Math 2.0

Convex and discrete geometry Tilings Problem

Hilbert's eighteenth problem

Builds on Crystallographic groupTessellationSphere packingBieberbach theoremsWallpaper groupKepler conjecture

Problem

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

  1. For each nn, are there only finitely many crystallographic groups of dimension nn, up to a suitable notion of equivalence?
  2. Is there a polyhedron that tiles ℝ3\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 ℝ3\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. In dimension 22 there are 1717 wallpaper groups, and in dimension 33 there are 219219 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, proved by Thomas Hales, who announced his proof in 1998. The densest packing of balls in ℝ3\R^3 has density π/18\pi/\sqrt{18}, attained by the face-centred cubic packing. See sphere packing for the general setting.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Affine map, concept, no evidence levelAffine mapconceptCrystallographic group, concept, no evidence levelCrystallographic groupconceptEuclidean isometry, concept, no evidence levelEuclidean isometryconceptFree abelian group, concept, no evidence levelFree abelian groupconceptGroup isomorphism, concept, no evidence levelGroup isomorphismconceptIndex of a subgroup, concept, no evidence levelIndex of a subgroupconceptNormal subgroup, concept, no evidence levelNormal subgroupconceptPacking density, concept, no evidence levelPacking densityconceptSphere packing, concept, no evidence levelSphere packingconceptTessellation, concept, no evidence levelTessellationconceptTranslation, concept, no evidence levelTranslationconceptWallpaper group, example, no evidence levelWallpaper groupexampleEuclidean group, object, no evidence levelEuclidean groupobjectKepler conjecture, statement, no evidence levelKepler conjecturestatementBieberbach theorems, statement, no evidence levelBieberbach theoremsstatementHilbert's eighteenth problem, problem, no evidence levelHilbert's eighteenth …problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Affine map concept
  2. Crystallographic group concept
  3. Euclidean isometry concept
  4. Free abelian group concept
  5. Group isomorphism concept
  6. Index of a subgroup concept
  7. Normal subgroup concept
  8. Packing density concept
  9. Sphere packing concept
  10. Tessellation concept
  11. Translation concept
  12. Wallpaper group example
  13. Euclidean group object
  14. Kepler conjecture statement
  15. Bieberbach theorems statement
  16. Hilbert's eighteenth problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_eighteenth_problem
About this page
TagSDSGN4B6JD
URIm2://SDSGN4B6JD
Kindproblem
WritingW0 stub
MSC52C22, 52C17, 20H15
StatusThis copy of the site has no build data. Run m2 status SDSGN4B6JD.
AboutCrystallographic group, Tessellation, Sphere packing
UsesCrystallographic group, Tessellation, Sphere packing, Bieberbach theorems, Wallpaper group, Kepler conjecture
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