The Fibonacci Spiral and Pseudospirals

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NCB Deposit # 152

Dr. Cye Waldman
cye@att.net

"Gnomons Land"

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.

Click here for the full article including equations and code.

A Dignomonic Tiling and Associated Spirals

A Monognomonic Tiling of the Golden Rectangle and Spiral

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,

"Whorled figures are well known, albeit perhaps not by that name. The Fibonacci tiling is an example. The figure is built up by adding squares in a spiral manner such that there are no unfilled spaces. In this case, the tiles grow in size as the Fibonacci sequence. In general, the first tile is called the seed and the remaining tiles are all of the same shape and grow in size in a manner as to leave no unoccupied territory; they are tessellated."

The "pdf" file provides extensive Matlab code for your browsing pleasure.

A Matlab computer program to produce the mono and dignomonic tilings is provided in the Appendix. It is quite simple to use. If there are no input parameters beyond the number of tiles desired, the program oriduces a monognomic tiling.

References

M.J. Gazalé, Gnomon: From Pharaohs to Tractals. Princeton University Press (1999).

Other Waldman spiral animations in the NCB:
Sinusoidal Spirals: < ..//waldman/waldman.htm >
Bessel Functions    < ..//waldman2/waldman2.htm >
Gamma Funcions   < ..//waldman3/waldman3.htm >
Polynomial Spirals and Beyond   < ..//waldman4/waldman4.htm >
Fibonacci and Binet Spirals with a touch of Mondrian < ..//waldman6/waldman6.htm >
"Other" Fibonacci Spirals and Binet Spirals < ..//waldman7/waldman7.htm >
Voderberg Tiling   < ..//waldman9/waldman9.htm >
Cornu-Voderberg Tilings < ..//waldman10/waldman10.htm >
Aleph Animation < ..//waldman11/waldman11.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.

2016