Sierpinski Carpet-Triangle
Subdividing an equilateral triangle into four equilateral triangles,
removing the middle triangle, and recursing leads to the Sierpinski
triangle. In three dimensions, a similar construction based on cubes
produces the Sierpinski sponge a nd the Menger sponge.
A variation of the Sierpinski carpet, called the Wallis sieve, starts in
the same way, by subdividing the unit triangle into nine smaller
triangle and removing the middle of them. At the next level of
subdivision, it subdivides each of the triangle into 25 smaller
triangle and removes the middle one, and it continues at the ith step
by subdividing each triangle into (2i + 1)2 smaller triangle and
removing the middle one.