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by carapace
2346 days ago
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Ooo, that footnote: "Flat tori in three-dimensional space and convex integration" https://www.pnas.org/content/109/19/7218 Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration", Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, 109 (19): 7218–7223, doi:10.1073/pnas.1118478109, PMC 3358891, PMID 22523238 > It was a long-held belief that ... no isometric embedding of the square flat torus—a differentiable injective map that preserves distances—could exist into three-dimensional space. In the mid 1950s Nash (1) and Kuiper (2) amazed the world mathematical community by showing that such an embedding actually exists. > ... > In this article, we convert convex integration theory into an explicit algorithm. We then provide an implementation leading to images of an embedded square flat torus in three-dimensional space. This visualization has led us in turn to discover a unique geometric structure. This structure, described in the corrugation theorem below, reveals a remarkable property: The normal vector exhibits a fractal behavior. |
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