Hypercubic honeycomb

<table><tbody><tr><td>A <a href="/facts/Square_tiling/3ym86mvv">regular square tiling</a>.1 color</td><td>A <a href="/facts/Cubic_honeycomb/eMxySFur">cubic honeycomb</a> in its regular form.1 color</td></tr><tr><td>A checkboard square tiling2 colors</td><td>A <a href="/facts/Cubic_honeycomb/eMxySFur">cubic honeycomb</a> checkerboard.2 colors</td></tr><tr><td>Expanded square tiling3 colors</td><td><a href="/facts/Expansion_(geometry)/eMpoRlnm">Expanded</a> cubic honeycomb4 colors</td></tr><tr><td>4 colors</td><td>8 colors</td></tr></tbody></table>
<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a hypercubic honeycomb is a family of <a href="/facts/List_of_regular_polytopes/Kng25Ymq">regular honeycombs</a> (<a href="/facts/Tessellation/7BWDywXC">tessellations</a>) in n-<a href="/facts/Dimension/0ktmUyac">dimensional</a> spaces with the <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbols</a> {4,3...3,4}  and containing the symmetry of <a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter group</a> R<i>n</i> (or B~<i>n</i>–1) for <i>n</i> ≥ 3.
</p><p>The tessellation is constructed from 4 n-<a href="/facts/Hypercube/q2mRPbIz">hypercubes</a> per <a href="/facts/Ridge_(geometry)/T2E6BTU4">ridge</a>. The <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> is a <a href="/facts/Cross-polytope/erk0byF6">cross-polytope</a> {3...3,4}.
</p><p>The hypercubic honeycombs are <a href="/facts/Self-dual_polytope/rkIHbmAf">self-dual</a>.
</p><p><a href="/facts/Coxeter/D2l5jWGG">Coxeter</a> named this family as δn+1 for an n-dimensional honeycomb.
</p>

Hypercubic honeycomb open-in-new

Hypercubic honeycomb