I was toying around with Desmos and I discovered something intriguing...

If you take a random polygon with X amount of sides...
Get the midpoint of each side...
And then draw a polygon using each midpoint a vertex...
And then continue these steps over and over...
No matter how random the first shape is you will always eventually end up with an equilateral version with the same amount of sides.

https://www.desmos.com/geometry/ws4lajz86s

Sure it isn't just an approximation 'cause floating percision issues?

This does seem to be known: "if r = 1⁄2, then the derived polygons are called midpoint polygons and tend to a shape with opposite sides parallel and equal in length." That's not quite what you've claimed (only opposite sides will be equal, rather than the entire shape being equilateral), but it is pretty close.

It seems pretty obvious that this is only true for polygons with an even number of sides, though, since a shape with an odd number of sides will always have one side that has no partner to be parallel with.

Apparently the limit shape of such a recurrence is an affine transformation of the regular polygon, if you want to pin it down with even more precision than just "opposite sides parallel and equal."

I dunno I just thought it was cool,¯\_(ツ)_/¯