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2 22 honeycomb
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<table><tbody><tr><th colspan="2">222 honeycomb</th></tr><tr><td colspan="2">(no image)</td></tr><tr><td>Type</td><td><a href="/facts/Uniform_7-polytope/suCJSvTN">Uniform tessellation</a></td></tr><tr><td><a href="/facts/Coxeter_symbol/PfHvVPBg">Coxeter symbol</a></td><td>222</td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>{3,3,32,2}</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td>6-face type</td><td><a href="/facts/2_21_polytope/nF3nYaQk">221</a> </td></tr><tr><td>5-face types</td><td><a href="/facts/5-orthoplex/vU1TvdQ1">211</a><a href="/facts/5-simplex/KIfBFfhl">{34}</a></td></tr><tr><td>4-face type</td><td><a href="/facts/5-cell/Rh6lwDcc">{33}</a></td></tr><tr><td>Cell type</td><td><a href="/facts/Tetrahedron/7mkSrl6j">{3,3}</a></td></tr><tr><td>Face type</td><td><a href="/facts/Triangle/cRXUwsMn">{3}</a></td></tr><tr><td>Face figure</td><td>{3}×{3} <a href="/facts/Duoprism/D73YMRth">duoprism</a></td></tr><tr><td>Edge figure</td><td><a href="/facts/Birectified_5-simplex/H5kx0rFW">{32,2}</a> </td></tr><tr><td>Vertex figure</td><td><a href="/facts/Gosset_1_22_polytope/7XXjWGQ8">122</a> </td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td> E ~ 6 {\displaystyle {\tilde {E}}_{6}} , [[3,3,32,2]]</td></tr><tr><td>Properties</td><td><a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a>, <a href="/facts/Facet-transitive/ukLVQlK2">facet-transitive</a></td></tr></tbody></table> <p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the 222 honeycomb is a <a href="/facts/Uniform_honeycomb/fGXtrgGF">uniform tessellation</a> of the six-dimensional Euclidean space. It can be represented by the <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> {3,3,32,2}. It is constructed from <a href="/facts/2_21_polytope/nF3nYaQk">221</a> <a href="/facts/Facet_(geometry)/IreRHGJ7">facets</a> and has a <a href="/facts/1_22_polytope/7XXjWGQ8">122</a> <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>, with 54 221 polytopes around every vertex. </p><p>Its <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a> is the <i>E6 lattice</i>, and the <a href="/facts/Root_system/TFaEdwMU">root system</a> of the <a href="/facts/E6_(mathematics)/U9YEMtWj">E6</a> <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> so it can also be called the E6 honeycomb. </p>