The Sierpinski Pyramid is a 3-dimensional version of a fractal cluster. It is formed by repeatedly cutting out pieces of a pyramid. To form it, you start with a pyramid (a tetrahedron to be more exact) and divide it into 8 identical pyramids. You then cut out all of the smaller pyramids except for the ones at the vertices:
You then do the same for the 4 pyramids left and continue infinitely. At the end, you will get the following picture:
You can easily calculate the fractal dimension of this fractal using the similarity method. Since there are 4 identical pyramids, each of which has to be magnified twice to get the entire figure, the dimension is log 4 / log 2, which is equal to 2. After an infinite number of iterations, the fractal becomes an infinite network of lines, whose topological dimension is 1. Thus, the Sierpinski Pyramid is indeed a fractal, since the fractal dimension is greater than the topological dimension.
After an infinite number of iterations, the fractal will be an infinitely complex network of lines. The overall length of the lines will be infinite. This is true for all fractals whose fractal dimension is greater than 1.
As you can see from the picture, the figure formed at the surface of every face of the cube is the Sierpinski Triangle. The area of the Sierpinski Triangle is equal to 0. Because of this, the surface area of the Sierpinski Pyramid also has to be 0.
After every iteration, we leave only 1/2 of the volume of the pyramid. If we continue taking away 1/2 of the volume an infinite number of times, the volume will be approaching 0. That is the volume of the Sierpinski Pyramid.