{"id":206,"date":"2013-09-15T05:56:13","date_gmt":"2013-09-15T05:56:13","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=206"},"modified":"2016-12-09T05:13:04","modified_gmt":"2016-12-09T05:13:04","slug":"633-honeycomb-in-upper-half-space","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2013\/09\/15\/633-honeycomb-in-upper-half-space\/","title":{"rendered":"{6,3,3} Honeycomb in Upper Half Space"},"content":{"rendered":"
\"{6,3,3}<\/a>

{6,3,3} Honeycomb in Upper Half Space – Roice Nelson<\/p><\/div>\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.<\/p>\n

However, besides honeycombs in Euclidean space, we can also have honeycombs in hyperbolic space<\/a>, which is a 3-dimensional Riemannian manifold with constant negative curvature. The {6,3,3} honeycomb lives in hyperbolic space. Here Roice Nelson<\/a> has drawn it in the upper half space model of hyperbolic space, which is the 3d analogue of Poincar\u00e9’s famous upper half-plane model<\/a> of the hyperbolic plane. As usual, you can click the image for a better view!<\/p>\n

The {6,3,3} honeycomb is also called the hexagonal tiling honeycomb<\/b><\/a>. The reason is that three hexagonal tilings of the plane meet at any edge of this honeycomb. This fact is recorded in the notation {6,3,3}, which 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. Similarly, the symbol for the hexagonal tiling honeycomb is {6,3,3} because 3 hexagonal tilings meet along each edge.<\/p>\n

The {6,3,3} honeycomb is one of 15 regular honeycombs in hyperbolic space. For a complete list, with links to pictures, see:<\/p>\n

\u2022 Tesselations of hyperbolic 3-space<\/a>, Wikipedia.<\/p>\n

Roice Nelson is a software developer with a passion for exploring mathematics through visualization:<\/p>\n

\u2022 Roice<\/a>.<\/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":"

A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiiling of the plane. However, not only can we have honeycombs in Euclidean space, we can also have them in hyperbolic space. The {6,3,3} honeycomb lives in hyperbolic space.<\/p>\n

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