Math 2.0

Calculus of variations and optimal control; optimization Geometric measure theory and optimization on manifolds Problem

Kelvin problem

Builds on HoneycombSurface areaPlateau's lawsHoneycomb theorem

Problem

Among all partitions of ℝ3\R^3 into cells of equal volume, find one whose interfaces have the least surface area per cell.

To make the question precise, fix the cell volume to 11. Then ask for the least value of the quantity lim sup⁡R→∞ℋ2(Σ∩BR)∣BR∣.\limsup_{R \to \infty} \frac{\mathcal{H}^2(\Sigma \cap B_R)}{\abs{B_R}}. Here Σ\Sigma is the union of the cell boundaries, ℋ2\mathcal{H}^2 is area, and BRB_R is the ball of radius RR about the origin.

Remarks

Kelvin proposed as the answer a partition into congruent cells shaped like truncated octahedra, arranged as in the bitruncated cubic honeycomb. His cells are not exactly polyhedra: their hexagonal faces are slightly curved, so that they satisfy Plateau's laws. In 1993 Denis Weaire and Robert Phelan found a partition into two kinds of cells of equal volume with about 0.3%0.3\% less area than Kelvin's. So Kelvin's proposal is not optimal. The problem is open: no partition is known to be optimal, and it is not known whether the Weaire–Phelan structure is. The two-dimensional analogue is solved: by the honeycomb theorem of Thomas Hales, the regular hexagonal tiling is optimal in the plane.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Honeycomb, concept, no evidence levelHoneycombconceptMean curvature, concept, no evidence levelMean curvatureconceptMinimal surface, concept, no evidence levelMinimal surfaceconceptParametrised surface, concept, no evidence levelParametrised surfaceconceptRegular polygon, concept, no evidence levelRegular polygonconceptTessellation, concept, no evidence levelTessellationconceptPlateau's laws, statement, no evidence levelPlateau's lawsstatementSurface area, concept, no evidence levelSurface areaconceptHoneycomb theorem, statement, no evidence levelHoneycomb theoremstatementKelvin problem, problem, no evidence levelKelvin problemproblem
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. Honeycomb concept
  2. Mean curvature concept
  3. Minimal surface concept
  4. Parametrised surface concept
  5. Regular polygon concept
  6. Tessellation concept
  7. Plateau's laws statement
  8. Surface area concept
  9. Honeycomb theorem statement
  10. Kelvin problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure
About this page
TagCVTKSCP55A
URIm2://CVTKSCP55A
Kindproblem
WritingW0 stub
MSC49Q10, 52C22, 49Q20
StatusThis copy of the site has no build data. Run m2 status CVTKSCP55A.
AboutHoneycomb
UsesHoneycomb, Surface area, Plateau's laws, Honeycomb theorem

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