User:Tomruen/Point group symmetries

Point group symmetries can be defined as discrete Coxeter groups and continuous orthogonal groups that leave one point unchanged. Both include rotations and reflections, while chiral half groups exist with only rotations. Symmetries with reflections are called full symmetry, while without reflections are called rotational or proper symmetry.

Introduction[edit]

Orthogonal groups[edit]

All point groups of n-dimensions can be seen as subgroups of orthogonal groups O(n) or special orthogonal groups SO(n) = O(n)/O(1), disallowing reflections. Orientation in space is represented by n orthonormal basis vectors u₁, u₂... u_n. Put into a matrix, this basis determinant can be +1 or -1 representing direct and mirrored transformations.

Many subgroups can be represented as Cartesian products of lower dimensional symmetries. For example O(a)×O(b) is a subgroup of O(a+b).

A point in n-dimensions has O(n) symmetry. A segment in n-dimensions has O(n-1) symmetry. A k-dimensional object in n dimension will have O(n-k) symmetry beyond its own symmetry in k-dimension.

Coxeter groups and notation[edit]

Coxeter notation is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

There is a direct correspondence between Coxeter (bracket) notation and Coxeter-Dynkin diagram. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors.

The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the A_n group is represented by [3ⁿ⁻¹], to imply n nodes connected by n−1 order-3 branches. Example A₂ = [3,3] = [3²] represents diagrams .

Chiral subgroups are represented with a + symbol. [3] is the dihedral group, Dih₃, order 6, while [3]⁺ is the cyclic group order 3.

Chiral subgroups can be applied to portions of a Coxeter diagram that are isolated by all even-order branches. [3⁺,4], is the pyritohedral group, order 24, while [3⁺,4,1⁺], , becomes chiral tetrahedral group [3,3]⁺, , order 12.

Coxeter graphs that are unconnected can be expressed as direct products, while connected rotational groups can be expressed as semidirect products.

Objects defined by composite[edit]

Johnson defined product, sum, and join operators for constructing higher dimensional polytopes from lower. Johnson defines ( ) as a point (0-polytope), { } is a line segment defined between two points (1-polytope). Many vertex figures for uniform polytopes can be expressed with these operators.

A product operator, ×, defines rectangles and prisms with independent proportions. dim(A×B) = dim(A)+dim(B).

For instance { }×{ } is a rectangle, symmetry [2], (a lower symmetry form of a square), and {4}×{ } is a square prism, symmetry [4,2] (a lower symmetry form of a cube), and {4}×{4} is called a duoprism in 4-dimensions, symmetry [4,2,4] (a lower symmetry form of a tesseract).

A sum operator, +, makes duals to the prisms. dim(A+B) = dim(A)+dim(B).

For instance, { }+{ } is a rhombus or fusil in general, symmetry [2], {4}+{ } is a square bipyramid, symmetry [4,2] (lower symmetry form of a regular octahedron), and {4}+{4} is called a duopyramid in 4-dimensions, symmetry [4,2,4] (a lower symmetry form of the 16-cell).

The product and sum operators are related by duality: !(A×B)=!A+!B and !(A+B)=!A×!B, where !A is dual polytope of A.

A join operator, ∨, makes pyramidal composites, orthogonal orientations with an offset direction as well, with edges between all pairs of vertices across the two. dim(A∨B) = dim(A)+dim(B)+1.

The isosceles triangle can be seen as ( )∨{ }, symmetry [ ], and tetragonal disphenoid is { }∨{ }, symmetry [2]. A square pyramid is {4}∨( ), symmetry [4,1]. A 1 branch is symbolic, representing [4,2,1⁺], or , having an orthogonal mirror inactivated by an alternation.

The join operator is self-related by duality: !(A∨B)=!A∨!B. More generally any expression of these operators can be dualed by replacing polytopes by dual, and swapping product and sum operators.

Polytopes constructed by:

Products are orthotopes (prisms)
Sums are orthoplexes (bipyramids or fusils)
Products and sums are Hanner polytopes
Join are simplexes (pyramids)

Continuous symmetry objects are constructed by:

Products are cylinders: circle×segment (3D), circle×circle (4D), sphere×segment (4D), sphere×circle (5D), etc.
Sums are bicones: circle+segment (3D), circle+circle (4D), sphere+segment (4D), sphere+circle (5D), etc.
Joins are cones: circle∨point (3D), circle∨segment (4D), sphere∨point (5D), circle∨circle (5D), sphere∨segment (5D), sphere∨circle (6D), etc.

