© 1997 by London Mathematical Society
© The London Mathematical Society
Reciprocating the Regular Polytopes
Department of Mathematics, University of Toronto Toronto, Ontario M5S 1A1, Canada
Received 2 May 1995. Revision received 22 August 1995.
For reciprocation with respect to a sphere 
x2=c in Euclidean n-space, there is a unitary analogue: Hermitian reciprocation with respect to an antisphere 
u=c. This is now applied, for the first time, to complex polytopes.
When a regular polytope 
has a palindromic Schläfli symbol, it is self-reciprocal in the sense that its reciprocal ', with respect to a suitable concentric sphere or antisphere, is congruent to . The present article reveals that and ' usually have together the same vertices as a third polytope + and the same facet-hyperplanes as a fourth polytope – (where + and – are again regular), so as to form a ‘compound’, +[2]–. When the geometry is real, + is the convex hull of and ', while – is their common content or ‘core’. For instance, when is a regular p-gon {p}, the compound is
 |
The exceptions are of two kinds. In one, + and – are not regular. The actual cases are when 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 are the poles of its own facet-hyperplanes, so that , ', + and – all coincide. Then is said to be strongly self-reciprocal.