Dodecadodecahedron

The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles.

It has four Wythoff constructions between four Schwarz triangle families: 2 | 5 5/2, 2 | 5 5/3, 2 | 5/2 5/4, 2 | 5/3 5/4, but represent identical results.

A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets:

However, this construction replaces the crossing pentagonal faces of the dodecadodecahedron with non-crossing sets of rhombi, so it does not produce the same internal structure.

As such, it is topologically a regular polyhedron of index two:[2][3] Graphs of the dodecadodecahedron (left) and its dual (right) drawn in Bring's curve.

3D model of a dodecadodecahedron
Animated truncation sequence from {5/2, 5} to {5, 5/2}