Example coordinates with operators:

{ }×{ } = (±1; ±1) = rectangle or square in 2D, symmetry [2], , order 4
- {4}×{ } = (±1, ±1; ±1) = rectangular parallelepiped or cube in 3D, symmetry [4,2], , order 16
- {4}×{4} = (±1, ±1; ±1, ±1) = 4-4 duoprism or tesseract in 4D, symmetry [4,2,4], , order 64
{ }+{ } = (±1, 0), (0, ±1) = rhombus or square in 2D, , order 4
- {4}+{ } = (±1, ±1; 0), (0, 0; ±1) = square-segment duopyramid = octahedron in 3D, symmetry [4,2], , order 16
- {4}+{4} = (±1, ±1; 0, 0), (0, 0; ±1, ±1) = square-square duopyramid = 16-cell in 4D, symmetry [4,2,4], , order 64
{ }∨{ } = (±1; -1; 0), (0; +1; ±1) = segment-segment pyramid = tetragonal disphenoid in 3D, symmetry [2,1,2],
- ( )∨( ) = { } = (±1) = segment in 1D, symmetry [1], , order 2
- { }∨( ) = (±1; -1), (0; +1) = isosceles triangle in 2D, symmetry [1], , order 2
- {4}∨( ) = (±1, ±1; -1), (0, 0; +1) = square pyramid in 3D, symmetry [4,1], , order 16
- {4}∨{ } = (±1, ±1; -1; 0), (0, 0; +1; ±1) = square-segment pyramid in 4D, symmetry [4,2,1], , order 16
- {4}∨{4} = (±1, ±1; -1; 0, 0), (0, 0; +1; ±1, ±1) = square-square pyramid in 5D, symmetry [4,2,4,1], , order 64

1D[edit]

O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih₁. SO(1) is just the identity.

#	Rank 1 groups	Order	Other names	Example geometry	Example finite subgroups
1	O(1)	2	Dih₁	Full symmetry a line segment, point	[ ] =
1b	SO(1)	1	C₁	Direct symmetry, ray	[ ]⁺ =

2D[edit]

Continuous symmetries in 2-space can be classified as products of orthogonal groups O(2) or special orthogonal groups SO(2) = O(2)/O(1).

#	Rank 2 groups	Other names	Example geometry		Example finite subgroup
#	Rank 2 groups	Other names	Example geometry		Bracket	Graph	Order
1	O(2)	Dih_∞		Full symmetry a circle, $S^{1}$ , point	[p]		2p
1b	SO(2)	C_∞	Circle group, $S_{1}$	Direct	[p]⁺		p
1c	Dih(6) ⊂ O(2)	Dih₆		Hexagon, {6}	[[3]] = [6]		12
1d	C(6) ⊂ SO(2)	C₆		Hexagon, {6}	[[3]]⁺ = [6]⁺		6
1e	Dih(5) ⊂ O(2)	Dih₅		Pentagon, {5}	[5]		10
1f	C(5) ⊂ SO(2)	C₅		Pentagon, {5}	[5]⁺		5
1g	Dih(3) ⊂ O(2)	Dih₃		Equilateral triangle, {3} reuleaux triangle	[3]		6
1h	C(3) ⊂ SO(2)	C₃		Equilateral triangle, {3} reuleaux triangle	[3]⁺		3
2	O(1)×O(1)×2!	Dih₄		Square, {4}	[[2]] = [4]		8
2b	(O(1)×O(1)×2!)/O(1)	C₄		Square, {4}	[[2]]⁺ = [4]⁺		4
2c	O(1)×O(1)	Dih₁×Dih₁=Dih₂		Rectangle, { }², rhombus ellipse, stadium, lens	[ ]×[ ] = [2]		4
2d	(O(1)×O(1))/O(1)	C₂	Half turn	Parallelogram, zonogons	[2]⁺		2
3	O(1)×SO(1)	Dih₁		Isosceles triangle, { }∨( ) kite, isosceles trapezoids oval, crescent, parabola, circular segment	[1]		2
3b	SO(1)×SO(1)	C₁		Scalene triangle, ( )∨( )∨( ) irregular quadrilateral, arbelos	[1]⁺		1

3D[edit]

Continuous symmetries in 3-space can be classified as products of orthogonal groups O(n) or special orthogonal groups SO(n) = O(n)/O(1), along with semidirect products with half turns, C₂.

