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%I #46 Feb 20 2021 09:57:21
%S 1,1,2,5,52,103769
%N Number of combinatorial types of n-dimensional parallelohedra.
%C a(n) is the number of topologically distinct shapes the Voronoi cell (or Vocell) of an n-dimensional lattice can have.
%C a(n) is the number of combinatorially distinct parallelotopes that tile R^n. Dirichlet proved a(2) = 2, Fedorov showed a(3) = 5, while a(4) = 52 is due to Delone as corrected by Stogrin, and a(5) = 103769 to Engel. - _Jonathan Sondow_, May 26 2017
%C The papers by Dutuor Sikiric, Garber et al say that actually a(5) = 110244. The claim that every parallelotope is a Voronoi cell of some lattice in R^n up to an affine transformation is a conjecture open for n > 5. - _Andrey Zabolotskiy_, Feb 20 2021
%D J. H. Conway, The Sensual Quadratic Form.
%D E. S. Fedorov, An Introduction to the Theory of Figures. Notices of the Imperial Petersburg Mineralogical Society, 2nd series, vol. 21, 1-279, 1885. (English translation in Symmetry of crystals, ACA Monograph no. 7, 50-131, 1971.)
%H David Austin, <a href="http://www.ams.org/samplings/feature-column/fc-2013-11">Fedorov's Five Parallelohedra</a>, AMS Feature Column, 2017.
%H B. N. Delaunay, <a href="http://mi.mathnet.ru/eng/izv5329">Sur la partition régulière de l'espace à 4-dimensions. Première partie</a>, Izv. Akad. Nauk SSSR Otdel. Fiz.-Mat. Nauk, 79-110, 1929.
%H B. N. Delaunay, <a href="http://mi.mathnet.ru/eng/izv5333">Sur la partition régulière de l'espace à 4-dimensions. Deuxième partie</a>, Izv. Akad. Nauk SSSR Otdel. Fiz.-Mat. Nauk, 145-164, 1929.
%H Lejeune G. Dirichlet, <a href="https://eudml.org/doc/147457">Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen</a>, J. reine angew. Math., 40 209-227 (1850); [Oeuvre Vl. II, p. 41-59].
%H Mathieu Dutour Sikirić, Alexey Garber, Achill Schürmann, Clara Waldmann, <a href="https://doi.org/10.1107/S2053273316011682">The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices</a>, Acta Crystallographica A72 (2016), 673-683; arXiv:<a href="https://arxiv.org/abs/1507.00238">1507.00238</a> [math.MG], 2015-2016.
%H Mathieu Dutour Sikirić, Alexey Garber, and Alexander Magazinov, <a href="https://doi.org/10.1137/18M1235004">On the Voronoi Conjecture for Combinatorially Voronoi Parallelohedra in Dimension 5</a>, SIAM J. Discrete Math., 34(4), 2481-2501 (2020).
%H P. Engel, <a href="https://doi.org/10.1107/S0108767300007145">The contraction types of parallelohedra in E^5</a>, Acta Cryst. A 56 (2000), 491-496.
%H Alexey Garber and Alexander Magazinov, <a href="https://arxiv.org/abs/1906.05193">Voronoi conjecture for five-dimensional parallelohedra</a>, arXiv:1906.05193 [math.CO], 2019-2020.
%H M. I. Stogrin, <a href="http://mi.mathnet.ru/tm3121">Regular Dirichlet-Voronoi partitions for the second triclinic group</a>, Trudy Matematicheskogo Instituta imeni V. A. Steklova, 123 (1973) [in Russian] = Proceedings of the Steklov Institute of Mathematics, 123 (1973).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Parallelohedron">Parallelohedron</a>
%e In 1 dimension: the Vocell is an interval (1 possible shape)
%e In 2 dimensions: a hexagon or rectangle (2 possible shapes)
%e In 3 dimensions: truncated octahedron, hexarhombic dodecahedron, rhombic dodecahedron, hexagonal prism, cuboid (5 possible shapes)
%Y Cf. A071881, A071882, A321015.
%K nonn,hard,nice,more
%O 0,3
%A _N. J. A. Sloane_, Jun 10 2002
%E Corrected by _J. H. Conway_, Dec 25 2003