Scalar and vector problems of diffraction by regular polyhedrons (Platonic
bodies)-tetrahedron, cube, octahedron, dodecahedron, and icosahedron-are co
nsidered. On the basis of structural relationships of the geometric analysi
s of invariant operators on regular polyhedrons and formalism of induced re
presentations, the canonical forms of boundary equations are determined, wh
ich are similar to the equations of the classical method of separation of v
ariables in linear boundary value problems with symmetries. The results of
realization of canonical forms in the scalar problem of diffraction of a pl
ane wave by an octahedron with the wave dimensions ka less than or equal to
50 are presented.