#	Rank 3 groups	Other names	Example geometry		Example finite subgroup
#	Rank 3 groups	Other names	Example geometry		Bracket	Product	Graph	Order
1	O(3)		Full symmetry of the sphere, $S^{2}$ , point		[4,3]			48
1b	SO(3)	Sphere group	Rotational symmetry		[4,3]⁺			24
2	O(2)×O(1) O(2)⋊C₂	Dih_∞×Dih₁ Dih_∞⋊C₂	Full symmetry of a circle $S^{1}$ , segment, { } spheroid, torus, cylinder, bicone, hyperboloid, capsule, lemon Full circular symmetry with half turn		[p,2]	[p]×[ ]		4p
2b	SO(2)×O(1)	C_∞×Dih₁	Rotational symmetry with reflection		[p⁺,2]	[p]⁺×[ ]		2p
2c	SO(2)⋊C₂	C_∞⋊C₂	Rotational symmetry with half turn		[p,2]⁺	([p]⁺×[ ])⁺		2p
3	O(2)×SO(1)	Dih_∞ Circular symmetry	Full symmetry of a hemisphere, spherical segment cone, paraboloid or any surface of revolution	, ,	[p,1]	[p]×[ ]⁺		2p
3b	SO(2)×SO(1)	C_∞ Circle group	Rotational symmetry	, ,	[p,1]⁺	([p]×[ ]⁺)⁺		p
4	O(1)×O(1)×O(1)×3!	[4,3]	Full of symmetry of cube, {4,3}		[3[2,2]] = [4,3]			48
4b	(O(1)×O(1)×2!)×O(1)	[4,2]	Full symmetry of square prism, {4}×{ }		[[2],2] = [4,2]	[4]×[ ]⁺		16
4c	O(1)×O(1)×O(1)	Dih³ ₁	Full of symmetry of rectangular cuboid, { }³		[2,2]	[ ]×[ ]×[ ]		8
4d	(O(1)×O(1)×O(1))/O(1)		Direct rectangular cuboid Rhombic disphenoid		[2,2]⁺	([ ]×[ ]×[ ])⁺		4
5	(O(1)×O(1)×2!)×SO(1)	Dih₄	Full symmetry of square pyramid, {4}∨( )		[[2],1] = [4,1]	[4]×[ ]⁺		8
5b	O(1)×O(1)×SO(1)	Dih² ₁ = Dih₂	Full symmetry of rectangle pyramid, { }²∨( )		[2,1]	[ ]×[ ]×[ ]⁺		4
5c	(O(1)×O(1)×SO(1))/O(1)	C₂	Direct rectangular pyramid		[2,1]⁺	([2]×[ ])⁺		2
6	(O(1)×(SO(1)×SO(1)×2!))×2!		Full symmetry of tetragonal disphenoid, Aut({ }∨{ })		[4,2⁺]			8
6b	O(1)×(SO(1)×SO(1)×2!)	Dih₁×Dih₁ = Dih₂	Full symmetry of tetragonal disphenoid, { }∨{ }		[2,1]	[ ]×[ ]×[ ]⁺		4
6c	O(1)×SO(1)×SO(1)	Dih₁	Full symmetry of mirrored sphenoid, { }∨( )∨( )		[1,1]	[ ]×[ ]⁺×[ ]⁺		2
6d	SO(1)×SO(1)×SO(1)	C₁	Scalene tetrahedron, ( )∨( )∨( )∨( )		[1,1]⁺	[ ]⁺×[ ]⁺×[ ]⁺		1

4D[edit]

Continuous symmetries in 4-space can be classified as products of orthogonal groups O(4) or special orthogonal groups SO(4) = O(4)/O(1), along with semidirect products with half turns, C₂.

