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 NCB Deposit  # 155

Dr. Cye Waldman
cye@att.net

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"The Fibonacci Spiral Reimagined"

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 . . . .

fibonaccistamp
Leonardo de Pisa, a.k.a. Fibonacci

classic


The classic Fibonacci spiral

reimagined
Fibonacci spiral reimagined
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:
"The image of the Fibonacci spiral shown on the left is so ingrained in the human psyche that the image to its right seems like an error, a fluke, or perhaps simply a trick, a trompe l'oeil."

The animation below shows that not only does the reimagined Fibonacci spiral circumscribe the tiling but that it also is an inscribed Fibonacci spiral in its own right.  More simply put, the spiral on the right is exactly and unmistakeably a true Fibonacci spiral.

fibanimation

The concept depicted in the above animation can be carried on in the same spirit:  the figures below show the second and fourth iterations.


threespirals

Three Fibonacci spirals


image4
Five Fibonacci spirals

The animation below shows sixty iterations.  In each frame the new spiral increases by one Fibonacci number, but is simultaneously increased and diminished in size.
phieq
After sixty iterations, it is over three orders of magnitude greater than the original.

swirl

We have previously reimagined the Fibonacci spiral as circumventing 45-45-90 degree triangles, as shown below.  But this mosaic is not a tessellation, as is the original.


image5


The classic Fibonacci spiral

image6

Fibonacci spiral reimagined

References
Other Waldman spirals in the NCB:
Sinusoidal Spirals:  < http://curvebank.calstatela.edu/waldman/waldman.htm >
Polynomial Spirals and Beyond:  < http://curvebank.calstatela.edu/waldman4/waldman4.htm >
The Fibonacci and Pseudospirals: < http://curvebank.calstatela.edu/waldman6/waldman6.htm >
The "Other" Fibonacci and Binet Spirals: < http://curvebank.calstatela.edu/waldman7/waldman7.htm >
Gnomons:  < http://curvebank.calstatela.edu/waldman16/waldman16.htm >

Other spiral Deposits in the NCB:
< http://curvebank.calstatela.edu/spiral/spiral.htm >
< http://curvebank.calstatela.edu/log/log.htm >

Other Fibonacci, Liber Abaci, and Pisa Deposits in the NCB:
< http://curvebank.calstatela.edu/fibonacci/fibonacci.htm >
Binet

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:

Golden ratio

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.


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