Triakis icosahedron

In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron. It has also been called the kisicosahedron. 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. The capsid of the Hepatitis A virus has the shape of a triakis icosahedron. The triakis icosahedron can be formed by gluing triangular pyramids to each face of a regular icosahedron. Depending on the height of these pyramids relative to their base, the result can be either convex or non-convex. This construction, of gluing pyramids to each face, is an instance of a general construction called the Kleetope; the triakis icosahedron is the Kleetope of the icosahedron. This interpretation is also expressed in the name, triakis, which is used for the Kleetopes of polyhedra with triangular faces. When depicted in Leonardo's form, with equilateral triangle faces, it is an example of a non-convex deltahedron, one of the few known deltahedra that are isohedral (meaning that all faces are symmetric to each other). 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. Alternatively, for the same form of the triakis icosahedron, the triples of coplanar isosceles triangles form the faces of the first stellation of the icosahedron. 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. Each edge of the triakis icosahedron has endpoints of total degree at least 13. By Kotzig's theorem, this is the most possible for any polyhedron.