Reciprocating the regular polytopes

For reciprocation with respect to a sphere Sigma x(2) = c in Euclidean n-space, there is a unitary analogue: Hermitian reciprocation with respect to an antisphere Sigma <(u)over bar u> = c. This is now applied, for the first time, to complex polytopes. When a regular polytope Pi has a palindromic Schlafli symbol, it is self-reciprocal in the sense that its reciprocal Pi', with respect to a suitable concentric sphere or antisphere, is congruent to Pi. The present article reveals that Pi and Pi' usually have together the same vertices as a third polytope Pi(+) and the same facet-hyperplanes as a fourth polytope Pi(-) (where Pi(+) and Pi(-) are again regular), so as to form a 'compound', Pi(+)[2 Pi]Pi(-). When the geometry is real, Pi(+) is the convex hull of Pi and Pi', while Pi(-) is their common content or 'core'. For instance, when Pi is a regular p-gon {p}, the compound is {2p}[2{p}]{2p}. The exceptions are of two kinds. In one, Pi(+) and Pi(-) are not regular. The actual cases are when Pi is an n-simplex {3, 3,...,3} with n greater than or equal to 4 or the real 4-dimensional 24-cell {3, 4, 3} = 2{3}2{4}2{3}2 or the complex 4-dimensional Witting polytope 3{3}3{3}3(3)3. The other kind of exception arises when the vertices of Pi are the poles of its own facet-hyperplanes, so that Pi, Pi', Pi(+) and Pi(-) all coincide. Then Pi is said to be strongly self-reciprocal.