Discrete & Computational Geometry

, Volume 13, Issue 3, pp 573-583

First online:

Polytopes that fill ℝn and scissors congruenceand scissors congruence

  • J. C. LagariasAffiliated withAT & T Bell Laboratories
  • , D. MoewsAffiliated withDepartment of Mathematics, University of California

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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.