# How can this fractal shape perfectly cover a certain platonic solid?

The following fractal shape has a surprising property:

This two-dimensional shape can be folded onto the surface of a regular polyhedron (one of the five platonic solids) in a way that perfectly covers the entire surface of the polyhedron with no gaps and no overlaps.

How can this be done, and which one of the platonic solids can it be done to?

• That shape's west coast looks like a fractal Wales...
– smci
Jun 30, 2019 at 18:36

Preliminary analysis.

The triangular symmetry shows it has to be a polyhedron made of triangles. The icosahedron seems too complex compared to the fractal shown. It must be the tetrahedron or the octahedron.

After some playing around with Acorn, I came up with the answer:

It is the octahedron.

You see in red the ouline of the unfolded octahedron. The black regions A, B and C nicely fill in the white regions with the same name. The three black regions at the tips combine to form the face opposite to the center. That face is actually missing from the red outline.

• The final face consists of a 1/4 of a triangle at each tip, with a 1/4 of that triangle on an edge, with a 1/4 of that triangle on an edge, etc.... (1/4+1/16+...=1/(1-1/4)-1=1/3 and we have 3 such constructs).
– JMP
Jun 29, 2019 at 10:56