A cube can be considered as having square two polar caps and a belt of 4 squares around its equator. The belt can be conveniently unfolded, in the style of a polyhedral net, into a strip, which is nice for a map projection. Also chess on a cube.
Similarly, a regular octahedron "on its side" can be seen as two triangular caps and a belt of six triangles in a strip. In the tetrahedron, the caps are line segments and a belt of 4 triangles. Probably not so good for a map projection, though I haven't seen it. The equator should go through the middle of the belt.
The regular dodecahedron has two pentagonal caps and two rows of pentagons as its belt. The belt no longer unfolds nicely into a contiguous strip. Choose the unfolding which keeps the equator continuous.
Icosahedron: two caps of 4 triangles each, a center one and 3 neighbors. The belt is a single row of triangles that weaves up and down between the caps.
This net of the truncated icosahedron (soccer ball) strongly suggests a belt and 2 pentagonal caps.
We can also consider points as caps, then all the faces as rows of belts.
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