THIS IS A SPECIAL ALGEBRAIC TOPOLOGY SEMINAR.  PLEASE NOTE DIFFERENT DAY (TUESDAY). \noindent Consider $F:\mathbf{R}^n\rightarrow\mathbf{R}^k$ given by $k$ quadratic forms and the varieties $V=F^{-1}(0)$ and $Z=V\cap S^{n-1}$. In 1984-86 I described the topology of the generic $Z$ when $k=2$ and the quadratic forms are simultaneously diagonalizable. In 2007 Sam Gitler showed me a construction called the \textit{polyhedral product functor} (due to him, Tony Bahri, Martin Bendersky, and Fred Cohen) that includes $Z$ as a special case and this old work gained new life: Sam and I obtained in 2008 new results about the case $k>2$ and after that many old and new problems have been solved and the work continues intensely until today. At the recent Conference in honour of Martin and Sam I told this story in some detail and presented new results about the singular case.  \medskip \noindent After recalling the parts of this story which are necessary to understand the rest of the talk, this time I will concentrate on recent work about cases where $Z$ has dihedral symmetry. In it I have run into unexpected difficulties of a number-theoretic character having to do with the Vandermonde matrix on the $n$-th roots of unity.