RECIPROCATING THE REGULAR POLYTOPES

Abstract

For reciprocation with respect to a sphere [sum ]x²=c in Euclidean n-space, there is a unitary analogue: Hermitian reciprocation with respect to an antisphere [sum ]uu=c. This is now applied, for the first time, to complex polytopes.

When a regular polytope Π has a palindromic Schläfli symbol, it is self-reciprocal in the sense that its reciprocal Π′, with respect to a suitable concentric sphere or antisphere, is congruent to Π. The present article reveals that Π and Π′ usually have together the same vertices as a third polytope Π⁺ and the same facet-hyperplanes as a fourth polytope Π⁻ (where Π⁺ and Π⁻ are again regular), so as to form a ‘compound’, Π⁺[2Π]Π⁻. When the geometry is real, Π⁺ is the convex hull of Π and Π′, while Π⁻ is their common content or ‘core’. For instance, when Π is a regular p-gon {p}, the compound is

formula here

The exceptions are of two kinds. In one, Π⁺ and Π⁻ are not regular. The actual cases are when Π is an n-simplex {3, 3, ..., 3} with n[ges ]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 Π are the poles of its own facet-hyperplanes, so that Π, Π′, Π⁺ and Π⁻ all coincide. Then Π is said to be strongly self-reciprocal.