A family of uniform polytopes with symmetric shadows (Q1090939)

\textit{H. S. M. Coxeter}'s paper ''The 600-cell \(\{\) 3,3,5\(\}\) as a shadow of \(4_{21}\), the \(E_ 8\) polytope'' [Math. Forsch. Oberwolfach, Tagungsber. 31 (1981)] stimulated the author to generalize the ideas used there. This paper mentioned above is an example of orthogonal projections from 2n-space to n-space. Therefore manipulating Coxeter diagrams with the help of Wythoff's construction, described e.g. by \textit{H. S. M. Coxeter} in ''Regular polytopes'', 2nd ed. (1963; Zbl 0118.359), 3rd ed. (1973), p. 86, 196, the author can provide a family of orthogonal projections of one uniform polytope onto another. After theoretical expositions there are presented a lot of interesting examples concerning uniform polytopes or honeycombs in Euclidean and non-Euclidean spaces. The main theorem is the author's proposition (3.4) which makes assertions about the images of a special map concerning Coxeter graphs with some nodes ringed according to Wythoff's construction.