Here is a way to create a shape:
For convex regular shape S with n sides: add a convex regular shape with n+1 sides to the outer edge of each open side (I'll explain what an open side is in a moment) of S, with one of the sides of the shape being added having its midpoint on the midpoint of the side it's being added to, the sides of the two shapes being parallel, and the side of the added shape being no longer than the side of the shape you're adding to. Then, one could repeat the process for each shape of n+1 sides, adding shapes of n+2 sides. This could continue.
As an example, one could start with n=3, an equilateral triangle. On each side of the triangle, the 4-sided convex regular shape, a square, is added, with the midpoint of one side being on the midpoint of a side of the triangle. You can decide the length (above 0) of the sides of the square.
After this step, you could add pentagons to the sides of each square that do not rest on the triangle (these sides are open).
The shape will be considered complete if no separate convex regular shapes overlap.
My question is this: Given n=3, and the initial side length is 1, what is the maximum side length in a complete shape, iterated up to n=7, of the heptagons (7-sided shape)?
You will have: 1 triangle, 3 squares, 9 pentagons, 36 hexagons, and 180 heptagons.
How can you maximize the side length of the heptagons to keep the shape complete (ensure that nothing is overlapping)?