John Conway discovered a right triangle that can be partitioned into five similar triangles. The sides are in proportion 1 : 2 : √5.
You can make a larger similar triangle by making the entire triangle the central (green) triangle of a new triangle.
Here’s the same image with the small triangles filled in as in the original.
Repeating this process creates an aperiodic tiling of the plane.
The tiling was discovered by Conway, but Charles Radin was the first two describe it in a publication [1]. Radin attributes the tiling to Conway.
Related posts
[1] Charles Radin. “The Pinwheel Tilings of the Plane.” Annals of Mathematics, vol. 139, no. 3, 1994, pp. 661–702.