A071881 -id:A071881 - cp's OEIS frontend

A071880 Number of combinatorial types of n-dimensional parallelohedra.

1, 1, 2, 5, 52, 103769

Offset: 0

N. J. A. Sloane, Jun 10 2002

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

			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)

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.)

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

Cf. A071881, A071882, A321015.

Corrected by J. H. Conway, Dec 25 2003

A071882 Number of contraction types of n-dimensional parallelohedra.

Original entry on oeis.org

1, 1, 2, 5, 52, 179372

Offset: 0

N. J. A. Sloane, Jun 10 2002

Or, number of inequivalent Delaunay [Delone] decompositions of R^n.

Sikiric et al. say that actually a(5) = 181394. - Andrey Zabolotskiy, Mar 02 2019

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.
P. Engel, The contraction types of parallelohedra in E^5, Acta Cryst. A 56 (2000), 491-496.
M. I. Stogrin, Regular Dirichlet-Voronoi partitions for the second triclinic group, Trudy Matematicheskogo Instituta imeni V. A. Steklova, 123 (1973) = Proceedings of the Steklov Institute of Mathematics, 123 (1973).

Cf. A071880, A071881.

A321015 Number of isohedral Voronoi parallelotopes in R^n.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 7, 4, 6, 3, 10, 3, 7, 6, 9, 3, 10, 3, 10, 7, 6, 3, 15, 5, 6, 6, 11, 3, 14, 3, 11, 6, 6, 8, 16, 3, 6, 6, 15, 3, 15, 3, 10, 10, 6, 3, 19, 6, 10, 6, 10, 3, 14, 7, 16, 6, 6, 3, 22, 3, 6, 11, 13, 7, 14, 3, 10, 6, 15, 3, 23, 3, 6, 10, 10, 8, 14, 3, 19, 8, 6, 3, 23, 7

Offset: 1

N. J. A. Sloane, Nov 04 2018

nonn

			Of the five different Voronoi cells of 3-dimensional lattices, only two are isohedral, so a(3) = 2: the cube and the rhombic dodecahedron, the Voronoi cells of the primitive cubic and the face-centered cubic lattices.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Marjorie Senechal, Introduction to lattice geometry. In M. Waldschmidt et al., eds., From Number Theory to Physics, pp. 476-495. Springer, Berlin, Heidelberg, 1992. See Cor. 3.7.

Cf. A000005, A001620, A321013, A321014, A071880, A071881, A071882.

d2:=proc(n) local c; if n <= 3 then return(0); fi;
c:=NumberTheory[tau](n)-1;
if (n mod 2)=0 then c:=c-1; fi;
if (n mod 3)=0 then c:=c-1; fi; c; end; # A321014
d3:=proc(n) local c;  c:=0;
if (n mod 6)=0 then c:=c+1; fi;
if (n mod 7)=0 then c:=c+1; fi;
if (n mod 8)=0 then c:=c+1; fi; c; end; # A321013
[seq(NumberTheory[tau](n)+d2(n)+d3(n),n=1..120)];

a(n) = 2*numdiv(n) + sum(k = 6, 8, !(n % k)) + n%2 + (n%3>0) - 3; \\ Amiram Eldar, Feb 02 2025

a(n) = d(n) + A321013(n) + A321014(n), where d(n) = A000005(n) is the number of divisors of n.

Sum_{k=1..n} a(k) ~ n * (2*log(n) + 4*gamma - 571/168), where gamma is Euler's constant (A001620). - Amiram Eldar, Feb 02 2025

A089301 Number of C-types (or contiguity types) of n-dimensional lattices.

Original entry on oeis.org

1, 1, 1, 1, 3, 76

Offset: 0

N. J. A. Sloane, following a suggestion of J. H. Conway, Dec 26 2003

Ryskov, S. S., C-types of n-dimensional parallelohedra, Dokl. Akad. Nauk SSSR 212 (1973), 46-49.
Ryskov, S. S. and Baranovskii, E. P., C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings), Trudy Mat. Inst. Steklov 137 (1976). English translation: Proc. Steklov Inst. Math. 1978, no. 4, 140 pp.

Cf. A071880, A071881, A071882.

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A140441 Erroneous version of A071881.

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A071880 Number of combinatorial types of n-dimensional parallelohedra.

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A071882 Number of contraction types of n-dimensional parallelohedra.

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A321015 Number of isohedral Voronoi parallelotopes in R^n.

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A089301 Number of C-types (or contiguity types) of n-dimensional lattices.

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