I was thinking about Sierpinski triangles recently, and trying to relate it to the 3 4 5 triangle. The 3 side has a Sierpinski triangle, with 9 segments, equaling 3^2. The 4 side has an equivalent "Sierpinski square", with 16 segments, equaling 4^2 (basically four squares touching at the corners with an empty square space in the middle, forming a cross shape). Then I was going to show how the two shapes could be combined to equal 5^2, or come up with its own shape (a house seems best, as the regular pentagon tiles the plane so the line totals don't work out).

As of yet, fruitless. It's a cute little project that shades of the standard presentation of three squares forming a center 3 4 5 triangle, with an interesting twist.