	96	{}	starry polygon
G10 4[4]3	288	24	4(288)3	4{4}3		96	72	4{}
		12		4{8/3}3					starry polygon
		24	3(288)4	3{4}4		72	96	3{}
		12		3{8/3}4					starry polygon
G20 3[5]3	360	30	3(360)3	3{5}3		120	120	3{}	self-dual, same as ${\displaystyle \mathbb {R} ^{4}}$ representation as {3,3,5}
				3{5/2}3					self-dual, starry polygon
G16 5[3]5	600	30	5(600)5	5{3}5		120	120	5{}	self-dual, same as ${\displaystyle \mathbb {R} ^{4}}$ representation as {3,3,5}
		10		5{5/2}5					self-dual, starry polygon
G21 3[10]2	720	60	3(720)2	3{10}2		360	240	3{}	same as
				3{5}2					starry polygon
				3{10/3}2					starry polygon, same as
				3{5/2}2					starry polygon
			2(720)3	2{10}3		240	360	{}
				2{5}3					starry polygon
				2{10/3}3					starry polygon
				2{5/2}3					starry polygon
G17 5[6]2	1200	60	5(1200)2	5{6}2		600	240	5{}	same as
		20		5{5}2					starry polygon
		20		5{10/3}2					starry polygon
		60		5{3}2					starry polygon
		60	2(1200)5	2{6}5		240	600	{}
		20		2{5}5					starry polygon
		20		2{10/3}5					starry polygon
		60		2{3}5					starry polygon
G18 5[4]3	1800	60	5(1800)3	5{4}3		600	360	5{}
		15		5{10/3}3					starry polygon
		30		5{3}3					starry polygon
		30		5{5/2}3					starry polygon
		60	3(1800)5	3{4}5		360	600	3{}
		15		3{10/3}5					starry polygon
		30		3{3}5					starry polygon
		30		3{5/2}5					starry polygon

Visualizations of regular complex polygons

Polygons of the form _p{2r}_q can be visualized by q color sets of p-edge. Each p-edge is seen as a regular polygon, while there are no faces.

2D orthogonal projections of complex polygons ₂{r}_q

Polygons of the form ₂{4}_q are called generalized

• 
  ₂{4}₂, , with 4 vertices, and 4 edges
• 
  ₂{4}₃, , with 6 vertices, and 9 edges^[13]
• 
  ₂{4}₄, , with 8 vertices, and 16 edges
• 
  ₂{4}₅, , with 10 vertices, and 25 edges
• 
  ₂{4}₆, , with 12 vertices, and 36 edges
• 
  ₂{4}₇, , with 14 vertices, and 49 edges
• 
  ₂{4}₈, , with 16 vertices, and 64 edges
• 
  ₂{4}₉, , with 18 vertices, and 81 edges
• 
  ₂{4}₁₀, , with 20 vertices, and 100 edges

Complex polygons _p{4}₂

Polygons of the form _p{4}₂ are called generalized hypercubes (squares for polygons). They share vertices with the 4D p-p duoprisms, vertices connected by p-edges. Vertices are drawn in green, and p-edges are drawn in alternate colors, red and blue. The perspective is distorted slightly for odd dimensions to move overlapping vertices from the center.

• 
  ₂{4}₂, or , with 4 vertices, and 4 2-edges
• 
  ₃{4}₂, or , with 9 vertices, and 6 (triangular) 3-edges^[13]
• 
  ₄{4}₂, or , with 16 vertices, and 8 (square) 4-edges
• 
  ₅{4}₂, or , with 25 vertices, and 10 (pentagonal) 5-edges
• 
  ₆{4}₂, or , with 36 vertices, and 12 (hexagonal) 6-edges
• 
  ₇{4}₂, or , with 49 vertices, and 14 (heptagonal)7-edges
• 
  ₈{4}₂, or , with 64 vertices, and 16 (octagonal) 8-edges
• 
  ₉{4}₂, or , with 81 vertices, and 18 (enneagonal) 9-edges
• 
  ₁₀{4}₂, or , with 100 vertices, and 20 (decagonal) 10-edges

3D perspective projections of complex polygons _p{4}₂. The duals ₂{4}_p

are seen by adding vertices inside the edges, and adding edges in place of vertices.

