Spirals and Vortices pp 91-112 | Cite as

# Spirals, Their Types and Peculiarities

## Abstract

In this chapter, we provide mathematical data concerning the description of spirals. Before starting with mathematical equations, Albrecht Dürer’s pioneering works are briefly introduced. Subsequently, we discuss some properties of different spirals in a plane which make them important in nature and for technical applications. Smooth spirals are usually described by equations which are formulated either in terms of the polar coordinates radius and angle, such spirals being called algebraic, or in terms of curvature and arc length; then they are referred to as pseudo-spirals. We consider in detail a number of spirals of both classes emphasizing their most essential features. Besides 2D spirals we also discuss examples of 3D spirals, usually referred to as helices. To conclude the chapter we mention non-smooth spirals and fractal spirals.

## References

- 1.M. Hazewinkel (ed.),
*Encyclopaedia of Mathematics*(Kluwer, Alphen aan den Rijn, 1994), http://www.encyclopediaofmath.org - 2.A.A. Savelov,
*Planar Curves*(Moscow, 1960) (In Russian)Google Scholar - 3.J.D. Lawrence,
*A Catalog of Special Plane Curves*. Dover Books on Mathematics (Courier Dover, Mineola, 2013), p. 186Google Scholar - 4.A. Schioner,
*Albrecht Dürer: Genie zwischen Mittelalter und Neuzeit*(Pustet, Regensburg, 2011)Google Scholar - 5.A. Dürer,
*Instructions for Measuring with Compass and Ruler*(Nuremberg, 1525)Google Scholar - 6.J. Havil,
*Nonplussed! Mathematical Proof of Implausible Ideas*(Princeton University, Princeton, 2007), p. 109Google Scholar - 7.G.J. Chin, Flying along a logarithmic spiral. Science
**290**, 1857 (2000)Google Scholar - 8.M. Cortie, The form, function, and synthesis of the molluscan shell, in
*Spiral Symmetry*, ed. by I. Hargittai, C.A. Pickover (World Scientific, Singapore, 1992), p. 370Google Scholar - 9.G. Bertin, C.C. Lin,
*Spiral Structure in Galaxies: A Density Wave Theory*(MIT, Cambridge, 1996), p. 78Google Scholar - 10.M. Livio,
*The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number*(Broadway Books, New York, 2002)Google Scholar - 11.R.A. Dunlap,
*The Golden Ratio and Fibonacci Numbers*(World Scientific Publishing, Singapore, 1997)Google Scholar - 12.N.B. Kellogg,
*The Transition Curve or Curve of Adjustment as Applied to the Alignment of Railroads*, 3rd edn. (BiblioBazaar, Charleston, 2008)Google Scholar - 13.V.G.A. Goss, Application of analytical geometry to the form of gear teeth. Resonance
**18**, 817–831 (2013)Google Scholar - 14.A. James, Loxodromes: a rhumb way to go. Math. Mag.
**77**, 349–356 (2004)Google Scholar - 15.B. Ernst,
*The Magic Mirror of M. C. Escher*(Random House, New York, 1976)Google Scholar - 16.B.B. Mandelbrot,
*The Fractal Geometry of Nature*(W. H. Freemann, New York, 1982)Google Scholar - 17.H.-O. Peitgen, P. Richter,
*The Beauty of Fractals*(Springer, Berlin, 1986)Google Scholar