In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as . It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol
Let be the smallest (most negative) zero of the polynomial , where is the golden ratio. Let the point be given by
Let the matrix be given by
is the rotation around the axis by an angle of , counterclockwise. Let the linear transformations be the transformations which send a point to the even permutations of with an even number of minus signs. The transformations constitute the group of rotational symmetries of a regular tetrahedron. The transformations , constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points are the vertices of a great snub icosahedron. The edge length equals , the circumradius equals , and the midradius equals .
For a great snub icosidodecahedron whose edge length is 1, the circumradius is
Its midradius is
The four positive real roots of the sextic in ,
are the circumradii of the snub dodecahedron (U<sub>29</sub>), great snub icosidodecahedron (U<sub>57</sub>), great inverted snub icosidodecahedron (U<sub>69</sub>), and great retrosnub icosidodecahedron (U<sub>74</sub>).