• 
  ₃{4}₂, or with 9 vertices, 6 3-edges in 2 sets of colors
• 
  ₂{4}₃, with 6 vertices, 9 edges in 3 sets
• 
  ₄{4}₂, or with 16 vertices, 8 4-edges in 2 sets of colors and filled square 4-edges
• 
  ₅{4}₂, or with 25 vertices, 10 5-edges in 2 sets of colors

Other Complex polygons _p{r}₂

• 
  ₃{6}₂, or , with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blue^[14]
• 
  ₃{8}₂, or , with 72 vertices in black, and 48 3-edges colored in 2 sets of 3-edges in red and blue^[15]

2D orthogonal projections of complex polygons, _p{r}_p

Polygons of the form _p{r}_p have equal number of vertices and edges. They are also self-dual.

• 
  ₃{3}₃, or , with 8 vertices in black, and 8 3-edges colored in 2 sets of 3-edges in red and blue^[16]
• 
  ₃{4}₃, or , with 24 vertices and 24 3-edges shown in 3 sets of colors, one set filled^[17]
• 
  ₄{3}₄, or , with 24 vertices and 24 4-edges shown in 4 sets of colors^[17]
• 
  ₃{5}₃, or , with 120 vertices and 120 3-edges^[18]
• 
  ₅{3}₅, or , with 120 vertices and 120 5-edges^[19]

Regular complex polytopes

In general, a regular complex polytope is represented by Coxeter as _p{z₁}_q{z₂}_r{z₃}_s… or Coxeter diagram …, having symmetry _p[z₁]_q[z₂]_r[z₃]_s… or ….^[20]

There are infinite families of regular complex polytopes that occur in all dimensions, generalizing the

cross polytopes in real space. Shephard's "generalized orthotope" generalizes the hypercube; it has symbol given by γ^p
_n = _p{4}₂{3}₂…₂{3}₂ and diagram …. Its symmetry group has diagram _p[4]₂[3]₂…₂[3]₂; in the Shephard–Todd classification, this is the group G(p, 1, n) generalizing the signed permutation matrices. Its dual regular polytope, the "generalized cross polytope", is represented by the symbol β^p
_n = ₂{3}₂{3}₂…₂{4}_p and diagram ….^[21]

A 1-dimensional regular complex polytope in ${\displaystyle \mathbb {C} ^{1}}$ is represented as , having p vertices, with its real representation a regular polygon, {p}. Coxeter also gives it symbol γ^p
₁ or β^p
₁ as 1-dimensional generalized hypercube or cross polytope. Its symmetry is _p[] or , a cyclic group of order p. In a higher polytope, _p{} or represents a p-edge element, with a 2-edge, {} or , representing an ordinary real edge between two vertices.^[21]

A dual complex polytope is constructed by exchanging k and (n-1-k)-elements of an n-polytope. For example, a dual complex polygon has vertices centered on each edge, and new edges are centered at the old vertices. A v-valence vertex creates a new v-edge, and e-edges become e-valence vertices.^[22] The dual of a regular complex polytope has a reversed symbol. Regular complex polytopes with symmetric symbols, i.e. _p{q}_p, _p{q}_r{q}_p, _p{q}_r{s}_r{q}_p, etc. are self dual.

Enumeration of regular complex polyhedra

Coxeter enumerated this list of nonstarry regular complex polyhedra in ${\displaystyle \mathbb {C} ^{3}}$, including the 5 platonic solids in ${\displaystyle \mathbb {R} ^{3}}$.^[23]

A regular complex polyhedron, _p{n₁}_q{n₂}_r or , has faces, edges, and vertex figures.

A complex regular polyhedron _p{n₁}_q{n₂}_r requires both g₁ = order(_p[n₁]_q) and g₂ = order(_q[n₂]_r) be finite.

