The element number of the convex regular polytopes (Q2430730)

In this paper the focus is on sets \(\Sigma\) of \(n\)-dimensional polytopes which are obtained as unions of a finite number of \(n\)-dimensional polytopes selected in a set \(\Omega\) (the element set), and identified along \((n-1)\)-dimensional faces. The element number of the set \(\Sigma\) of polyhedra is the minimum cardinality of the element sets for \(\Sigma\). It is shown that in \(4\)-dimensional space the element number of the six convex regular polyhedra is at least \(2\), and at most \(6\) since \(6\) is the number of regular \(4\)-polytopes. Also, if \(n\geq 5\), the element number of the \(n\)-dimensional regular polytopes is \(3\), unless \(n+1\) is a square number. In proving the results it plays an important role whether the dichoral angle of each polytope is \(\frac{2\pi}{n}\) (\(n\) integer), as well as to check whether the facet angles are \(Q\)-linear independent or not.