Say I have a polygon drawn on the face of a clock. Is there a way to quantify how symmetrical is the polygon and its subset polygons?
In the attached image I illustrate the intuition with quadrilateral, but ideally I'd like to generalize this to any kind of polygon.
Can anyone think of a clever way to solve this?
EDIT:
I have managed to solve this problem via computer where I simply express the ratio between the number of total edges and polygons of the shape, with the number of distinct edges and polygons of the shape.
So for shape 1 of my example, there are 4 kinds of edges and polygons ([0 3], [0 6], [0 3 6], and [0 3 6 9]). Each of these distinct elements each manifests 4 times in the shape, to make a total of 16 sub-polygons/edges that can be constructed from the whole. For instance, [0 3] appears by drawing clockwise from nodes 0 to 3, 3 to 6, 6 to 9, and 9 to 0.
Therefore, dividing the total number of polygons and edges (16) by the number of distinct polygons and edges (4), then dividing by the cardinality of the shape expresses what I am attempting to quantify quite well: 16/4 * 1/4 =1 (perfectly symmetrical).
In my 5th example, in the polygon [0 1 3 7] there are 11 distinct edges and polygons, and 12 total edges and polygons ([0,6] can be drawn moving clockwise from 1 to 7, or from 7 to 1). Therefore that polygon I get 11/12 * 1/4 = ~0.0303
This measure as far as I can tell perfectly captures the kind of measure I am after. With that being said, it is very computationally intensive.
Perhaps there's a more clever way to calculate this without iterating?