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 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:

  • a=asymmetric (no symmetry),
  • g=gyrosymmetric (rotational symmetry only),
  • r=regular (reflections on vertices and edges),
  • p=persymmetric (reflections passing through edges),
  • d=diasymmetric (reflections pass through vertices), and
  • i=isosymmetry (reflections pass through both vertices and edges).

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.B.c.B^-1, where B is a chain of zero or more vertices, B^-1 is the same chain in reverse order, and a and c are zero or one vertices. A diasymmetric form will include both a and c, an isosymmetric form only have one of them, and persymmetric form has neither.

For achiral polygons, polytile partition notation uses a pipe operator |, as 3 partitions, |a|B|c|^n, allowing the reverse polychain B^-1 or bm.bm-1…b1 to be suppressed:

  1. rn symmetry: p:|a|||^p
  2. dn symmetry: p:|a|b1.b2…bm|c|^n
  3. in symmetry: p:|a|b1.b2…bm||^n
  4. pn symmetry: p:||b1.b2…bm||^n

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