This story is a supplement to the feature "Using Causality to Solve the Puzzle of Quantum Spacetime" which was printed in the July 2008 issue of Scientific American.

A Whole New Dimension to Space
In everyday life the number of dimensions refers to the minimum number of measurements required to specify the position of an object, such as latitude, longitude and altitude. Implicit in this definition is that space is smooth and obeys the laws of classical physics.

But what if space is not so well behaved? What if its shape is determined by quantum processes in which everyday notions cannot be taken for granted? For these cases, physicists and mathematicians must develop more sophisticated notions of dimensionality. The number of dimensions need not even be an integer, as in the case of fractals—patterns that look the same on all scales.

Cantor Set : Take a line, chop out the middle third and repeat ad infinitum. The resulting fractal is larger than a solitary point but smaller than a continuous line. Its Hausdorff dimension [see next page] is 0.6309.

Sierpinski Gasket: A triangle from which ever smaller subtriangles have been cut, this figure is intermediate between a one-dimensional line and a 2-D surface. Its Hausdorff dimension is 1.5850.

Menger Sponge: A cube from which subcubes have been cut, this fractal is a surface that partially spans a volume. Its Hausdorff dimension is 2.7268, similar to that of the human brain.