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Kstati vot dokazali taki etot vopros s zatyagivaniem poliedrami iz ravnostoronnih treugol'nikov. Tam poluchaetsya chto lyubuyu lomanuyu mozhno posheveliti i ona staynet zatyagivaemoy. I chto dlya chetyrehzvennoy lomanoy esli dliny diagonaley algebraicheski nezvisimy,to zatyanut' nel'zya s giptezoy o lagranzhevosti eto nikak ne protivorechit vprochem https://arxiv.org/abs/2005.02555 Domes over curves Alexey Glazyrin, Igor Pak A closed PL-curve is called integral if it is comprised of unit intervals. Kenyon's problem asks whether for every integral curve γ in R3, there is a dome over γ, i.e. whether γ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when γ is a quadrilateral, thus giving a negative solution to Kenyon's problem in full generality. We then prove that domes exist over a dense set of integral curves. Finally, we give an explicit construction of domes over all regular n-gons. Добавить комментарий: |
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