Some bendings of a long cylinder
- V. A. Zalgaller
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Elementary tools are applied to describe piecewise-linear isometric embeddings of cylindrical surfaces in ℝ3. Let T2 be a flat torus, let γ⊂T2 be the shortest closed geodesic of length lo, and let k be a fixed positive integer. We assume that if l is the length of any closed geodesic on T2 which is homotopic neither to γ nor to any power of γ, then l>kl0. It is shown how to embed T2 in ℝ3 if k is sufficiently large. The same problem is solved for a flat skew torus T2. It is also shown that if a knot of arbitrary type in ℝ3 is fixed and k is sufficiently large, then T2 can be isometrically embedded in ℝ3 as a tube knotted according to the type of fixed knot. Bibliography; 4 titles.
- S. Yu. Afon'kin and E. Yu. Afon'kina, Origami: Games and Tricks with Paper [in Russian], Petersburg Origami Center, Saint Petersburg (1994).
- C. P. Rourke and B. J. Sandersen, Introduction to Piecewise-Linear Topology (Ergeb. Math., 69), Springer, Berlin (1972).
- Yu. D. Burago and V. A. Zalgaller, “Isometric piecewise-linear embeddings of surfaces with polyhedral metric in ℝn,” Algebra Analiz, 7, No. 3, 76–95 (1995).
- A. D. Milka, “Linear bendings of regular convex polyhedra,” Mat. Fiz., Anal., Geom., 1, No. 1, 116–130 (1994).
- Some bendings of a long cylinder
Journal of Mathematical Sciences
Volume 100, Issue 3 , pp 2228-2238
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