(20 equilateral; 60 isosceles)
|Edges||120 (2 types)|
|Vertices||42 (2 types)|
|Vertex configurations||(12) 35
|Symmetry group||Icosahedral (Ih)|
|Dual polyhedron||Truncated rhombic triacontahedron|
Its name comes from a topological construction from the icosidodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general icosahedral symmetry can be maintained even with the 12 order-5 vertices at a different distance from the center as the other 30.
It can also be topologically constructed from the icosahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices. From this construction, all 80 triangles will be equilateral, but faces will be coplanar.
Tripentakis icosidodecahedron, the Kleetope of the icosahedron, can be obtained by raising low pyramids on each equilateral triangular face on a pentakis icosidodecahedron. It has 120 isosceles triangle faces (2 types), 180 edges (3 types) and 62 vertices (3 types).
- George W. Hart, Sculpture based on Propellorized Polyhedra, Proceedings of MOSAIC 2000, Seattle, WA, August, 2000, pp. 61–70 
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
- Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings (p 284)