{"id":2939,"date":"2016-12-01T01:00:05","date_gmt":"2016-12-01T01:00:05","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=2939"},"modified":"2016-12-09T05:06:20","modified_gmt":"2016-12-09T05:06:20","slug":"truncated-633-honeycomb","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2016\/12\/01\/truncated-633-honeycomb\/","title":{"rendered":"Truncated {6,3,3} Honeycomb"},"content":{"rendered":"
\n
\"Truncated<\/a>

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

This is an image of the truncated {6,3,3} honeycomb<\/a>, created by Roice Nelson<\/a>. This honeycomb lives in a curved 3-dimensional space called hyperbolic space<\/a>.<\/p>\n

To understand the truncated {6,3,3} honeycomb, we need to start with the {6,3,3} honeycomb: <\/p>\n

\n
\"{6,3,3}<\/a>

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

The {6,3,3} honeycomb is also called the hexagonal tiling honeycomb<\/a>, because it contains sheets of hexagons tiling flat Euclidean planes embedded in hyperbolic space. <\/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 above 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

To obtain the truncated {6,3,3} honeycomb, we replace each vertex of {6,3,3} honeycomb by a tetrahedron. We can think of this process as chopping off the vertices of the {6,3,3} tiling, or truncating<\/a> it. <\/p>\n

The the {6,3,3} honeycomb has this Coxeter diagram<\/a>:<\/p>\n

\n●—6—o—3—o—3—o<\/b>\n<\/div>\n

  <\/p>\n

while the Coxeter diagram of the truncated {6,3,3} honeycomb is this:<\/p>\n

\n●—6—●—3—o—3—o<\/b>\n<\/div>\n

  <\/p>\n

The extra black dot here indicates that each vertex in the truncated {6,3,3} honeycomb corresponds to a vertex-edge flag<\/b> in the {6,3,3} honeycomb: that is, a pair consisting of a vertex and an edge incident to that vertex.<\/p>\n

Both these honeycombs have the same symmetry group, a discrete subgroup of the isometry group<\/a> 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 here:<\/p>\n

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

For more on the hexagonal tiling honeycomb, see:<\/p>\n

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

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

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

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

The image of the truncated {6,3,3} honeycomb was created by Roice Nelson and put on on Wikicommons<\/a> under a Creative Commons Attribution-Share Alike 3.0 Unported<\/a> license. The image of the {6,3,3} honeycomb was created by him and put on Wikicommons<\/a> under the same type of license.<\/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 an image of the truncated {6,3,3} honeycomb<\/a> in hyperbolic space<\/a>. <\/p>\n

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