[1] It was first depicted, in a non-convex form with equilateral triangle faces, by Leonardo da Vinci in Luca Pacioli's Divina proportione, where it was named the icosahedron elevatum.

[2] The capsid of the Hepatitis A virus has the shape of a triakis icosahedron.

Depending on the height of these pyramids relative to their base, the result can be either convex or non-convex.

[2] This interpretation is also expressed in the name, triakis, which is used for the Kleetopes of polyhedra with triangular faces.

[4] In another of the non-convex forms of the triakis icosahedron, the three triangles adjacent to each pyramid are coplanar, and can be thought of as instead forming the visible parts of a convex hexagon, in a self-intersecting polyhedron with 20 hexagonal faces that has been called the small triambic icosahedron.

[6] Yet another non-convex form, with golden isosceles triangle faces, forms the outer shell of the great stellated dodecahedron, a Kepler–Poinsot polyhedron with twelve pentagram faces.

[7] Each edge of the triakis icosahedron has endpoints of total degree at least 13.

[9] The triakis icosahedron is a Catalan solid, the dual polyhedron of the truncated dodecahedron.

The truncated dodecahedron is an Archimedean solid, with faces that are regular decagons and equilateral triangles, and with all edges having unit length; its vertices lie on a common sphere, the circumsphere of the truncated decahedron.