Conway symmetry for polygons
Polytile notation only concerns itself with cyclic/rotational symmetry. We may also be interested in reflectional symmetry. John H. Conway presented a system that applies to all planar polygons.
It uses a letter class followed by the symmetry order.
Conway defines 5 symmetry classes:
Reflectional symmetry orders are doubled from a gyration symmetry. We may add these names to qualify a given polytile: a1, gn, p2n, i2n, d2n symmetric.
We can inspect polytile notation to determine the symmetry. A p-tile, p:a1.a2 am^n, explicitly expresses the gyrosymmetric as gn, representing repeated sequences of angle indices.
A p-tile with reflectional symmetry will have the form: p:A.b1.A^-1.b2, where A is a chain of zero or more vertices, A^-1 is the same chain in reverse order, and b1 and b2 are zero or one vertices. A diasymmetric form will include both b1 and b2, an isosymmetric form only have one of them, and persymmetric form has neither.
© 2020 Created by Tom Ruen