Given g = order(_p[n₁]_q[n₂]_r), the number of vertices is g/g₂, and the number of faces is g/g₁. The number of edges is g/pr.

Space	Group	Order	Coxeter number	Polygon		Vertices	Edges		Faces		Vertex figure	Van Oss polygon	Notes
${\displaystyle \mathbb {R} ^{3}}$	G(1,1,3) 2[3]2[3]2 = [3,3]	24	4	α3 = 2{3}2{3}2 = {3,3}		4	6	{}	4	{3}	{3}	none	Real tetrahedron Same as
${\displaystyle \mathbb {R} ^{3}}$	G23 2[3]2[5]2 = [3,5]	120	10	2{3}2{5}2 = {3,5}		12	30	{}	20	{3}	{5}	none	Real icosahedron
				2{5}2{3}2 = {5,3}		20	30	{}	12	{5}	{3}	none	Real dodecahedron
${\displaystyle \mathbb {R} ^{3}}$	G(2,1,3) 2[3]2[4]2 = [3,4]	48	6	β2 3 = β3 = {3,4}		6	12	{}	8	{3}	{4}	{4}	Real octahedron Same as {}+{}+{}, order 8 Same as , order 24
${\displaystyle \mathbb {R} ^{3}}$				γ2 3 = γ3 = {4,3}		8	12	{}	6	{4}	{3}	none	Real cube Same as {}×{}×{} or
${\displaystyle \mathbb {C} ^{3}}$	G(p,1,3) 2[3]2[4]p p=2,3,4,...	6p3	3p	βp 3 = 2{3}2{4}p		3p	3p2	{}	p3	{3}	2{4}p	2{4}p	Generalized octahedron Same as p{}+p{}+p{}, order p3 Same as , order 6p2
${\displaystyle \mathbb {C} ^{3}}$				γp 3 = p{4}2{3}2		p3	3p2	p{}	3p	p{4}2	{3}	none	Generalized cube Same as p{}×p{}×p{} or
${\displaystyle \mathbb {C} ^{3}}$	G(3,1,3) 2[3]2[4]3	162	9	β3 3 = 2{3}2{4}3		9	27	{}	27	{3}	2{4}3	2{4}3	Same as 3{}+3{}+3{}, order 27 Same as , order 54
${\displaystyle \mathbb {C} ^{3}}$				γ3 3 = 3{4}2{3}2		27	27	3{}	9	3{4}2	{3}	none	Same as 3{}×3{}×3{} or
${\displaystyle \mathbb {C} ^{3}}$	G(4,1,3) 2[3]2[4]4	384	12	β4 3 = 2{3}2{4}4		12	48	{}	64	{3}	2{4}4	2{4}4	Same as 4{}+4{}+4{}, order 64 Same as , order 96
${\displaystyle \mathbb {C} ^{3}}$				γ4 3 = 4{4}2{3}2		64	48	4{}	12	4{4}2	{3}	none	Same as 4{}×4{}×4{} or
${\displaystyle \mathbb {C} ^{3}}$	G(5,1,3) 2[3]2[4]5	750	15	β5 3 = 2{3}2{4}5		15	75	{}	125	{3}	2{4}5	2{4}5	Same as 5{}+5{}+5{}, order 125 Same as , order 150
${\displaystyle \mathbb {C} ^{3}}$				γ5 3 = 5{4}2{3}2		125	75	5{}	15	5{4}2	{3}	none	Same as 5{}×5{}×5{} or
${\displaystyle \mathbb {C} ^{3}}$	G(6,1,3) 2[3]2[4]6	1296	18	β6 3 = 2{3}2{4}6		36	108	{}	216	{3}	2{4}6	2{4}6	Same as 6{}+6{}+6{}, order 216 Same as , order 216
${\displaystyle \mathbb {C} ^{3}}$				γ6 3 = 6{4}2{3}2		216	108	6{}	18	6{4}2	{3}	none	Same as 6{}×6{}×6{} or
${\displaystyle \mathbb {C} ^{3}}$	G25 3[3]3[3]3	648	9	3{3}3{3}3		27	72	3{}	27	3{3}3	3{3}3	3{4}2	Same as . ${\displaystyle \mathbb {R} ^{6}}$ representation as 221 Hessian polyhedron
	G26 2[4]3[3]3	1296	18	2{4}3{3}3		54	216	{}	72	2{4}3	3{3}3	{6}
				3{3}3{4}2		72	216	3{}	54	3{3}3	3{4}2	3{4}3	Same as [24] ${\displaystyle \mathbb {R} ^{6}}$ representation as 122

