Sierpiński fractal | Sierpiński sponge | Sierpiński tetrahedron

The tetrix is the three-dimensional analog of the Sierpiński sieve illustrated above, also called the Sierpiński sponge or Sierpiński tetrahedron. The nth iteration of the tetrix is implemented in the Wolfram Language as SierpinskiMesh[n, 3]. Let N_n be the number of tetrahedra, L_n the length of a side, and A_n the fractional volume of tetrahedra after the nth iteration. Then N_n | = | 4^n L_n | = | (1/2)^n = 2^(-n) A_n | = | L_n^3 N_n = (1/2)^n.

chaos game | Menger sponge | Sierpiński sieve