Rhombic Triacontahedron

Dual of the IcosiDodecahedron

The Rhombic Triacontahedron is difficult to see, because the 3-edged vertices tend to jump in and jump out. We therefore show an anaglyph image (use red-green stereo glasses).
On the usual images of the Rhombic Triacontahedron the 3-edged vertices jump in and out. This phenomenon occurs naturally with rhombic tessalations and has been used to create optical illusions showing paradoxical pictures.

The Rhombic Triacontahedron can be obtained from the dodecahedron by raising pentagon pyramids on its pentagon faces and it can be obtained from the icosahedron by raising triangular pyramids on its triangular faces. The resulting triacontahedra have the same size if the icosahedron edges are by the golden ratio (sqrt(5)+1)/2 longer than the dodecahedron edge lengths.
In the animation, face colors depend on the face normals; therefore one can see that in the last step neighbouring pyramid faces become coplanar, thus creating the rhombic faces of the triacontahedron with diagonal length ratio (sqrt(5)+1)/2.
The Rhombic Triacontahedron inside its dual, the icosidodecahedron (wireframe).
Its 5-edged vertices touch the midpoints of the icosidodecahedron pentagon faces. The length of the long diagonals of the rhombic faces equals the distance between the midpoints of neighboring pentagon faces of the icosidodecahedron.
The endpoints of the short diagonals, i.e. the 3-edged vertices, stay well inside the icosidodecahdron. To obtain these consider the inscribed sphere of the icosidodecahedron, it touches at the midpoints of the pentagons and the midpoints of the triangles are outside of this sphere. Inversion of the triangle midpoints in this sphere gives the 3-edged vertices of the rhombic triacontahedron. The pyramid construction above is simpler.