Soap films and Kelvin's curved, truncated octahedron
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Here, the Weierstrass representation theorem for minimal surfaces is used to derive parametrizations of two soap films spanning a rectangular prism with square base. The parametrizations are explicit, although the exact values of the height of the prism and the edge length of the square base can only be determined numerically from the formulas. The more famous of these two soap films can be extended to a partition of ℝ3 by curved, truncated octahedra. Furthermore, if the height of the prism is taken to be\(1/\sqrt 2 \), then the partition is that discovered by Lord Kelvin in the late 19th century.
Math Subject ClassificationsPrimary: 49Q05 secondary: 32G15
Key Words and PhrasesMinimal surface flat structure soap film triply periodic Kelvin
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