OFFSET
0,3
COMMENTS
a(n) is the number of topologically distinct shapes the Voronoi cell (or Vocell) of an n-dimensional lattice can have.
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
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
REFERENCES
J. H. Conway, The Sensual Quadratic Form.
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.)
LINKS
David Austin, Fedorov's Five Parallelohedra, AMS Feature Column, 2017.
B. N. Delaunay, Sur la partition régulière de l'espace à 4-dimensions. Première partie, Izv. Akad. Nauk SSSR Otdel. Fiz.-Mat. Nauk, 79-110, 1929.
B. N. Delaunay, Sur la partition régulière de l'espace à 4-dimensions. Deuxième partie, Izv. Akad. Nauk SSSR Otdel. Fiz.-Mat. Nauk, 145-164, 1929.
Lejeune G. Dirichlet, Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen, J. reine angew. Math., 40 209-227 (1850); [Oeuvre Vl. II, p. 41-59].
Mathieu Dutour Sikirić, Alexey Garber, Achill Schürmann, Clara Waldmann, The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices, Acta Crystallographica A72 (2016), 673-683; arXiv:1507.00238 [math.MG], 2015-2016.
Mathieu Dutour Sikirić, Alexey Garber, and Alexander Magazinov, On the Voronoi Conjecture for Combinatorially Voronoi Parallelohedra in Dimension 5, SIAM J. Discrete Math., 34(4), 2481-2501 (2020).
P. Engel, The contraction types of parallelohedra in E^5, Acta Cryst. A 56 (2000), 491-496.
Alexey Garber and Alexander Magazinov, Voronoi conjecture for five-dimensional parallelohedra, arXiv:1906.05193 [math.CO], 2019-2020.
M. I. Stogrin, Regular Dirichlet-Voronoi partitions for the second triclinic group, Trudy Matematicheskogo Instituta imeni V. A. Steklova, 123 (1973) [in Russian] = Proceedings of the Steklov Institute of Mathematics, 123 (1973).
Wikipedia, Parallelohedron
EXAMPLE
In 1 dimension: the Vocell is an interval (1 possible shape)
In 2 dimensions: a hexagon or rectangle (2 possible shapes)
In 3 dimensions: truncated octahedron, hexarhombic dodecahedron, rhombic dodecahedron, hexagonal prism, cuboid (5 possible shapes)
CROSSREFS
KEYWORD
nonn,hard,nice,more
AUTHOR
N. J. A. Sloane, Jun 10 2002
EXTENSIONS
Corrected by J. H. Conway, Dec 25 2003
STATUS
approved