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Based on an approach of Affentranger&Schneider we give an asymptotic formula for the expected number of $k$-faces of the orthogonal projection of a regular $n$-crosspolytope onto a randomly chosen isotopic subspace of fixed dimension, as $n$ tends to infinity. In particular, we present a precise asymptotic formula for the (spherical) volume of spherical regular simplices, which generalizes Daniel's formula.