In the context of the theory and computation of fixed points of continuous mappings, researchers have developed combinatorial analogs of Brouwer's fixed-point theorem on the simplex and on the n-cube. Although the simplex and the cube have different combinatorial properties regarding their boundaries, they are both instances of a simplotope, which is the cross-product of simplices. This paper presents three combinatorial theorems on the simplotope, and shows how each translates into some known and new results on the simplex and cube, including various forms of Sperner's lemma. Each combinatorial theorem also implies set covering lemmas on the simplotope, the simplex, and the cube, including the Generalized Covering lemma, the Knaster–Kuratowski–Mazurkiewicz Lemma, and a lemma of Freidenfelds.
