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  "body": "\n## Definition\n\nLet $n \\ge 3$.\nA *honeycomb* in $\\R^n$ is a [[tiling|GKJK293DXS]] of $\\R^n$ whose tiles, called *cells*, are $n$-dimensional [[convex polytopes|300K6CB1KH]].\nIn dimension $3$ the cells are convex [[polyhedra|0325SGSJAR]].\n\n## Remarks\n\nExample: the cubes $[a, a+1] \\times [b, b+1] \\times [c, c+1]$ with $a, b, c \\in \\Z$ form the *cubic honeycomb* in $\\R^3$.\nThe cubes are convex polyhedra, every point $(x, y, z)$ lies in the cube with $(a, b, c) = (\\floor{x}, \\floor{y}, \\floor{z})$, and distinct cubes have disjoint interiors.\n\nNon-example: copies of a regular tetrahedron cannot form a honeycomb in $\\R^3$.\nTake a generic point $p$ on an edge of one cell.\nThe cells through $p$ each fill either a wedge with the dihedral angle $\\alpha = \\arccos(1/3)$ or a half-space, and these angles add up to $2\\pi$.\nSo $k \\alpha = 2\\pi$ or $k \\alpha = \\pi$ for some integer $k \\ge 1$.\nBoth fail, since $\\alpha \\approx 70.53^\\circ$ and neither $360^\\circ$ nor $180^\\circ$ is an integer multiple of it.\n\nMany authors also require the honeycomb to be face-to-face, so that two cells meet in a common face of both or not at all.\nThe term is also used for tilings of hyperbolic space and of spheres by polytopes.\n",
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