A071880 Number of combinatorial types of n-dimensional parallelohedra.

1, 1, 2, 5, 52, 103769

Offset: 0

Author

N. J. A. Sloane, Jun 10 2002

Keywords

• nonn
• hard
• nice
• more

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

Examples

			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)
		

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

Crossrefs

Cf. A071881, A071882, A321015.

Extensions

Corrected by J. H. Conway, Dec 25 2003