recursive creation of an unlimited 7-gon

Question

hey, i wondered if there's an algorithm to create a polygon, composed of triangles, to look like this:

http://homepages.wmich.edu/~drichter/images/mathieu/numberedvertices.jpg

the numbering of the vertices is not important, just the method how to get the points. notice that every point is connected to exactly 7 others.

Answer 1

What you're looking for is something like this.

It's an example of an order-7 triangular tiling and closely related heptagonal tiling, which are in turn hyperbolic tilings/tessellations. You can compute these to arbitrary resolution as this video shows. (you can take pairs of white and black triangles in the video to get the equilateral-like triangles in your picture)

The basic idea is to set up three circles that intersect at the appropriate angle -- 2π/7 in this case -- and then to reflect the triangle you get ad infinitum. This is the basic construction behind Escher's famous Circle Limit pictures.

Let me know if you need more details.