---
tag: SDSGN4B6JD
kind: problem
name: Hilbert's eighteenth problem
msc: [52C22, 52C17, 20H15]
uses: [7KTJKFQT2N, GKJK293DXS, HF32CBN5MA, YE87R5CDJY, MY3CBHARYQ, AFMBQT06ZV]
about: [7KTJKFQT2N, GKJK293DXS, HF32CBN5MA]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%27s_eighteenth_problem
same_as:
  wikidata: Q11059889
  wikipedia: Hilbert's eighteenth problem
history:
  posed_by: "David Hilbert"
  first_stated: "1900"
---

## Problem

The problem consists of three questions about filling Euclidean space $\R^n$.

1. For each $n$, are there only finitely many [[crystallographic groups|7KTJKFQT2N]] of dimension $n$, up to a suitable notion of equivalence?
2. 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?
3. What is the densest way to pack congruent balls in $\R^3$, and more generally to pack copies of a given solid?

## Remarks

David Hilbert posed the problem in his list of 1900.
The first question was answered positively by Ludwig Bieberbach in 1910 and 1911; see the [[Bieberbach theorems|YE87R5CDJY]].
In dimension $2$ there are $17$ [[wallpaper groups|MY3CBHARYQ]], and in dimension $3$ there are $219$ space groups up to affine equivalence.
The second question was answered positively by Karl Reinhardt in 1928, who found such a polyhedron; Heinrich Heesch gave a plane example in 1935.
Tiles of this kind are called anisohedral.
The ball case of the third question is the [[Kepler conjecture|AFMBQT06ZV]], proved by Thomas Hales, who announced his proof in 1998.
The densest packing of balls in $\R^3$ has density $\pi/\sqrt{18}$, attained by the face-centred cubic packing.
See [[sphere packing|HF32CBN5MA]] for the general setting.

## Explanations

### L1 human

Hilbert asked three questions about filling space with copies of one shape.
How many kinds of symmetry can a crystal have?
Can a shape fill space even though its copies cannot all be placed in the same way relative to the pattern?
And how tightly can equal balls, like oranges in a crate, be packed?
All three have been answered: finitely many, yes, and the grocer's pyramid stacking is the best.