Suppose we want to partition the space $\mathbb R^3$ into a countably infinite number of "bubbles" (connected components), so that the volume of every bubble does not exceed $1$, and the average area of the "film" (the surface separating the bubbles) per a unit of volume is as low as possible. How should we shape the bubbles? What is the lowest average area of the film per a unit of volume that we can attain?
Partitioning space into "bubbles" using as low surface area as possible
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optimization
combinatorial-geometry
minimal-surfaces
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1Do these [Weaire-Phelan](https://en.wikipedia.org/wiki/Weaire–Phelan_structure) guys have anything to do with what you're looking for? I can't tell if the problem they're designed to solve is the same as the problem you're trying to solve. – 2017-01-15
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0Yes, it seems a very close problem, although I do not require all bubbles to be of the same volume (not sure if it is significant). Wikipedia also mentions that "It has not been proved that the Weaire–Phelan structure is optimal". – 2017-01-15
1 Answers
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It is the 14 sided truncated octahedron. Names to look up include Frank Morgan, Rob Kusner, Nicos Kapouleas. Morgan has a book for beginners on geometric measure theory, this may be in it.
The cell-transitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the body-centered cubic lattice.
https://en.wikipedia.org/wiki/Truncated_octahedron
Apparently, if we allow more than one type of cell, we can do a little better.
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0The [Weire-Phelan structure](https://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure), mentioned in a comment under the question, is said to have a lower average area of film. – 2017-01-15
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0@DavidK alright, two different pieces. Good to know. – 2017-01-15