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.

Orthogonal groups
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... un. 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
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 An 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
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:
 * 1) Products are orthotopes (prisms)
 * 2) Sums are orthoplexes (bipyramids or fusils)
 * 3) Products and sums are Hanner polytopes
 * 4) Join are simplexes (pyramids)

Continuous symmetry objects are constructed by:
 * 1) Products are cylinders: circle×segment (3D), circle×circle (4D), sphere×segment (4D), sphere×circle (5D), etc.
 * 2) Sums are bicones: circle+segment (3D), circle+circle (4D), sphere+segment (4D), sphere+circle (5D), etc.
 * 3) 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
O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih1. SO(1) is just the identity.

2D
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).

3D
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.

4D
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.

5D
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).