File:Truncated Platonic solid like a ball.svg

English: A Platonic icosahedron is truncated in a same manner on each vertex, each edge losing finally two thirds of its initial length. And the icosahedron becomes a polyhedron of thirty‑two faces and ninety edges of the same length denoted ${\displaystyle {\text{by}}\,t:\ }$ one third of the length of the thirty initial edges. The image shows ten of the twenty faces of the truncated solid: twenty regular hexagons, each on an equilateral face of the initial solid. The projection onto the drawing plane gives as outline to the icosahedron a regular decagon. The image shows a regular cross section of the initial icosahedron, not distorted by the projection, which contains five edges, and the five vertices of which are on the outline of the icosahedron. A homothety shrinks this cross section by a scale factor ${\displaystyle {\text{of }}\,{\tfrac {1}{3}},\,}$ in yellow at the drawing center, it is a face of the truncated solid not distorted by the projection. Twelve faces of the truncated icosahedron are convex regular pentagons, out of which six are visible in yellow.