  The principle of hierarchical design is a prominent theme in many natural
systems where mechanical efficiency is of importance. Here we establish the
properties of a particular hierarchical structure, showing that high mechanical
efficiency is found in certain loading regimes. We show that in the limit of
gentle loading, the optimal hierarchical order increases without bound. We show
that the scaling of material required for stability against loading to be
withstood can be altered in a systematic, beneficial manner through
manipulation of the number of structural length scales optimised upon. We
establish the relationship between the Hausdorff dimension of the optimal
structure and loading for which the structure is optimised. Practicalities of
fabrication are discussed and examples of hierarchical frames of the same
geometry constructed from solid beams are shown.
