Fractal structure produces fractal electrons with fractal energies

Fractals are a source of endless fascination to me. Life itself relies on many of the concepts of fractals: trees are fractal, as are feathers, coast lines, and many other things in nature. Indeed, it was the realization that landscapes are fractal that made modern animated films possible.
But, what about fractional dimensions in quantum mechanics? It turns out that electrons in a Sierpiński gasket are also fractal, and that has some pretty cool consequences.
A fractal is a weird beast. A line is 1D, a square is 2D, and a cube is 3D: dimensions come in integer quantities. Except they don't. For instance, it is possible to create a shape that has a finite area, but a perimeter that is infinitely long (the construction of such a shape is pictured below). A shape with these properties does not behave like a 2D object, but it's not a 3D object. Instead, it is a two-and-a-bit-D object. That is a fractal.
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