There is a strong feeling among many researchers that certain physical phenomena are fractal in nature. The difficulty, in any given case, is to make precise what one means when one says that something is fractal. An example might help. The Cantor middle third set, discussed in detail here, is a fractal in the sense that its Hausdorff (now often called the fractal) dimension is between zero and 1; it is ln2/ln3, in fact. Now the spectral lines of the hydrogen atom seem to be fractal also, just from a casual observation of the self-similar nature of the various series. They certainly do not have all the properties of the Cantor set and so, we may well ask, in what sense are they fractal? The purpose of this paper is to give one answer to this question.

• Received 17 June 1988

DOI:https://doi.org/10.1103/PhysRevA.38.6362