The two-dimensional Sierpinski gasket can be created as follows:

1. Assign the value 0 to i.
2. Define the (three) vertices of a triangle.
3. Select a point inside of the triangle, p_i, "at random".
4. Select a vertex, v, "at random".
5. Increase i.
6. Defined p_i as the mid-point of the line defined by v and p_i-1.
7. If not done, return to step 3.

In three dimensions the process is essentially the same except that you start with a tetrahedron. In both cases, the result is a successive subdivision of the original shape.

Letting n denote the the "subdivision factor" and k denote the number of similar objects created by the subdivision process, he fractal dimension is usually defined as:

Intuitively, the fractal dimension of an object is how many similar objects we create when we subdivide it.

In the case of the two-dimensional Sierpinski gasket, each time we subdivide a side by a factor of 2 we keep 3 of the 4 triangles created. Thus, d = (ln 3)/(ln 2) = 1.58496. In the case of the three-dimensional Sierpinski gasket, each time we subdivide a side by a factor of 2 we keep 4 of the tetrahedra. Thus, d = (ln 4)/(ln 2) = 2.