Visualizations of regular complex polyhedra

2D orthogonal projections of complex polyhedra, _p{s}_t{r}_r

• 
  Real {3,3}, or has 4 vertices, 6 edges, and 4 faces
• 
  ₃{3}₃{3}₃, or , has 27 vertices, 72 3-edges, and 27 faces, with one face highlighted blue.^[25]
• 
  ₂{4}₃{3}₃, has 54 vertices, 216 simple edges, and 72 faces, with one face highlighted blue.^[26]
• 
  ₃{3}₃{4}₂, or , has 72 vertices, 216 3-edges, and 54 vertices, with one face highlighted blue.^[27]

Generalized octahedra have a regular construction as and quasiregular form as . All elements are simplexes.

• 
  Real {3,4}, or , with 6 vertices, 12 edges, and 8 faces
• 
  ₂{3}₂{4}₃, or , with 9 vertices, 27 edges, and 27 faces
• 
  ₂{3}₂{4}₄, or , with 12 vertices, 48 edges, and 64 faces
• 
  ₂{3}₂{4}₅, or , with 15 vertices, 75 edges, and 125 faces
• 
  ₂{3}₂{4}₆, or , with 18 vertices, 108 edges, and 216 faces
• 
  ₂{3}₂{4}₇, or , with 21 vertices, 147 edges, and 343 faces
• 
  ₂{3}₂{4}₈, or , with 24 vertices, 192 edges, and 512 faces
• 
  ₂{3}₂{4}₉, or , with 27 vertices, 243 edges, and 729 faces
• 
  ₂{3}₂{4}₁₀, or , with 30 vertices, 300 edges, and 1000 faces

Generalized cubes

Generalized cubes have a regular construction as and prismatic construction as , a product of three p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.

• 
  Real {4,3}, or has 8 vertices, 12 edges, and 6 faces
• 
  ₃{4}₂{3}₂, or has 27 vertices, 27 3-edges, and 9 faces^[25]
• 
  ₄{4}₂{3}₂, or , with 64 vertices, 48 edges, and 12 faces
• 
  ₅{4}₂{3}₂, or , with 125 vertices, 75 edges, and 15 faces
• 
  ₆{4}₂{3}₂, or , with 216 vertices, 108 edges, and 18 faces
• 
  ₇{4}₂{3}₂, or , with 343 vertices, 147 edges, and 21 faces
• 
  ₈{4}₂{3}₂, or , with 512 vertices, 192 edges, and 24 faces
• 
  ₉{4}₂{3}₂, or , with 729 vertices, 243 edges, and 27 faces
• 
  ₁₀{4}₂{3}₂, or , with 1000 vertices, 300 edges, and 30 faces

Enumeration of regular complex 4-polytopes

Coxeter enumerated this list of nonstarry regular complex 4-polytopes in ${\displaystyle \mathbb {C} ^{4}}$, including the 6

${\displaystyle \mathbb {R} ^{4}}$

Space	Group	Order	Coxeter number	Polytope	Vertices	Edges	Faces	Cells	Van Oss polygon	Notes
${\displaystyle \mathbb {R} ^{4}}$	G(1,1,4) 2[3]2[3]2[3]2 = [3,3,3]	120	5	α4 = 2{3}2{3}2{3}2 = {3,3,3}	5	10 {}	10 {3}	5 {3,3}	none	Real 5-cell (simplex)
${\displaystyle \mathbb {R} ^{4}}$	G28 2[3]2[4]2[3]2 = [3,4,3]	1152	12	2{3}2{4}2{3}2 = {3,4,3}	24	96 {}	96 {3}	24 {3,4}	{6}	Real 24-cell
	G30 2[3]2[3]2[5]2 = [3,3,5]	14400	30	2{3}2{3}2{5}2 = {3,3,5}	120	720 {}	1200 {3}	600 {3,3}	{10}	Real 600-cell
				2{5}2{3}2{3}2 = {5,3,3}	600	1200 {}	720 {5}	120 {5,3}		Real 120-cell
${\displaystyle \mathbb {R} ^{4}}$	G(2,1,4) 2[3]2[3]2[4]p =[3,3,4]	384	8	β2 4 = β4 = {3,3,4}	8	24 {}	32 {3}	16 {3,3}	{4}	Real 16-cell Same as , order 192
${\displaystyle \mathbb {R} ^{4}}$				γ2 4 = γ4 = {4,3,3}	16	32 {}	24 {4}	8 {4,3}	none	Real tesseract Same as {}4 or , order 16
${\displaystyle \mathbb {C} ^{4}}$	G(p,1,4) 2[3]2[3]2[4]p p=2,3,4,...	24p4	4p	βp 4 = 2{3}2{3}2{4}p	4p	6p2 {}	4p3 {3}	p4 {3,3}	2{4}p	Generalized 4- orthoplex Same as , order 24p3
${\displaystyle \mathbb {C} ^{4}}$				γp 4 = p{4}2{3}2{3}2	p4	4p3 p{}	6p2 p{4}2	4p p{4}2{3}2	none	Generalized tesseract Same as p{}4 or , order p4
${\displaystyle \mathbb {C} ^{4}}$	G(3,1,4) 2[3]2[3]2[4]3	1944	12	β3 4 = 2{3}2{3}2{4}3	12	54 {}	108 {3}	81 {3,3}	2{4}3	Generalized 4- orthoplex Same as , order 648
${\displaystyle \mathbb {C} ^{4}}$				γ3 4 = 3{4}2{3}2{3}2	81	108 3{}	54 3{4}2	12 3{4}2{3}2	none	Same as 3{}4 or , order 81
${\displaystyle \mathbb {C} ^{4}}$	G(4,1,4) 2[3]2[3]2[4]4	6144	16	β4 4 = 2{3}2{3}2{4}4	16	96 {}	256 {3}	64 {3,3}	2{4}4	Same as , order 1536
${\displaystyle \mathbb {C} ^{4}}$				γ4 4 = 4{4}2{3}2{3}2	256	256 4{}	96 4{4}2	16 4{4}2{3}2	none	Same as 4{}4 or , order 256
${\displaystyle \mathbb {C} ^{4}}$	G(5,1,4) 2[3]2[3]2[4]5	15000	20	β5 4 = 2{3}2{3}2{4}5	20	150 {}	500 {3}	625 {3,3}	2{4}5	Same as , order 3000
${\displaystyle \mathbb {C} ^{4}}$				γ5 4 = 5{4}2{3}2{3}2	625	500 5{}	150 5{4}2	20 5{4}2{3}2	none	Same as 5{}4 or , order 625
${\displaystyle \mathbb {C} ^{4}}$	G(6,1,4) 2[3]2[3]2[4]6	31104	24	β6 4 = 2{3}2{3}2{4}6	24	216 {}	864 {3}	1296 {3,3}	2{4}6	Same as , order 5184
${\displaystyle \mathbb {C} ^{4}}$				γ6 4 = 6{4}2{3}2{3}2	1296	864 6{}	216 6{4}2	24 6{4}2{3}2	none	Same as 6{}4 or , order 1296
${\displaystyle \mathbb {C} ^{4}}$	G32 3[3]3[3]3[3]3	155520	30	3{3}3{3}3{3}3	240	2160 3{}	2160 3{3}3	240 3{3}3{3}3	3{4}3	Witting polytope ${\displaystyle \mathbb {R} ^{8}}$ representation as 421