Disdyakis triacontahedron

Projected into a sphere, the edges of a disdyakis triacontahedron define 15 great circles.

Normalizing all vertices to the unit sphere gives a spherical disdyakis triacontahedron, shown in the adjacent figure.

The edges of the polyhedron projected onto a sphere form 15 great circles, and represent all 15 mirror planes of reflective Ih icosahedral symmetry.

Combining pairs of light and dark triangles define the fundamental domains of the nonreflective (I) icosahedral symmetry.

The edges of a compound of five octahedra also represent the 10 mirror planes of icosahedral symmetry.

The disdyakis triacontahedron has three types of vertices which can be centered in orthogonally projection: The disdyakis triacontahedron, as a regular dodecahedron with pentagons divided into 10 triangles each, is considered the "holy grail" for combination puzzles like the Rubik's cube.

[4] Since 2016, the Dice Lab has used the disdyakis triacontahedron to mass-market an injection-moulded 120-sided die.

[5] It is claimed that 120 is the largest possible number of faces on a fair die, aside from infinite families (such as right regular prisms, bipyramids, and trapezohedra) that would be impractical in reality due to the tendency to roll for a long time.

[6] A disdyakis triacontahedron projected onto a sphere is used as the logo for Brilliant, a website containing series of lessons on STEM-related topics.