dihedral

A dihedral group D_n is a mathematical group structure representing the symmetries acting on the vertices of a regular n-gon. For example, D₃ represents the symmetries of a triangle.

This python class generates the group structure of D_n for any n, and contains methods for generating + verifying subgroups as well as applying transformations to the vertices.

Class structure

Rotation actions are stored in the class variable r as an array r[0], ..., r[n / 2] where the identity is r[0]. Reflections are stored in the same way in the class variable s. Vertices are stores as numpy arrays in v, but can be accessed in a more readable integer format using the method vertices.

Acting on the vertices

Vertices and actions of the group D₃ are shown below as an example. The dotted lines denote reflections, and the arrow denoting R₁ is a 120 degree turn. R₂ would then be a 240 degree turn.

from dihedral import D

d3 = D(3)

# List the vertices in the group
d3.vertices()
# [0, 1, 2]

# Perform a rotation of 240 degrees on the vertices
d3.apply(d3.r[2])
# [2, 0, 1]

# Rotate vertex 0 by 120 degrees
d3.apply(d3.r[1], [0])
# [1]

Composing symmetries

Symmetries can be combined using the compose method.

# Compose a rotation with reflection (s_2(r_1))
composed_symmetry = d3.compose([d3.s[2], d3.r[1]])

# Apply the symmetry to vertex 1
d3.apply(composed_symmetry, [1])
# [2]

Finding subgroups

The method subgroups returns all subgroups in the dihedral group by generating all possible subsets of the group and verifying relevant group properties (identity, closure).

d3.subgroups()
# [['r0'], ['r0', 's0'], ['r0', 's1'], ['r0', 's2'], ['r0', 'r1', 'r2'], ['r0', 'r1', 'r2', 's0', 's1', 's2']]

Specific subsets can also be verified as being a subgroup or not using the has_subgroup method. This method takes a list of actions or another D instance.

d6 = D(6)
d3 = D(3)

d6.has_subgroup(d3)
# True

d6.has_subgroup([d6.r[0], d6.r[1]])
# False

d6.has_subgroup([d6.r[0]])
# True

Reducing operations

While compose combines operations, they can be reduced to a readable format using reduce_operations. For example,

# Reduce r_1(s_2)
d3.reduce_operations(([d3.r[1], d3.s[2]]))
# s0