#	Rank 4 groups	Other names	Example geometry	Example finite subgroup
#	Rank 4 groups	Other names	Example geometry	Bracket	Product	Graph	Order
1	O(4)		Full symmetry of the 3-sphere, $S^{3}$ , point	[4,3,3]			384
1b	SO(4)		Direct 3-sphere	[4,3,3]⁺			192
2	O(3)×O(1)		Full symmetry of spherinder, $S^{2}$ ×{ }, segment, { }	[4,3,2]	[4,3]×[ ]		96
2b	SO(3)×O(1)			[(4,3)⁺,2]	[4,3]⁺×[ ]		48
2c	SO(3)⋊C₂		Direct spherinder	[4,3,2]⁺	([4,3]×[ ])⁺		48
3	O(3)×SO(1)	O(3)	Full symmetry of hypercone, $S^{2}$ ∨( )	[4,3,1]	[4,3]×[ ]⁺		48
3b	SO(3)×SO(1)	SO(3)	Direct hypercone	[4,3,1]⁺	[4,3]⁺×[ ]⁺		24
4	O(2)×O(2)×2!		Extended full symmetry of duocylinder, aut( $S^{1}$ × $S^{1}$ )	[[p,2,p]]			8p²
4b	O(2)×O(2)	Dih² _∞	Full symmetry of duocylinder, $S^{1}$ × $S^{1}$	[p,2,q]	[p]×[q]		4pq
4c	O(2)×SO(2)	Dih_∞×C_∞		[p,2,q⁺]	[p]×[q]⁺		2pq
4d	SO(2)×SO(2)	C_∞×C_∞	Direct duocylinder	[p⁺,2,q⁺]	[p]⁺×[q]⁺		pq
4e	O(2)×Dih_n	Dih_∞×Dih_n	Full symmetry of circle-n-gon duoprism, $S^{1}$ ×{n}	[p,2,q]	[p]×[q]		4pq
4f	(O(2)×O(2))/O(1)	Dih² _∞/Dih₁	Direct duocylinder	[p,2,q]⁺	([p]×[q])⁺		2pq
4g	(O(2)×Dih_n)/O(1)	C_∞⋊C_n	Direct symmetry of circle-n-gon duoprism	[p,2,q]⁺	([p]×[q])⁺		2pq
4h	O(2)×(O(1)×O(1)×2!)	Dih_∞×Dih₄	Full symmetry of circle-square duoprism, $S^{1}$ ×{4}	[p,2,4]	[p]×[4]		16p
4i	O(2)×O(1)×O(1)	Dih_∞××Dih₂	Full symmetry of circle-rectangle duoprism, $S^{1}$ ×{ }²	[p,2,2]	[p]×[ ]×[ ]		8p
4j	(O(2)×O(1)×O(1))/O(1)	(Dih_∞×Dih² ₁)/O(1)	Direct	[p,2,2]⁺	([p]×[ ]×[ ])⁺		4p
4h	SO(2)×O(1)×O(1)	C_∞×Dih² ₁		[p⁺,2,2]	[p]⁺×[ ]×[ ]		4p
4i	SO(2)×(O(1)×O(1))/O(1)	C_∞×C₂	Direct	[p⁺,2,2⁺]	[p]⁺×[2]⁺		2p
5	O(2)×O(1)×SO(1)	Dih_∞×Dih₁	Full symmetry of cylinder pyramid, ( $S^{1}$ ×{ })∨( )	[p,2,1]	[p]×[ ]×[ ]⁺		4p
