{"id":745,"date":"2014-03-15T10:01:05","date_gmt":"2014-03-15T10:01:05","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=745"},"modified":"2016-12-09T05:15:17","modified_gmt":"2016-12-09T05:15:17","slug":"633-honeycomb","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2014\/03\/15\/633-honeycomb\/","title":{"rendered":"{6,3,3} Honeycomb"},"content":{"rendered":"
\n
\"{6,3,3}<\/a>

{6,3,3} Honeycomb – Roice Nelson<\/p><\/div>\n<\/div>\n

This is the {6,3,3} honeycomb, drawn by Roice Nelson<\/a>.<\/p>\n

A 3-dimensional honeycomb<\/a> is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiling<\/a> of the plane. Besides honeycombs in 3d Euclidean space, we can also have honeycombs in 3d hyperbolic space<\/a>, a non-Euclidean geometry with constant negative curvature. The {6,3,3} honeycomb lives in hyperbolic space. For another view of it, also by Roice Nelson, see:<\/p>\n

• {6,3,3} honeycomb in upper half space<\/a>, Visual Insight<\/i>.<\/p>\n

When I said a honeycomb is a way of filling 3d space with polyhedra, I was lying slightly. The {6,3,3} honeycomb is also called the hexagonal tiling honeycomb<\/b><\/a>. The reason is that it contains many sheets of hexagons, tiling planes in the usual way hexagons do. These are flat Euclidean planes lying in 3d hyperbolic space, called horospheres<\/a>. The sheets of hexagons are not exactly polyhedra, because they have infinitely many<\/i> polygonal faces! So, the {6,3,3} honeycomb is an example of a ‘paracompact’ honeycomb.<\/p>\n

The notation {6,3,3} is an example of a Schl\u00e4fli symbol<\/a>. The Schl\u00e4fli symbol is defined in a recursive way. The symbol for the hexagon is {6}. The symbol for the hexagonal tiling of the plane is {6,3} because 3 hexagons meet at each vertex. Finally, the hexagonal tiling honeycomb has symbol {6,3,3} because 3 hexagonal tilings meet at each edge.<\/p>\n

Just as the {6,3} inside {6,3,3} describes the hexagonal tilings inside the {6,3,3} honeycomb, the {3,3} describes the vertex figure<\/a> of this honeycomb: that is, the way the edges meet at each vertex. {3,3} is the Schl\u00e4fli symbol for the regular tetrahedron, and if you look at the picture you can can see that each vertex has 4 edges coming out, just like the edges going from the center of a tetrahedron to its corners.<\/p>\n

The hexagonal tiling honeycomb is an example of a uniform honeycomb in 3d hyperbolic space<\/a>. It is called regular<\/b> because it is especially symmetrical: its symmetry group acts transitively on the set of flags, where a flag<\/b> is a vertex lying on an edge lying on a face lying on cell (in this case, a sheet of hexagons). <\/p>\n

As already mentioned, the hexagonal tiling honeycomb is called paracompact<\/b> because it has infinite cells, which in this case are the hexagonal tilings {6,3}. There are 15 regular honeycombs in 3d hyperbolic space, of which 11 are paracompact. For more, see:<\/p>\n

• Paracompact uniform honeycomb<\/a>, Wikipedia.<\/p>\n

The Coxeter diagram<\/a> of the {6,3,3} honeycomb is<\/p>\n

\n●—6—o—3—o—3—o<\/b>\n<\/div>\n

  <\/p>\n

The symmetry group of the {6,3,3} honeycomb is a discrete subgroup of the symmetry group of hyperbolic space. This discrete group has generators and relations summarized by the unmarked Coxeter diagram:<\/p>\n

\no—6—o—3—o—3—o<\/b>\n<\/div>\n

 <\/p>\n

This diagram says there are four generators $s_1, \\dots, s_4$ obeying relations encoded in the edges of the diagram:<\/p>\n

$$ (s_1 s_2)^6 = 1 $$
\n$$ (s_2 s_3)^3 = 1 $$
\n$$ (s_3 s_4)^3 = 1 $$<\/p>\n

together with relations <\/p>\n

$$s_i^2 = 1$$<\/p>\n

and <\/p>\n

$$ s_i s_j = s_j s_i \\; \\textrm{ if } \\; |i – j| > 1 $$<\/p>\n

Marking the Coxeter diagram in different ways lets us describe many honeycombs with the same symmetry group as the hexagonal tiling honeycomb—in fact, $2^4 – 1 = 15$ of them, since there are 4 dots in the Coxeter diagram! You can see some of these uniform honeycombs here:<\/p>\n

• Hexagonal tiling honeycomb<\/a>, Wikipedia.<\/p>\n

Puzzle:<\/b> The symmetry group of 3d hyperbolic space, not counting reflections, is $\\mathrm{PSL}(2,\\mathbb{C})$. Can you explicitly describe the subgroup that preserves the hexagonal tiling honeycomb? <\/p>\n

Roice Nelson, the creator of this image, is a software developer with a passion for exploring mathematics through visualization:<\/p>\n

\u2022 Roice<\/a>.<\/p>\n

\n

Martin Weissman answered the puzzle on G+<\/a>:<\/p>\n

Well, it’s $\\mathrm{PSL}_2(\\mathbb{Z}[e^{2 \\pi i \/ 3}])$, of course! :)\ufeff\n<\/p><\/blockquote>\n

Since he’s an expert on arithmetic Coxeter groups, this must be about right! Theorem 10.2 here:<\/p>\n

• Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups<\/a>, Canad. J. Math. Vol.<\/i> 51<\/b> (1999), 1307\u20131336.<\/p>\n

is a bit more precise. It gives a nice description of the even part<\/b> of the Coxeter group discussed in this article, that is, the part generated by products of pairs<\/i> of reflections. To get this group, we start with 2 × 2 matrices with entries in the Eisenstein integers<\/a><\/b>: the integers with a cube root of -1 adjoined. We look at the matrices where the absolute value<\/i> of the determinant is 1, and then we ‘projectivize’ it, modding out by its center. That does the job!<\/p>\n

They call the even part of the Coxeter group $[3,3,6]^+$, and they call the group it’s isomorphic to $\\mathrm{P\\overline{S}L}_2(\\mathbb{E}])$, where $\\mathbb{E}$ is their notation for the Eisenstein integers, also called $\\mathbb{Z}[e^{2 \\pi i \/ 3}]$. The funny little line over the \"S\" is a notation of theirs: $\\mathrm{SL}_2$ stands for 2 × 2 matrices with determinant 1, but $\\mathrm{\\overline{S}L}_2$ is their notation for 2 × 2 matrices whose determinant has absolute value 1.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This is the {6,3,3} honeycomb, drawn by Roice Nelson<\/a>. A 3-dimensional honeycomb<\/a> is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiling<\/a> of the plane. Besides honeycombs in 3d Euclidean space, we can also have honeycombs in 3d hyperbolic space<\/a>. The hexagonal tiling honeycomb lives in hyperbolic space, and each vertex has 4 edges coming out, just as if we drew edges from the middle of a tetrahedron to its 4 corners.<\/p>\n

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