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?
2026-03-25 11:12:19.1774437139
Partitioning space into "bubbles" using as low surface area as possible
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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.
https://en.wikipedia.org/wiki/Truncated_octahedron
Apparently, if we allow more than one type of cell, we can do a little better.