| |
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 .
. . .
|
Leonardo
de Pisa, a.k.a. Fibonacci
|
|
The classic
Fibonacci spiral
|
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.
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
|
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.
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
|
Fibonacci
spiral
reimagined
|
|
|
|
|
|
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 >
|
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
|
|
|
