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Foam problems are about how to best partition spaceinto bubbles of minimal surface area. We investigate thecase where one unit-volume bubble is required to tiled-dimensional space in a periodic fashion according to thestandard, cubical lattice. While a cube requires surface area2d, we construct such a bubble having surface area very closeto that of a sphere; that is, proportional to (the minimumpossible even without the constraint of being periodic). Ourmethod for constructing this “spherical cube” is inspired byfoundational questions in the theory of computation relatedto the concept of hardness amplification. Our methods givenew algorithms for “coordinated discretization” of high-dimensional data points, which have near-optimal noiseresistance. We also provide the most efficient known cubicalfoam in three dimensions.