Abstract       The purpose of this article is to develop the parametric equation for the plane curve of the equiangular spiral – also known as the logarithmic spiral or the logistique –  from its geometric definition.  Through analysis, it was found to be    where a is an arbitrary constant and a is a constant angle made between the radial vector to any point on the curve and the tangent line at that point.

Introduction

     Most readers will have had at least some informal introduction to the elegant beauty of the spiral prior to reading this. Certainly, most will have witnessed the sublime geometry of a spiral in the vortex of water rushing down a drain, the tempest of a hurricane, the hypnotic pattern of a magician’s spiral, the shape of a snail’s shell, the inscrutable movement of the arms of a spiral galaxy like our own... The formal mathematical study of spirals – as did so many other rich intellectual traditions present today – began with the ancient Greeks. Building upon the work of Conon of Samos, Archimedes of Syracuse wrote a definitive early work on the subject in 250 BC entitled On Spirals. Indeed, the Archimedes’ spiral, as the name implies, is so named in his honor.

     Fascination with spirals did not end with the Greeks. In 1638, while studying dynamics, the great Rene¢ Descartes – for whom the Cartesian coordinate system is named – discovered a spiral that forms a constant angle between a line drawn from the origin to any point on the curve and the tangent line at that point: the equiangular spiral. The rich geometry of this spiral has inspired for it an almost mystical reverence. Jacob Bernoulli, who later demonstrated its "reproductive" properties (what contemporary students of fractals might call self-similarity), referred to it as spira mirabilis, "the wonderful spiral." Moreover, his reverence for it was such that he arranged for it to be inscribed upon his tombstone with the epitaph Eadem mutata resurgo, "I shall arise the same, though changed." The parallel between the self-similarity of the equiangular spiral and his epitaph is obvious – the same geometry, though changed magnitude. The "wonderful" properties of the spiral are further enhanced by its connection to Golden mean and Fibonacci numbers, themselves objects of mystery and intrigue. The Golden mean, f , involves the scenario where one is given a line segment to arbitrarily divide. It turns out that the most aesthetic division – so-called dynamic symmetry – is not in equal halves or a quarter and a third, but rather in a irrational ratio given by f , where f is

Fibonacci numbers are the sequence generated if a pair of rabbits, one male and one female, are released into some isolated habitat. After one month, the rabbits become sexual mature and may procreate. The gestation period for the pregnant rabbit is one month, and she always gives birth to a pair of rabbits, one male, one female. Further, suppose these rabbits are immortal. Then the number of pairs of rabbits at the end of each month produces the sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, ...

     The connection between the equiangular spiral, the Golden mean, and Fibonacci numbers is best explained in the following way:
 

Starting with a single Golden rectangle (of length f and width 1), there is a natural sequence of nested Golden rectangles obtained by removing the leftmost square from the first rectangle, the topmost square from the second rectangle, etc. The length and width of the nth Golden rectangle can be written as linear expressions a+bf, where the coefficients a and b are always Fibonacci numbers (Golden Mean)!

Following this pattern, the Golden rectangles are inscribed on the interior of the spiral. However, the equiangular spiral is not some purely mathematical abstraction. It has been observed in nature in the septa of the Nautilus (as seen in the accompanying image), the development of pinecones, the pattern of seeds in a sunflower, and many organisms where their rate of growth and size are proportional.
 

Spira Mirabilis

     The equiangular spiral is defined in the following manner:
 

1. A spiral – that is, a member of the family of functions r=f(q), where f is a monotonic      increasing function.  2. Draw a line is from the origin to any point P on the spiral, a so-called radial vector.  3. The angle made between the radial vector and the tangent line of the curve at P is constant.  4. Let the constant angle be a.

     Due to the intractability of directly developing a parametric equation from the sophisticated geometry of this spiral, the approach taken here is an indirect one. Namely, the polar form will be developed first, which can be translated into the parametric form more conveniently.

     Consider the figure of the equiangular spiral. If we take the limit as d q approaches zero, the angle Ð PQM approximately equals a .
 

Correspondingly, r then becomes approximately equal to the length of the radial vector. Therefore, as r sweeps through q , the change in r equals rcota , a relationship that allows for the development of a separable differential equation.
 

Let C be an arbitrary constant a.
 

 

Thereby, the polar equation for the equiangular spiral has been derived. However, it is the parametric form that preoccupies us. Therefore, recall that x and y are given by
 

It is then a small matter to make the coordinate conversion.
 

As confirmation that this indeed produces the equiangular spiral, a graph generated using the parametric equation developed is included, where the constant a=1 and a =80^o.
 

 

References

The Golden Mean. Web page at Mathsoft. http://pauillac.inria.fr/algo/bsolve/constant/gold/gold.html

Lee, Xah. Equiangular Spiral. Web page. http://www.best.com/~xah/SpecialPlaneCurves_dir/EquiangularSpiral_dir/EquiangularSpiral.html

Brand, Matthew. Logarithmic Spirals. Web page at Massachusetts Institute of Technology. http://brand.www.media.mit.edu/people/brand/logspiral.html

Weisstein, Eric. Logarithmic Spiral. Web page. http://www.astro.virginia.edu/~eww6n/math/LogarithmicSpiral.html

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, 1967.

Yates, Robert C. Curves and Their Properties. National Council of Teachers of Mathematics, 1974.

Lawrence, Dennis J. Special Plane Curves. New York, New York: Dover Books, 1972.