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# The Hexagonal Tiling Honeycomb

Communicated by *Notices* Associate Editor Han-Bom Moon

This picture by Roice Nelson shows the “hexagonal tiling honeycomb.” But what is it? Roughly speaking, a honeycomb is a way of filling three-dimensional space with polyhedra. The most familiar one is the usual way of filling Euclidean space with cubes. This cubic honeycomb is denoted by the symbol because a square has 4 edges, 3 squares meet at each corner of a cube, and 4 cubes meet along each edge of this honeycomb. We can also define honeycombs in hyperbolic space, which is a three-dimensional Riemannian manifold with constant negative curvature. For example, there is a hyperbolic honeycomb denoted , which is a way of filling hyperbolic space with cubes where 5 meet along each edge. ,

Coxeter has classified the most symmetrical hyperbolic honeycombs *infinitely many* faces. The symbol for the hexagonal tiling honeycomb is because a hexagon has 6 edges, 3 hexagons meet at each corner in a plane tiled by regular hexagons, and 3 such planes meet along each edge of this honeycomb. ,

The hexagonal tiling honeycomb shows up naturally if we try to discretize spacetime while preserving as much symmetry as we can. In special relativity, **Minkowski spacetime** is equipped with the nondegenerate bilinear form

usually called the **Minkowski metric**. Hyperbolic space sits inside Minkowski spacetime as the hyperboloid of points with and Equivalently, we can think of Minkowski spacetime as the space . of hermitian complex matrices, using the fact that every such matrix is of the form

and In these terms, hyperbolic space is the hyperboloid .

How can we construct the hexagonal tiling honeycomb inside Sitting in the complex numbers we have the ring ? of **Eisenstein integers**: complex numbers of the form where are integers and is a nontrivial cube root of say , This lets us define a lattice in Minkowski spacetime, say . consisting of , hermitian matrices with entries that are Eisenstein integers. This lattice can be seen as a discretized version of Minkowski spacetime. Then comes a minor miracle: the points at the centers of hexagons in the hexagonal tiling honeycomb are precisely those points in the lattice that lie on the hyperboloid For two proofs see .

The hexagonal tiling honeycomb also arises in algebraic geometry. An **abelian variety** is a complex projective variety that is also an abelian group in a compatible way. The most famous are the one-dimensional ones, called **elliptic curves**. Any elliptic curve is of the form for some lattice We can get a highly symmetrical elliptic curve from the Eisenstein integers by forming the quotient . We can then form a two-dimensional abelian variety by taking the product . .

The set of isomorphism classes of complex line bundles over any complex projective variety becomes an abelian group thanks to our ability to tensor line bundles. This group, called the **Picard group**, has a natural topology, and it typically has many connected components. The set of connected components is an abelian group in its own right, called the **Néron–Severi group**. You can think of this as the group of equivalence classes of line bundles where two count as equivalent if one can be deformed to the other.

Thanks to some beautiful theorems on abelian varieties

Thus, the picture above gives a vivid example of some concepts from algebraic geometry. But it is also part of a larger story relating algebraic integers in the fields to regular honeycombs

## References

[ 1] - J. C. Baez, Line bundles on complex tori (part 5), The Café -Category, 2024. Available at https://golem.ph.utexas.edu/category/2024/04/line_bundles_on_complex_tori_p_2.html.
[ 2] - Christina Birkenhake and Herbert Lange,
*Complex abelian varieties*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004, DOI 10.1007/978-3-662-06307-1. MR2062673,## Show rawAMSref

`\bib{BL}{book}{ author={Birkenhake, Christina}, author={Lange, Herbert}, title={Complex abelian varieties}, series={Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]}, volume={302}, edition={2}, publisher={Springer-Verlag, Berlin}, date={2004}, pages={xii+635}, isbn={3-540-20488-1}, review={\MR {2062673}}, doi={10.1007/978-3-662-06307-1}, }`

[ 3] - H. S. M. Coxeter,
*Regular honeycombs in hyperbolic space*, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N. V., Groningen, 1956, pp. 155–169. MR87114,## Show rawAMSref

`\bib{Coxeter}{article}{ author={Coxeter, H. S. M.}, title={Regular honeycombs in hyperbolic space}, conference={ title={Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III}, }, book={ publisher={Erven P. Noordhoff N. V., Groningen}, }, date={1956}, pages={155--169}, review={\MR {87114}}, }`

[ 4] - C. W. L. Garner,
*Coordinates for vertices of regular honeycombs in hyperbolic space*, Proc. Roy. Soc. London Ser. A**293**(1966), 94–107, DOI 10.1098/rspa.1966.0160. MR196592,## Show rawAMSref

`\bib{Garner}{article}{ author={Garner, C. W. L.}, title={Coordinates for vertices of regular honeycombs in hyperbolic space}, journal={Proc. Roy. Soc. London Ser. A}, volume={293}, date={1966}, pages={94--107}, issn={0962-8444}, review={\MR {196592}}, doi={10.1098/rspa.1966.0160}, }`

[ 5] - Norman W. Johnson and Asia Ivić Weiss,
*Quadratic integers and Coxeter groups*, Canad. J. Math.**51**(1999), no. 6, 1307–1336, DOI 10.4153/CJM-1999-060-6. Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday. MR1756885,## Show rawAMSref

`\bib{JW}{article}{ author={Johnson, Norman W.}, author={Weiss, Asia Ivi\'{c}}, title={Quadratic integers and Coxeter groups}, note={Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday}, journal={Canad. J. Math.}, volume={51}, date={1999}, number={6}, pages={1307--1336}, issn={0008-414X}, review={\MR {1756885}}, doi={10.4153/CJM-1999-060-6}, }`

[ 6] - Suzana Milea, Christopher D. Shelley, and Martin H. Weissman,
*Arithmetic of arithmetic Coxeter groups*, Proc. Natl. Acad. Sci. USA**116**(2019), no. 2, 442–449, DOI 10.1073/pnas.1809537115. MR3904691,## Show rawAMSref

`\bib{MSW}{article}{ author={Milea, Suzana}, author={Shelley, Christopher D.}, author={Weissman, Martin H.}, title={Arithmetic of arithmetic Coxeter groups}, journal={Proc. Natl. Acad. Sci. USA}, volume={116}, date={2019}, number={2}, pages={442--449}, issn={0027-8424}, review={\MR {3904691}}, doi={10.1073/pnas.1809537115}, }`

## Credits

The figure is courtesy of Roice Nelson.

The author photo is courtesy of Lisa Raphals.