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.
Generalized Definitions Of Dimensions
Formulated by the early 20th-century German mathematician Felix Hausdorff, this definition is based on how the volume, V, of a region depends on its linear size, r. For ordinary three-dimensional space, V is proportional to r3. The exponent gives the number of dimensions. “Volume” can also refer to other measures of total size, such as area. For the Sierpi´nski gasket, V is proportional to r1.5850, reflecting the fact that this figure does not even fully cover an area.
This definition describes how things spread through a medium over time, be it an ink drop in a tank of water or a disease in a population. Each molecule of water or individual in the population has a certain number of closest neighbors, which determines the rate at which the ink or disease diffuses. In a three-dimensional medium, a cloud of ink grows in size as time to the 3/2 power. In the Sierpi´nski gasket, ink must ooze through a twisty shape, so it spreads more slowly—as time to the 0.6826 power, corresponding to a spectral dimension of 1.3652.
Applying the Definitions
In general, different ways to calculate the number of dimensions give different numbers, because they probe different aspects of the geometry. For some geometric figures, the number of dimensions is not fixed. For instance, diffusion may be a more complicated function than time to a certain power.
Quantum-gravity simulations focus on the spectral dimension. They imagine dropping a tiny being into one building block in the quantum spacetime. From there the being walks around at random. The total number of spacetime building blocks it touches over a given period reveals the spectral dimension.