A simplicial 4-arrangement of 33 hyperplanes (Q1093162)

We show that the projective d-arrangement \({\mathfrak C}^ d\) formed by the facet hyperplanes of a cross-polytope, its hyperplanes of mirror symmetry, and the hyperplane at infinity is simplicial precisely for \(d\leq 4.\) The arrangement \({\mathfrak C}^ 4\) is the only simplicial 4-arrangement presently known that does not lie in a natural sequence of analogous arrangements that are simplicial in each dimension. It has flat vector \({\mathfrak g}=<409,746\), \(290,33>\) and face vector \({\mathfrak f}=<409,4104,12336,14400,5760>\).