The Fibonacci Spiral and Pseudospirals

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NCB Deposit # 127

Dr. Cye Waldman
cye@att.net

"The Fibonacci Spiral and Pseudospirals"

More members of the Spiral Family of Plane Curves:
The Spirals of Archimedes, of Fermat, of Euler, of Cornu,
- Hyperbolic, Logarithmic, Spherical, Parabolic, Nielsen's, Seiffert . . . .

Click here to see a page from the Liber Abaci (1228) published by Fibonacci.

The Fibonacci Spiral and Pseuodospirals.
Click here for the full article including Equations.

The Fibonacci spiral is frequently regarded as an approximation to the golden spiral, which is a logarithmic spiral whose growth factor is ø, the golden ratio. Waldman writes,

"We find this amusing because an approximation should be easier than that which is being approximated. Calculation of the usual Fibonacci spiral is a 'cockamamie' process, whereas the logarithmic spiral is a straightforward equation."

The standard definition of a spiral is a curve on a plane that winds around a fixed point at a continuously increasing or decreasing distance from the point. The caveat is an auxiliary stipulation that the curvature is monotonic. Waldman demonstrates both a rational "step-and-arc" algorithm and a closed-form analytic solution for the Fibonacci spiral. Furthermore, he adds to the fusion of art and mathematics by animating a Fibonacci-Mondrian Curve.

The "pdf" file provides extensive Matlab code for your browsing pleasure.
A sampler is as follows:

References

Kappraff, J. and Adamson, G.W. (2004). "Generalized Binet Formulas, Lucas Polynomials and Cyclic Constants," Forma, 19,
355-366. < http://www.scipress.org/journals/forma/pdf/1904/19040355.pdf >

Maynard, P. (2008). "Generalized Binet Formulae," Applied Probability Trust; available at
< http://ms.appliedprobability.org/data/files/Articles%2040/40-3-2.pdf >

Other Waldman contributions to the NCB:
Sinusoidal Spirals: < ..//waldman/waldman.htm >
Bessel Functions    < ..//waldman2/waldman2.htm >
Gamma Funcions   < ..//waldman3/waldman3.htm >
Polynomial Spirals and Beyond   < ..//waldman4/waldman4.htm >

Other spiral Deposits in the NCB:
< ..//spiral/spiral.htm >
< ..//log/log.htm >

Other Fibonacci, Liber Abaci, and Pisa Deposits in the NCB:
< ..//fibonacci/fibonacci.htm >

b. Feb. 2, 1786
Renne

d. May 12, 1895
Paris

Jacques Philippe Marie Binet is linked to Fibonacci and the golden ratio by the following Binet function:

Moreover, others go so far as to suggest Binet might have been the first to have formulated matrix multiplication. Traditionally, this operation is credited to Cayley (1821-1895) who was far younger. In addition Binet overlapped in time and place with Cauchy (1789-1857) and shares credit with Cauchy for the Cauchy-Binet formula.

2013