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Cubic honeycomb
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<p>The cubic honeycomb or cubic cellulation is the only proper regular space-filling <a href="/facts/Tessellation/7BWDywXC">tessellation</a> (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>) in <a href="/facts/Euclidean_3-space/KAqKWUbS">Euclidean 3-space</a> made up of <a href="/facts/Cube/65lUaAqN">cubic</a> cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> is a regular <a href="/facts/Octahedron/G38q92xW">octahedron</a>. It is a <a href="/facts/Self-dual_tessellation/QVK02ksI">self-dual</a> tessellation with <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> {4,3,4}. <a href="/facts/John_Horton_Conway/g3PA1zoK">John Horton Conway</a> called this honeycomb a cubille. </p><p>A <a href="/facts/Honeycomb_(geometry)/seyRBcel">geometric honeycomb</a> is a <i>space-filling</i> of <a href="/facts/Polyhedron/UC0p9FTF">polyhedral</a> or higher-dimensional <i>cells</i>, so that there are no gaps. It is an example of the more general mathematical <i>tiling</i> or <i><a href="/facts/Tessellation/7BWDywXC">tessellation</a></i> in any number of dimensions. </p><p>Honeycombs are usually constructed in ordinary <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean</a> ("flat") space, like the <a href="/facts/Convex_uniform_honeycomb/78IkTl7A">convex uniform honeycombs</a>. They may also be constructed in <a href="/facts/Non-Euclidean_geometry/efKa7De2">non-Euclidean spaces</a>, such as <a href="/facts/Uniform_honeycombs_in_hyperbolic_space/n1c2vP9b">hyperbolic uniform honeycombs</a>. Any finite <a href="/facts/Uniform_polytope/nFcNAzLy">uniform polytope</a> can be projected to its <a href="/facts/Circumsphere/m6EPvP2E">circumsphere</a> to form a uniform honeycomb in spherical space. </p>