Like a circle in a spiral / Like a wheel within a wheel / Never ending or beginning / On an everspinning reel. The Windmills Of Your Mind
Broadly speaking, a spiral curve originates at a central point and gets further away (or closer) as it revolves around the point. Spirals abound in nature, being found at all scales from the whorls at our fingertips to vast rotating spiral galaxies. The seeds in a sunflower are arranged in spiral segments. In the technical world, the grooves of a gramophone record and the coils of a watch balancespring are spiral in form.
Spiral Equations
In polar coordinates, the radial distance r ( θ ) from the central point is a monotonic function of the azimuthal angle θ. There are several canonical spiral forms. The simplest is the Archimedean spiral, r = a θ. This is generated by a point moving with uniform speed along a ray that is rotating with constant angular speed. Since the points of intersection with a fixed ray from the origin are evenly spaced with separation 2 π a, it is also called an arithmetic spiral (left panel of figure above). It was described by Archimedes in his work On Spirlas.
More generally, we consider r^{k} = a^{k} θ. For k = 2 we get Fermat’s spiral r = a √ θ (centre panel of Fig). For k = –1 we have r = a / θ, which is a hyperbolic spiral (right panel of Fig).
An equiangular spiral is such that every ray from the origin cuts it at the same angle. Its equation is r = a exp ( b θ ). Since b θ = log (r / a ), it is also called a logarithmic spiral and, since the intersection points with a fixed ray form a geometric sequence, the name geometric spiral is also used. Christopher Wren observed that many seashells, such at the Nautilus, have logarithmic spiral crosssections.
Involutes
Suppose you wish to cut the grass. Here is an easy way:

Erect a stout column in the centre of the lawn;

Tie the mower to the column with a long rope;

Start it so that it winds inward in a spiral arc;

Relax and enjoy the magic of the automower.
The curve traced by the mower looks like an Archimedean spiral. It is actually slightly different: it is the involute of a circle (namely, the circular crosssection of the central column). The column should be chosen to have circumference 2 π a = D where D is the blade diameter of the mower.
If the column is described by the equations
x = a cos θ , y = a sin θ
where a is the radius of the circle, then the curve traced out by the mower is
x = a ( cos θ + θ sin θ ) , y = a ( sin θ – θ cos θ )
The radius vector is r^{2} = a^{2} ( 1 + θ ^{2} ) . For θ = 0 we have r = a, whereas, for the Archimedean spiral ( r^{2} = a^{2} θ ^{2} ) we have r = 0 when θ = 0. The two curves are close, but not identical:
Acknowledgement:
This article was inspired by correspondence with Stephen Richardson of Exeter NH.
Sources:
Wells, David, 1991: The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, ISBN: 9780140118131.
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