The groups\symmetries\dihedral.py module helps studying and visualising symmetries in a regular polygon. A polygon is defined with the following characteristics:
- its number of sides;
- its state: the current order of its vertices;
- its symmetries: rotations union reflections.
Assuming the current state of a square is [1, 2, 3, 4], then applying 2 unit rotations will make this current state into [3, 4, 1, 2]. Similarly other rotations and reflections can be applied.
Nomenchlature. The symmetries are named in a polygon as follows:
- rotations of n units: denoted by
n; - reflection about vertex at index n: denoted by
(n, ); - reflection about axis passing between vertices at index n and n + 1: denoted by
(n, n+1).
You can use that module to see how a polygon's vertices change upon applying various symmetries. Work will be continued on it.
Import the module:
>>> from groups.symmetries import *and define a polygon:
>>> d9 = Dihedral(9) # note the case in Dihedral
>>> d9
D[1, 2, 3, 4, 5, 6, 7, 8, 9]Now, you can apply the symmetries as defined in ¶Nomenchlature above. Here, 2 unit rotations will be 2, a reflection about 4th vertex will be (3, ), and so on...
>>> d9.apply(
... [2, (3, )]
... )
[9, 8, 7, 6, 5, 4, 3, 2, 1]
>>> # which is ...
>>> d9.determine(_)
(4,)Now, let us try to visualise this. Type
>>> d9.visualise(
... [2, (3, )]
... )
...A turtle window should appear. Maximise it to see:
