The whole of a self-similar object has the same shape as one or more of the parts; thus, self-similarity is a typical property of fractals. A state of constant proportion, or self-similarity, is called isometry. In geography, these similarities are not exact, but statistical; it’s therefore necessary to agree upper and lower limits to the scales involved, and sometimes the nature of the similarity will depend on the scale, since different processes may operate at different scales; see scaling law. Mandelbrot (1967) Science 156, 636 saw self-similarity in the curves of coastlines, but Andrle (1996) ESPL 21, 955 finds that the west coast of Britain is not statistically self-similar over the range of scale of measurement. Baas (2002) Geomorph. 48, 309 notes that theories of self-similarity are controversial, ‘especially with regard to their application to real physical systems’ and that ‘the peak of popular research interest into chaos, fractals, and self-organization has passed’. Nonetheless, Gaucherel et al. (2011) ESPL 36, 10, 1313 argue that self-similar dimension has been checked using either the whole channel network or a single channel link, and uses a new approach to check self-similarity at intermediate scales.