The soft condensed matter of biological organisms exhibits atomic motions whose properties depend strongly on temperature and hydration conditions. Due to the superposition of rapidly fluctuating alternative motions at both very low temperatures (quantum effects) and very high temperatures (classical Brownian motion regime), the dimension of an atomic "path" is in reality different from unity. In the intermediate temperature regime and under environmental conditions which sustain active biological functions, the fractal dimension of the sets upon which atoms reside is an open question. Measured values of the fractal dimension of the sets on which the hydrogen atoms reside within the azurin protein macromolecule are reported. The distribution of proton positions was measured employing thermal neutron elastic scattering from azurin protein targets. As the temperature was raised from low to intermediate values, a previously known and biologically relevant dynamical transition was verified for the azurin protein only under hydrated conditions. The measured fractal exponent of the geometrical sets on which protons reside in the biologically relevant temperature regime is given by D = (0.65+/-0.1). The relationship between fractal dimensionality and biological function is qualitatively discussed. (C) 2003 American Institute of Physics.