Vertex embeddings of regular polytopes
A polytope \(P\) admits a vertex embedding into a polytope \(Q\) if there exists a polytope \(P'\) similar to \(P\) such that \(V(P')\subseteq V(Q)\), where \(V(P)\) denotes the set of vertices of \(P\). Denote this relation by \(P\leq Q\). In this survey article, the authors present a unified proof of the following theorem, which classifies vertex embeddings among regular polytopes of the same dimension: Theorem. The only vertex embeddings of regular convex \(d\)-dimensional polytopes are the following: (a) For \(d=2\), a \(p\)-gon embeds into a \(q\)-gon if and only if \(p| q\). (b) For \(d=3\), simplex \(\leq\) cube \(\leq\) dodecahedron. (c) For \(d=4\), there is the following chain of vertex embeddings: cross-polytope \(\leq\) cube \(\leq\) \(24\)-cell \(\leq\) \(600\)-cell \(\leq\) \(120\)-cell; additionally, the cube embeds into the \(120\)-cell. (d) For \(d\geq2\), the following are equivalent: (i) the \((d-1)\)-dimensional simplex embeds into the \((d-1)\)-dimensional cube; (ii) the \(d\)-dimensional cross-polytope embeds into the \(d\)-dimensional cube; (iii) there exists a Hadamard matrix of degree \(d\), i.e., a \(d\times d\) matrix \(A\) with entries \(\pm1\) that satisfies \(AA^T=dI_d\), where \(I_d\) is the \(d\times d\) unit matrix. (e) For \(d>4\) there are no vertex embeddings of regular polytopes except those given in (d). The paper closes with some remarks on Hadamard matrices.
scientific article; zbMATH DE number 2060144