The icosidodecahedron is a uniform polyhedron bounded by 12 pentagons and 20 triangles. It is edge-uniform, and its two kinds of faces alternate around each vertex, so it is also a quasi-regular polyhedron. It may be constructed by truncating the dodecahedron or the icosahedron at the midpoints of its edges.
Pasting the two rotundae together in
(ortho orientation) produces the pentagonal
orthobirotunda (J34), another Johnson solid.
The icosidodecahedron may also be bisected with a plane parallel to two opposite triangles; in this case, the result will no longer have regular faces. However, it can be modified to have regular faces by shrinking the six vertices that lie on the bisecting plane towards the center by the Golden Ratio. This modified bisection produces two triangular hebesphenorotundae (J92), another of the Johnson solids:
In order to be able to identify the icosidodecahedron in various projections of 4D objects, it is useful to know how it appears from various viewpoints. The following are some of the commonly-encountered views:
Parallel projection centered on a pentagonal face.
Parallel projection centered on a triangular face.
Vertex-centered parallel projection. The edges of the projection envelope are images of 4 pentagons (the longer edges) and 4 triangles (the shorter edges).
Edge-centered projection. The top and bottom edges of the projection envelope are the images of two pentagonal faces.
The Cartesian coordinates of the icosidodecahedron, centered on the origin and having edge length 2, are all permutations of coordinates of:
- (0, 0, ±2φ)
together with the even permutations of coordinates of:
- (±1, ±φ2, ±φ)
where φ=(1+√5)/2 is the Golden Ratio.
It is evident from the first set of coordinates that an octahedron can be inscribed in an icosidodecahedron.