What space-filling arrangement of similar cells of equal volume has minimal surface area? This questions arises naturally in the theory
of foams when the liquid content is small. Kelvin (Thomson 1887) proposed that the
solution was a 14-sided truncated
octahedron having a very slight curvature of the hexagonal faces.
The isoperimetric quotient the uncurved truncated
octahedron is given by
while Kelvin's slightly curved variant has a slightly less optimal quotient of 0.757.
Despite one hundred years of failed attempts and Weyl's (1952) opinion that the curved truncated octahedron could
not be improved upon, Weaire and Phelan (1994) discovered a space-filling unit
cell consisting of six 14-sided polyhedra and two 12-sided polyhedra with irregular
faces and only hexagonal faces remaining planar. This structure has an isoperimetric quotient of 0.764, or approximately 0.3% less
that Kelvin's cell.
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Thomson, W. "On the Division of Space with Minimum Partitional Area." Philos.
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London: Taylor and Francis, 1996.
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2002.
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