called platonic solid fractals. In three dimensions, the fractals' volume is zero. Sierpinski tetrahedron A Sierpinski tetrahedron is formed by successive flakes of four regular tetrahedrons. Each flake is formed by placing a tetrahedron scaled by 1/2 in each corner. Its Hausdorff dimension is equal to , which is exactly equal to 2. On every face there is a Sierpinski triangle and infinitely many are contained within. Hexahedron flake A hexahedron, or cube, flake defined in the same way as the Sierpinski tetrahedron is simply a cube and is not interesting as a fractal. However, there are two pleasing alternatives. One is the Menger Sponge, where every cube is replaced by a three dimensional ring of cubes. Its Hausdorff dimension is ≈ 2.7268. Another hexahedron flake can be produced in a manner similar to the Vicsek fractal extended to three dimensions. Every cube is divided into 27 smaller cubes and the center cross is retained, which is the opposite of the Menger sponge where the cross is removed. However, it is not the Menger Sponge complement. Its Hausdorff dimension is ≈ 1.7712, because a cross of 7 cubes, each scaled by 1/3, replaces each cube. Octahedron flake An octahedron flake, or sierpinski octahedron, is formed by successive flakes of six regular octahedra. Each flake is formed by placing an octahedron scaled by 1/2 in each corner. Its Hausdorff dimension is equal to ≈ 2.5849. On every face there is a Sierpinski triangle and infinitely many are contained within. Dodecahedron flake A dodecahedron flake, or sierpinski dodecahedron, is formed by successive flakes of twenty regular dodecahedra. Each flake is formed by placing a dodecahedron scaled by in each corner. Its Hausdorff dimension is equal to ≈ 2.3296. Icosahedron flake An icosahedron flake, or sierpinski icosahedron, is formed by successive flakes of twelve regular icosahedra. Each flake is formed by placing an icosahedron scaled by in each corner. Its Hausdorff dimension is equal to ≈ 2.5819. See also List of fractals by Hausdorff dimension References External links Quadraflakes, Pentaflakes, Hexaflakes and more – includes Mathematica code to generate these fractals Javascript for covering the plane with 5-fold symmetric Pentaflake tiles. Fractals Fractal curves