5b	SO(2)×O(1)×SO(1)	C_∞×Dih₁		[p⁺,2,1]	[p]⁺×[ ]×[ ]⁺		2p
5c	(SO(2)⋊C₂)×SO(1)	C_∞⋊C₂	Direct cylinder pyramid	[p,2,1]⁺	[p,2]⁺×[ ]⁺		4p
6	O(2)×(SO(1)×SO(1)×2!)	Dih_∞×Dih₁	Extended full symmetry of circle-segment pyramid, $S^{1}$ ∨{ }	[p,[1,1]]	[p]×[ ]⁺×[ ]⁺		4p
6b	O(2)×SO(1)×SO(1)	Dih_∞	Full symmetry of circle-segment pyramid, $S^{1}$ ∨( )∨( )	[p,1,1]	[p]×[ ]⁺×[ ]⁺		2p
6c	SO(2)×SO(1)×SO(1)	C_∞	Direct	[p,1,1]⁺	[p]⁺×[ ]⁺×[ ]⁺		p
7	(O(1)×O(1)×O(1)×O(1))×4!		4-cube, {4,3,3}	[4,3,3]			384
7b	O(1)×O(1)×(O(1)×O(1)×2!)		Cubic prism, {4,3}×{ }	[4,3,2]	[4,3]×[ ]		96
7c	(O(1)×O(1)×2!)×(O(1)×O(1)×2!)	Dih⁴ ₂	Duoprism, {4}×{4}	[4,2,4]	[4]×[4]		64
7d	O(1)×O(1)×O(1)×O(1)	Dih⁴ ₁	4-orthotope, { }⁴	[2,2,2]	[ ]×[ ]×[ ]×[ ]		16
7e	(O(1)×O(1)×O(1)×O(1))/O(1)	Dih⁴ ₁/Dih₁	Direct	[2,2,2]⁺	([ ]×[ ]×[ ]×[ ])⁺		8
8	(O(1)×O(1)×O(1)×3!)×SO(1)		cube pyramid, {4,3}∨( )	[4,3,1]	[4,3]×[ ]⁺		48
8b	O(1)×O(1)×O(1)×SO(1)	Dih³ ₁	cuboid pyramid, { }³∨( )	[2,2,1]	[ ]×[ ]×[ ]×[ ]⁺		8
8c	(O(1)×O(1)×O(1)×SO(1))/O(1)	Dih³ ₁/Dih₁	Direct	[2,2,1]⁺	([ ]×[ ]×[ ])⁺×[ ]⁺		4
9	(O(1)×O(1)×2!)×(SO(1)×SO(1)×2!))	Dih₄×Dih₁	square-segment pyramid, {4}∨{ }	[[2],[1,1]]	[4]×[ ]×[ ]⁺		16
9b	(O(1)×O(1)×2!)×SO(1)×SO(1)	Dih₄	square-segment pyramid, {4}∨( )∨( )	[[2],1,1]	[4]×[ ]⁺×[ ]⁺		8
9c	O(1)×O(1)×SO(1)×SO(1)	Dih₂	rectangle-segment pyramid, { }²∨( )∨( )	[2,1,1]	[ ]×[ ]×[ ]⁺×[ ]⁺		4
9d	(O(1)×O(1)×SO(1)×SO(1))/O(1)	C₂	Direct, rectangle-segment pyramid	[2,1,1]⁺	([ ]×[ ])⁺×[ ]⁺×[ ]⁺		2
10	O(1)×SO(1)×SO(1)×SO(1)	Dih₁	bilateral symmetry, { }	[1,1,1]	[ ]×[ ]⁺×[ ]⁺×[ ]⁺		2
10b	SO(1)×SO(1)×SO(1)×SO(1)	C₁	No symmetry	[1,1,1]⁺	[ ]⁺×[ ]⁺×[ ]⁺×[ ]⁺		1

