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A primary parallelohedron is a polyhedron that can fill space with infinite translated copies.

It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, Tutton, A. E. H. Crystallography and Practical Crystal Measurement, 2nd ed. London: Lubrecht & Cramer, 1964.) that the primary parallelohedra are the cube, hexagonal prism, elongated dodecahedron, rhombic dodecahedron, and truncated octahedron.

Is there is a classification for any higher dimensions? What are the primary d-parallelotopes?

The following is a conjecture of mine regarding the case of $d=4$.

Conjecture: There are exactly 7 primary 4-parallelotopes:

(1) Hypercube

(2) 16-cell

(3) 24-cell

(4) Hexagonal Square Duoprism

(5) Prismatic Elongated Dodecahedron

(6) Prismatic Rhombic Dodecahedron

(7) Prismatic Truncated Octahedron

EDIT: Based upon the recent answer, an improved question would concern the classification of combinatorial equivalence classes of Voronoi cells of lattices. Have these been classified or are there known classifications which could prove or disprove my conjecture?

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  • $\begingroup$ The 16-cell isn’t a parallelotope… $\endgroup$
    – Daniel Sebald
    Oct 5, 2022 at 3:03

1 Answer 1

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Voronoi conjectured that every parallelotope is combinatorially equivalent to a Voronoi cell of a lattice. The conjecture was proved for $d\le 4$ by Delone.

Reference: Handbook of Convex Geometry, P. M. Gruber and J. M. Wills (eds.) page 1005

To answer the added question: there are 52 inequivalent Voronoi cells in 4 dimensions. I do not know whether this classification proves or refutes your stated conjecture, but it should not be hard to check.

Reference: Computational Geometry of Positive Definite Quadratic Forms, A. Schuermann, page 60

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  • $\begingroup$ In order to complete the question then, is there a classification of the combinatorial equivalence classes of Voronoi cells generated by lattices? Any comment on my conjecture? $\endgroup$
    – Samuel Reid
    Nov 17, 2015 at 21:47
  • $\begingroup$ Thank you for your very succinct and complete answer, I'll post another answer if I end up finishing a paper on the Voronoi conjecture for $d>4$ and the combinatorial classification of primary $d$-parallelotopes. $\endgroup$
    – Samuel Reid
    Nov 17, 2015 at 22:06

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