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Let P ⊂ R d be a d-dimensional polytope. The realization space of P is the space of all polytopes P ′ ⊂ R d that are combinatorially equivalent to P , modulo affine transformations. We report on work by the first author, which shows that realization spaces of 4-dimensional polytopes can be "arbitrarily bad": namely, for every primary semialgebraic set V defined over Z , there is a 4-polytope P (V ) whose realization space is "stably equivalent" to V . This implies that the realization space ofdoi:10.1090/s0273-0979-1995-00604-x fatcat:ndxu3zrrunhivljsl75rmfijbu