Icosidodecahedron

According to Wikipedia:

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, one of the quasi-regular polyhedra.

And, oh yes, this one is origami. No, really.

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