This is the first of three puzzlers for the Sonobe lover. (For partial background, see my earlier posts [1, 2, 3, gallery] on the Sonobe unit.)

**Here are two images of the same Sonobe construction. What is the underlying geometry?**

Keep reading for a few hints.

Hint #1: there are 90 units in the construction. Of these, 60 are purple and 30 are grey. (D would like to note that this color choice was her suggestion.) Also, all the units of the same color are interchangeable, in the sense that there is some rotation of the polyhedron that moves a given unit to the position of any other unit of the same color but leaves the overall object looking the same.

Hint #2: the symmetry group of the polyhedron is icosahedral.

Hint #3: I made a twin to the polyhedron from the same number of units connected in the same way. The difference is that the twin was made “inside-out,” so the pockets of the individual Sonobe units are on the inside, rather than the outside. The visual effect is that the little triangular pyramids that usually point outwards instead point inside the polyhedron. Also, the “ribs” of the individual units are readily visible.

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The underlying geometry could be a truncated icosahedron.

Hmm, if we truncate an icosahedron we do get 90 edges and icosahedral symmetry, but the faces are all hexagons and pentagons instead of triangles.

The truncated icosahedron I know is the soccer ball. The tiles of the surface are pentagons and hexagons.

Right, but in any Sonobe construction the underlying polyhedron has triangular faces. Not sure if you’re still pondering; if so, you should stop reading now (spoilers ahead :) ).

Truncated icosahedron is about as close a guess as possible — the actual underlying geometry is the dual polyhedron of the truncated icosahedron. Wikipedia calls it a Pentakis dodecahedron and it could also be called a stellated dodecahedron (using the sense of “stellated” that means “glue a pyramid to each face”, in this case a pyramid with equilateral triangular faces). In my model, the grey units are the edges of the dodecahedron and the purple units are the edges added by stellation.