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Discrete torsion and its application for a generalized Van der Waerden’s theorem. (English) Zbl 1238.51006

The authors study the discrete curvature and torsion for spatial polygonal lines of unit sides. Using inner products of oriented sides they express geometric conditions on a polygon. As an application and important result they give a generalization of van der Warden’s theorem, valid for the 5-dimensional case.

MSC:

51M04 Elementary problems in Euclidean geometries
51M20 Polyhedra and polytopes; regular figures, division of spaces
52C30 Planar arrangements of lines and pseudolines (aspects of discrete geometry)
53A04 Curves in Euclidean and related spaces
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References:

[1] J. D. Dunitz and J. Waser, The planarity of the equilateral, isogonal pentagon, Elem. Math. 27 (1972), no. 2, 25-32. · Zbl 0236.50003
[2] A. Goetz, Introduction to differential geometry , Addison Wesley Publishing Co., Reading, MA, 1970. · Zbl 0199.56101
[3] S. Hu, M. Lundgren and A. J. Niemi, Discrete Frenet frame, inflection point solitons, and curve visualization with applications to folded proteins, Phys. Rev. E 83 (2011), 061908.
[4] J. M. Sullivan, Curves of finite total curvature, in Discrete differential geometry , Oberwolfach Semin., 38, Birkhäuser, Basel, 2008, pp. 137-161. · Zbl 1151.53305
[5] B. L. van der Waerden, Ein Satz über räumliche Fünfecke, Elem. Math. 25 (1970), no. 4, 73-78. · Zbl 0196.24101
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