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  "body": "\n## Problem\n\nThe problem consists of three questions about filling Euclidean space $\\R^n$.\n\n1. For each $n$, are there only finitely many [[crystallographic groups|7KTJKFQT2N]] of dimension $n$, up to a suitable notion of equivalence?\n2. Is there a polyhedron that [[tiles|GKJK293DXS]] $\\R^3$ by congruent copies, such that no such tiling has a symmetry group transitive on the tiles?\n3. What is the densest way to pack congruent balls in $\\R^3$, and more generally to pack copies of a given solid?\n\n## Remarks\n\nDavid Hilbert posed the problem in his list of 1900.\nThe first question was answered positively by Ludwig Bieberbach in 1910 and 1911; see the [[Bieberbach theorems|YE87R5CDJY]].\nIn dimension $2$ there are $17$ [[wallpaper groups|MY3CBHARYQ]], and in dimension $3$ there are $219$ space groups up to affine equivalence.\nThe second question was answered positively by Karl Reinhardt in 1928, who found such a polyhedron; Heinrich Heesch gave a plane example in 1935.\nTiles of this kind are called anisohedral.\nThe ball case of the third question is the [[Kepler conjecture|AFMBQT06ZV]], proved by Thomas Hales, who announced his proof in 1998.\nThe densest packing of balls in $\\R^3$ has density $\\pi/\\sqrt{18}$, attained by the face-centred cubic packing.\nSee [[sphere packing|HF32CBN5MA]] for the general setting.\n\n## Explanations\n\n### L1 human\n\nHilbert asked three questions about filling space with copies of one shape.\nHow many kinds of symmetry can a crystal have?\nCan a shape fill space even though its copies cannot all be placed in the same way relative to the pattern?\nAnd how tightly can equal balls, like oranges in a crate, be packed?\nAll three have been answered: finitely many, yes, and the grocer's pyramid stacking is the best.\n",
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