  We discuss the construction of ADHM data for Yang-Mills instantons with the
symmetries of the regular polytopes in four dimensions. We show that the case
of the pentatope can be studied using a simple modification of the approach
previously developed for platonic data. For the remaining polytopes, we
describe a framework in which the building blocks of the ADHM data correspond
to the edges in the extended Dynkin diagram that arises via the McKay
correspondence. These building blocks are then assembled into ADHM data through
the identification of pairs of commuting representations of the associated
binary polyhedral group. We illustrate our procedure by the construction of
ADHM data associated with the pentatope, the hyperoctahedron and the 24-cell,
with instanton charges 4, 7 and 23, respectively. Furthermore, we show that
within our framework these are the lowest possible charges with these
symmetries. Plots of topological charge densities are presented that confirm
the polytope structure and the relation to JNR instanton data is clarified.
