  Sperner's lemma is a statement about labeled triangulations of a simplex.
McLennan and Tourky (2007) provided a novel proof of Sperner's Lemma by
examining volumes of simplices in a triangulation under time-linear
simplex-linear deformation. We adapt a similar argument to prove Tucker's Lemma
on a triangulated cross-polytope $P$. The McLennan-Tourky technique does not
directly apply because this deformation may distort the volume of $P$. We
remedy this by inscribing $P$ in its dual polytope, triangulating it, and
considering how the volumes of deformed simplices behave.
