Weaire-Phelan Structure

Weaire-Phelan structure made out of paper models

Lord Kelvin posed the question of how space could be divided into cells of equal volume and minimal surface area. This is equivalent to asking what shape equal-sized soap bubbles would take in foam since bubbles naturally assume shapes with minimal surface area. He conjectured that the solution was a lattice of truncated octahedra with slightly curved hexagonal faces. The polyhedral version of this structure is called the Kelvin structure. The physicists Denis Weaire and Robert Phelan applied a structure known from crystallography to the problem and found that the so-called Weaire-Phelan structure improves on the Kelvin structure.


  1. Cut out, score, fold and glue two copies of the dodecahedra and six copies of the 14-sided polyhedra. The dodecahedra are shown in green in the photo above.
  2. 0214-s02Glue pairs of 14-sided polyhedra together at their hexagonal faces. You will have three pairs of polyhedra.
  3. 0214-s03Glue a dodecahedron to one pair of 14-sided polyhedra at the top pair of pentagonal faces.
  4. 0214-s04Glue a pair of 14-sided polyhedra to the left-hand side of the dodecahedron.
  5. 0214-s05Glue a pair of 14-sided polyhedra to the front of the dodecahedron.
  6. 0214-s06Glue a dodecahedron to the top front of the group. This completes a unit that repeats to fill space.


  • 0214-n01The Beijing National Aquatics Center built for the 2008 Olympics has a design of "bubbles" enclosing the building that is based on the Weaire-Phelan structure.
  • The Weaire-Phelan structure has not been proven to be optimal.