  We investigate the hierarchies of half-supersymmetric branes in maximal
supergravity theories. By studying the action of the Weyl group of the
U-duality group of maximal supergravities we discover a set of universal
algebraic rules describing the number of independent 1/2-BPS p-branes, rank by
rank, in any dimension. We show that these relations describe the symmetries of
certain families of uniform polytopes. This induces a correspondence between
half-supersymmetric branes and vertices of opportune uniform polytopes. We show
that half-supersymmetric 0-, 1- and 2-branes are in correspondence with the
vertices of the $k_{21}$ , $2_{k1}$ and $1_{k2}$ families of uniform polytopes,
respectively, while 3-branes correspond to the vertices of the rectified
version of the $2_{k1}$ family. For 4-branes and higher rank solutions we find
a general behavior. The interpretation of half-supersymmetric solutions as
vertices of uniform polytopes reveals some intriguing aspects. One of the most
relevant is a triality relation between 0-, 1- and 2-branes.