5D[edit]

Continuous symmetries in 5-space can be classified as products of orthogonal groups O(5) or special orthogonal groups SO(5) = O(5)/O(1).

#	Rank 5 groups	Other names	Example geometry	Example finite subgroup
#	Rank 5 groups	Other names	Example geometry	Bracket	Product	Graph	Order
1	O(5)		Full symmetry of the 4-sphere, $S^{4}$ , point	[4,3,3,3]			3840
1b	SO(5)		Direct 4-sphere	[4,3,3,3]⁺			1920
2	O(4)×O(1)		Full symmetry of 3-sphere-segment prism, $S^{3}$ ×{ }, segment, { }	[4,3,3,2]	[4,3,3]×[ ]		768
2b	SO(4)×O(1)			[(4,3,3)⁺,2]	[4,3,3]⁺×[ ]		384
2c	SO(4)⋊C₂		Direct 3-sphere-segment prism	[4,3,3,2]⁺	([4,3,3]×[ ])⁺		384
3	O(4)×SO(1)	O(4)	Full symmetry of 3-sphere cone, $S^{3}$ ∨( )	[4,3,3,1]	[4,3,3]×[ ]⁺		384
3b	SO(4)×SO(1)	SO(4)	Direct 3-sphere cone	[4,3,3,1]⁺	[4,3,3]⁺×[ ]⁺		192
4	O(3)×O(1)×SO(1)		sphere-segment pyramid, $S^{2}$ ∨{ }	[4,3,2,1]	[4,3]×[ ]×[ ]⁺		96
4b	(O(3)×O(1)×SO(1))/O(1)		direct sphere-segment pyramid	[4,3,2,1]⁺	[4,3,2]⁺×[ ]⁺		48
4c	O(3)×SO(1)×SO(1)		sphere-segment pyramid	[4,3,1,1]	[4,3]×[ ]⁺×[ ]⁺		48
4d	(O(3)×SO(1)×SO(1))/O(1)		sphere-segment pyramid	[4,3,1,1]⁺	[4,3]⁺×[ ]⁺×[ ]⁺		24
5	O(3)×O(2)		sphere-circle prism, $S^{2}$ × $S^{1}$	[4,3,2,p]	[4,3]×[p]		96p
5b	(O(3)×O(2))/O(1)		Direct sphere-circle prism	[4,3,2,p]⁺	([4,3]×[p])⁺		48p
6	O(3)×(O(1)×O(1)×2!)		sphere-square prism, $S^{2}$ ×{4}	[4,3,2,4]	[4,3]×[4]		384
6b	O(3)×O(1)×O(1)		sphere-segment-segment prism, $S^{2}$ ×{ }×{ }	[4,3,2,2]	[4,3]×[ ]×[ ]		192
6c	O(3)×O(1)×SO(1)		(sphere-segment prism) pyramid, $S^{2}$ ×{ }∨( )	[4,3,2,1]	[4,3]×[ ]×[ ]⁺		96
6d	O(3)×(SO(1)×SO(1)×2!)		sphere-segment pyramid, $S^{2}$ ∨{ }	[4,3,1,2]	[4,3]×[ ]⁺×[ ]		96
6e	O(3)×SO(1)×SO(1)		sphere-point-point pyramid, $S^{2}$ ∨( )∨( )	[4,3,1,1]	[4,3]×[ ]⁺×[ ]⁺		48
7	O(2)×O(2)×O(1)		circle-circle-segment prism, aut( $S^{1}$ × $S^{1}$ )×{ }	[[p,2,p],2]	[p]×[p]×[ ]		16p²
7b	(O(2)×O(2)×2!)×O(1)		circle-circle-segment prism, $S^{1}$ × $S^{1}$ ×{ }	[p,2,q,2]	[p]×[q]×[ ]		8pq
7c	((O(2)×O(2)×2!)×O(1))/O(1)		Direct circle-circle-segment prism	[p,2,q,2]⁺	([p]×[q]×[ ])⁺		4pq
8	(O(2)×O(2)×2!)×SO(1)		circle-circle pyramid, aut( $S^{1}$ ∨ $S^{1}$ )	[[p,2,p],1]	[p]×[p]×[ ]⁺		8p²
