Reciprocating the Regular Polytopes

For reciprocation with respect to a sphere ∑ x 2 = c in Euclidean n ‐space, there is a unitary analogue: Hermitian reciprocation with respect to an antisphere ∑ ūu = 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 + [ 2 Π]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{ 2 p } [ 2 { p } ] { 2 p } .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 ⩾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.