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. The polar reciprocation of this solid through this sphere is a convex form of the triakis icosahedron, with all faces tangent to the same sphere, now an inscribed sphere, with coordinates and dimensions that can be calculated as follows. Let denote the golden ratio. The short edges of this form of the triakis icosahedron have length and the long edges have length Its faces are isosceles triangles with one obtuse angle of and two acute angles of As a Catalan solid, its dihedral angles are all equal, 160°36'45.188". One possible set of 32 Cartesian coordinates for the vertices of the triakis icosahedron centered at the origin (scaled differently than the one above) can be generated by combining the vertices of two appropriately scaled Platonic solids, the regular icosahedron and a regular dodecahedron: Twelve vertices of a regular icosahedron, scaled to have a unit circumradius, with the coordinates Twenty vertices of a regular dodecahedron, scaled to have circumradius with the coordinates and Symmetry In any of its standard convex or non-convex forms, the triakis icosahedron has the same symmetries as a regular icosahedron. The three types of symmetry axes of the icosahedron, through two opposite vertices, edge midpoints, and face centroids, become respectively axes through opposite pairs of degree-ten vertices of the triakis icosahedron, through opposite midpoints of edges between degree-ten vertices, and through opposite pairs of degree-three vertices. See also Triakis triangular tiling for other "triakis" polyhedral forms. Great triakis icosahedron References Catalan solids Deltahedra