Abstract
Suppose thatP is a (not necessarily convex) polytope in ℝn that can fill ℝn 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.
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References
A. D. Alexandrov, On completion of a space by polyhedra,Vestnik Leningrad Univ. Ser. Mat. Fiz. Him. 9 (1954), 33–43 (in Russian).
A. Bezdek and W. Kuperberg, Examples of space-tiling-polyhedra related to Hilbert's problem 18, question 2, in:Topics in Combinatorics and Graph Theory (R. Bodendick and R. Henn, eds.) Physica-Verlag, Heidelberg, 1990, pp. 87–92.
V. G. Boltianskii,Hilbert's Third Problem, Wiley, New York, 1978.
L. Danzer, Three-dimensional analogs of the Planar penrose tilings and quasicrystals,Discrete Math. 76 (1989), 1–7.
L. Danzer, Eine Schar von 1—Steinen, die den\(\mathbb{E}^3 \) seitentreu pflastern, aber weder periodisch, noch quasiperiodisch, Preprint, 1993.
M. Dehn, Ueber raumgleiche Polyeder,Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., 1900, pp. 345–354.
M. Dehn, Ueber den Rauminhalt,Math. Ann. 55 (1902), 465–478.
J. L. Dupont, Algebra of polytopes and homology of flag complexes,Osaka J. Math. 19 (1982), 599–641.
M. Goldberg, Tetrahedra equivalent to cubes by dissection,Elem. Math. 13 (1958), 107–109.
M. Goldberg, Two more tetrahedra equivalent to cubes by dissection,Elem. Math. 24 (1969), 130–132, Correction:25 (1970), 48.
M. Goldberg, Three infinite families of tetrahedral space-fillers,J. Combin. Theory Ser. A 16 (1974), 348–354.
H. Hadwiger, Ergänzungsgleichheitk-dimensionaler Polyeder,Math. Z. 55 (1952), 292–298.
H. Hadwiger,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin, 1957.
H. Hadwiger, Translative Zerlegungsgleichheit der Polyeder des gewöhnlichen Raumes,J. Reine Angew. Math. 233 (1968), 200–212.
D. Hilbert,Grundlagen der Geometrie, Teubner, Leipzig, 1899. (10th edition 1968.)
D. Hilbert, Mathematische Probleme,Göttinger Nachrichten, 1900, pp. 253–297. (Translation:Bull. Amer. Math. Soc. 8 (1902), 437–479. Reprinted in: Proceedings of Symposia in Pure Mathematics, Vol. 28, American Mathematical Society, Providence, RI, 1976, pp. 1–34).
M. J. M. Hill, Determination of the volumes of certain species of tetrahedra without employment of the method of limits,Proc. London Math. Soc. 27 (1896), 39–53.
M. Jaric (ed.),Aperiodicity and Order, Vol. 2, Academic Press, New York, 1989.
B. Jessen, The algebra of polyhedra and the Dehn-Sydler theorem,Math. Scand. 22 (1968), 241–256.
B. Jessen, Zur Algebra der Polytope,Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., 1972, pp. 47–53.
B. Jessen and A. Thorup, The algebra of polytopes in affine spaces,Math. Scand. 43 (1978), 211–240.
A. Katz, Theory of matching rules for the 3-dimensional Penrose tilings,Comm. Math. Phys. 118 (1988), 263–288.
R. B. Kershner, On paving the plane,Amer. Math. Monthly 75 (1968), 839–844.
P. Kramer, Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells,Acta. Cryst. Sect. A,38 (1982), 257–264.
W. Kuperberg, Knotted lattice-like space fillers, Preprint, 1993.
P. McMullen, Convex bodies which tile space by translation,Mathematika 27 (1980), 113–121. (Acknowledgment of priority,Mathematika 28 (1981) 192.)
J. Milnor, Hilbert's problem 18: on crystallographic groups, fundamental domains and on sphere packing, in:Mathematical Developments Arising from Hilbert Problems, Proceedings of Symposia on Pure Mathematics, Vol. 28, American Mathematical Society, Providence, RI, 1976, pp. 491–506.
K. Reinhardt, Zur Zerlegung der euklidische Räume in kongruente Polytope,Sitzungsb. Akad. Wiss. Berlin, 1928, pp. 150–155.
C.-H. Sah,Hilbert's Third Problem: Scissors Congruence, Pitman, San Francisco, 1979.
P. Schmitt, An aperiodic prototile in space, Unpublished notes, University of Vienna, 1988.
M. Senechal and J. Taylor, Quasicrystals: the view from Les Houches,Math. Intelligencer 12 (1990), 54–64.
D. M. Y. Sommerville, Space-filling tetrahedra in Euclidean space,Proc. Edinburgh Math. Soc. 41 (1923), 49–57.
D. M. Y. Sommerville, Division of space by congruent triangles and tetrahedra,Proc. Roy Soc. Edinburgh 43 (1923), 85–116.
S. K. Stein, A symmetric star body that tiles but not as a lattice,Proc. Amer. Math. Soc. 36 (1972), 543–548.
J.-P. Sydler, Sur la décomposition des polyèdres,Comment. Math. Helv. 16 (1943/44), 266–273.
J.-P. Sydler, Sur les tétraèdres équivalents à un cube,Elem. Math. 11 (1956), 78–81.
J.-P. Sydler, Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions,Comm. Math. Helv. 40 (1965), 43–80.
S. Szabó, A star polyhedron that tiles but not as a fundamental region, in: Colloquia Mathematica Societatis Janos Bolyai, Vol. 48, North-Holland, Amsterdam, 1985, pp. 531–544.
B. A. Venkov, On a class of Euclidean polyhedra,Vestnik Leningrad Univ. Ser. Math. Fiz. Him. 9 (1954), 11–31 (in Russian).
V. B. Zylev, Equicomposability of equicomplementable polyhedra,Soviet Math. Dokl. 161 (1965), 453–455.
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Lagarias, J.C., Moews, D. Polytopes that fill ℝn and scissors congruenceand scissors congruence. Discrete Comput Geom 13, 573–583 (1995). https://doi.org/10.1007/BF02574064
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DOI: https://doi.org/10.1007/BF02574064