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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 .
. . .
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Leonardo
de Pisa, a.k.a. Fibonacci |
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The classic
Fibonacci spiral
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Fibonacci spiral reimagined
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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."
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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.
The concept depicted
in the above animation can be carried on in the same spirit: the
figures below show the second and fourth iterations.
Three
Fibonacci spirals
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Five Fibonacci spirals
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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.
After sixty
iterations, it is over three orders of magnitude greater than the
original.
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.
The
classic Fibonacci spiral
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Fibonacci spiral reimagined
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Other
Waldman spirals in the NCB:
Sinusoidal Spirals: < ..//waldman/waldman.htm
>
Polynomial Spirals and Beyond: < ..//waldman4/waldman4.htm
>
The Fibonacci and Pseudospirals: < ..//waldman6/waldman6.htm
>
The "Other" Fibonacci and Binet Spirals: < ..//waldman7/waldman7.htm
>
Gnomons: < ..//waldman16/waldman16.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 >
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b. Feb. 2,
1786
Renne
d.
May 12, 1895
Paris
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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.
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