, Volume 13, Issue 1, pp 573-583

Polytopes that fill ℝn and scissors congruenceand scissors congruence

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Abstract

Suppose thatP is a (not necessarily convex) polytope in ℝ n that can fill ℝ n with congruent copies of itself. Then, except for its volume, all its classical Dehn invariants for Euclidean scissors congruence must be zero. In particular, in dimensions up to 4, any suchP is Euclidean scissors congruent to ann-cube. An analogous result holds in all dimensions for translation scissors congruence.