A power law for the distortion of planar sets
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We consider how to map the sites of a square region of planar lattice into a three-dimensional cube, so as to minimize the maximum distortion of distance. We consider the cube to be endowed with a “foliated” geometry in which horizontal distance is standard but vertical communication only occurs at the surface of the cube. These geometries may naturally arise if a planar data set is to be stored in a stack of chips. It is proved that any one-to-one map which fills the cube with a fixed “density” must produce a distortion of distance which grows as the one-sixth power of the diameter of the square and the two-thirds power of the density. Moreover, we explicitly define one-to-one maps with 100% density, one-sixth power stretching, and a small leading coefficient. As a final note, a high-dimensional analog is considered.
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