# Nested Helices

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Ropes, cords, strings, and twine are composed of threads. Many of these threads form helices around helices around helices, or "nested helices" [1].

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Contributed by: Erik Mahieu (June 2015)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

According to [2], an -tuple helix can be expressed recursively as , with , where the and the are the normal and binormal vectors of the Frenet–Serret frame.

The built-in *Mathematica* function FrenetSerretSystem computes the normal and binormal vectors and .

The are the parametric equations. is the angle of rotation along the helices and is the parameter of the . The are the winding directions ( for right, for left). The are the helix angles. The are the helix radii.

References

[1] C. Erdo¨nmez, C. E. İmrak, "Modeling Techniques of Nested Helical Structure Based Geometry for Numerical Analysis," *Journal of Mechanical Engineering,* 57(4), 2014 pp. 283–292. sl.sv-jme.eu/data/upload/2011/04/01_ 2009_ 006_Erdonmez _ 03.pdf.

[2] A. T. Ali, "Position Vectors of General Helices in Euclidean 3-Space," *Bulletin of Mathematical Analysis and Applications*, 3(2), 2011 pp. 198–205. www.emis.de/journals/BMAA/repository/docs/BMAA3-2-19.pdf.

[3] C. Erdo¨nmez, "-Tuple Complex Helical Geometry Modeling Using Parametric Equations," *Engineering with Computers*, 30(4), 2014 pp. 715–726. doi:10.1007/s00366-013-0319-9.

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