8b	O(2)×O(2)×SO(1)		circle-circle pyramid, $S^{1}$ ∨ $S^{1}$	[p,2,q,1]	[p]×[q]×[ ]⁺		4pq
9c	(O(2)×O(2)×SO(1))/O(1)		Direct circle-circle pyramid	[p,2,q,1]⁺	([p]×[q])⁺×[ ]⁺		2pq
9	O(2)×O(1)×O(1)×SO(1)		(circle-segment prism)-segment pyramid, ( $S^{1}$ ×{ })∨{ }	[p,2,2,1]	[p]×[ ]×[ ]×[ ]⁺		8p
9b	(O(2)×O(1)×O(1)×SO(1))/O(1)		direct (circle-segment prism)-segment pyramid	[p,2,2,1]⁺	([p]×[ ]×[ ])⁺×[ ]⁺		8p
9c	O(2)×O(1)×SO(1)×SO(1)		(circle-segment prism)-point-point pyramid, ( $S^{1}$ ×{ })∨( )∨( )	[p,2,1,1]	[p]×[ ]×[ ]⁺×[ ]⁺		4p
9d	(O(2)×O(1)×SO(1)×SO(1))/O(1)		Direct (circle-segment prism)-point-point pyramid	[p,2,1,1]⁺	[p,2]⁺×[ ]⁺×[ ]⁺		2p
9e	O(2)×SO(1)×SO(1)×SO(1)		circle-point-point-point pyramid, $S^{1}$ ∨( )∨( )∨( )	[p,1,1,1]	[p]×[ ]⁺×[ ]⁺×[ ]⁺		2p
9f	(O(2)×SO(1)×SO(1)×SO(1))/O(1)		circle-point-point-point pyramid	[p,1,1,1]⁺	[p]⁺×[ ]⁺×[ ]⁺×[ ]⁺		p
10	(O(1)×O(1)×O(1)×O(1)×O(1))×5!		5-cube, {4,3,3,3}	[4,3,3,3]			3840
10b	O(1)×O(1)×O(1)×O(1)×O(1)		Tesseract prism, {4,3,3}×{ }	[4,3,3,2]	[4,3,3]×[ ]		768
10c	O(1)×O(1)×O(1)×O(1)×O(1)		cubic prism prism, {4,3}×{4}	[4,3,2,4]	[4,3]×[4]		384
10d	O(1)×O(1)×O(1)×O(1)×O(1)		cubic prism prism, {4,3}×{ }×{ }	[4,3,2,2]	[4,3]×[ ]×[ ]		192
10e	(O(1)×O(1)×2!)×(O(1)×O(1)×2!)×O(1)	Dih⁴ ₂	Duoprism, {4}×{4}×{ }	[4,2,4,2]	[4]×[4]×[ ]		128
10f	(O(1)×O(1)×2!)×(O(1)×O(1)×2!)×O(1)	Dih⁴ ₂	Duoprism, {4}×{4}∨{ }	[4,2,4,1]	[4]×[4]×[ ]⁺		64
10g	O(1)×O(1)×O(1)×O(1)×O(1)	Dih⁵ ₁	5-orthotope, { }⁵	[2,2,2,2]	[ ]×[ ]×[ ]×[ ]×[ ]		32
10h	(O(1)×O(1)×O(1)×O(1)×O(1))/O(1)	Dih⁵ ₁/Dih₁	Direct	[2,2,2,2]⁺	([ ]×[ ]×[ ]×[ ]×[ ])⁺		8
11	(O(1)×O(1)×O(1)×O(1)×4!)×SO(1)		tesseract pyramid, {4,3,3}∨( )	[4,3,3,1]	[4,3,3]×[ ]⁺		384
11b	O(1)×O(1)×O(1)×O(1)×SO(1)	Dih⁴ ₁	4-orthotope pyramid, { }⁴∨( )	[2,2,2,1]	[ ]× ]×[ ]×[ ]×[ ]⁺		16
11c	(O(1)×O(1)×O(1)×O(1)×SO(1))/O(1)	Dih⁴ ₁/Dih₁	Direct	[2,2,2,1]⁺	([ ]×[ ]×[ ]×[ ])⁺×[ ]⁺		8
12	O(1)×O(1)×O(1)×SO(1)×SO(1)	Dih₃	box-segment pyramid, { }³∨( )∨( )	[2,2,1,1]	[ ]×[ ]×[ ]×[ ]⁺×[ ]⁺		8
12b	(O(1)×(O(1)×O(1)×SO(1)×SO(1))/O(1)	C₃	Direct, box-segment pyramid	[2,2,1,1]⁺	([ ]×[ ]×[ ])⁺×[ ]⁺×[ ]⁺		4
13	O(1)×O(1)×SO(1)×SO(1)×SO(1)	Dih₂	{2}	[1,1,1,1]	[ ]×[ ]×[ ]⁺×[ ]⁺×[ ]⁺		4
13b	SO(1)×SO(1)×SO(1)×SO(1)×SO(1)	C₁	No symmetry	[1,1,1,1]⁺	[ ]⁺×[ ]⁺×[ ]⁺×[ ]⁺×[ ]⁺		1