T H E M AT H E M AT I C S O F
B OTANY
• • • • •
by Gary Nored and Kate McKenna ~ artwork by Kate McKenna
Artist Kate McKenna of Fort Davis took two agave plants into her yard and dissected them to see if the plant revealed whether it had parts and
pieces that fit the Fibonacci number sequence.The results of her research included an oil painting that is 84 inches tall and 30 inches wide (shown
here) and a series of drawings and field notes.The painting is in a private collection, but the results of her research can be viewed through March at
the Chihuahuan Desert Nature Center’s Visitors’ Center 4 miles south of Fort Davis on Hwy. 118.
T
4
here are no num-
bers in all of
mathematics as
ubiquitous as these. First
described to Europeans
in the 13th century,
they’ve occupied the
minds of scientists, math-
ematicians and natural-
ists for almost 800 years.
In 1220, Leonardo of
Pisa, better known now
as “Fibonacci,” finished
his book Liber Abaci,
which introduced to
Europe for the first time
the method of number-
ing we now call “Arabic.”
In it, he also described a
series of numbers that
bear his name today
–
“The
Fibonacci
sequence.”
The Fibonacci se-
quence is comprised of
an infinite number of ele-
ments wherein each new
element is generated by
adding together the last
two numbers preceding
it. The first two numbers
of the series are 0 and 1.
Thus, 0 and 1 equals 1.
Then 1 and 2 equals 3,
then 2 and 3 equal 5. So
the series starts out like
this: 0, 1, 1, 2, 3, 5 then 8,
13, 21, 34, 55 – and
continues on forever.
The numbers them-
selves have interesting
properties, but the most
interesting thing to natu-
ralists is how often they
appear in nature. They
appear as the number of
petals in a flower, leaves
Cenizo
First Quarter 2010
in a spiral, seeds in a sunflower and in
the dividing of roots and branches. And
they appear just as frequently in animals.
Most of the really interesting ways we
see Fibonacci numbers are actually
more closely related to another remark-
able number – the Golden Ratio. Like
the Fibonacci sequence, mathematicians
have spent lifetimes studying this num-
ber – a number that can be derived from
the Fibonacci sequence. If we compute
the ratio of any number in the Fibonacci
sequence to its preceding number and
repeat this process up to larger numbers,
the value we obtain approaches the
value of the Golden Ratio forever.
The Golden Ratio is the basis of the
Golden Rectangle, which in turn, leads
us to the geometry of natural things.
The Golden Rectangle appears
everywhere in nature. For example, the
DNA molecule measures 34 angstroms
long by 21 wide for each full cycle of its
spiral. Twenty-one and 34 are Fibonacci
numbers, and their ratio closely approx-
imates the value of the Golden Ratio. If
you could draw a rectangle around this
section of DNA, you would have a
Golden Rectangle.
The Golden Rectangle has two
remarkable properties: If you cut a
square out of the rectangle, what’s left is
another Golden Rectangle. If you do
this several times, you create a pattern of
squares and rectangles. Now, if you
draw an arc between the opposite cor-
ners of the squares, you’ll begin to see
something quite beautiful and familiar –
a spiral – a Fibonacci spiral.
Like other appearances of the
Fibonacci sequence, this spiral is every-
where. It occurs in every size from galax-
ies to the fibers in cell walls of bacteria.
It’s the spiral of the Nautilus shell, of
curling waves, the unwinding of a fern
and of the human fingerprint. It
describes the arrangements of florets in
a blossom, bracts in a pine cone, spines
of a cactus, the proboscis of moths –
even the cochlea of the inner ear.
So what exactly accounts for all these
spirals? While their appearance in many
forms remains a mystery, we’re begin-
ning to understand how they happen in
plants.
A growing stem continually produces
a growth hormone called “auxin.”
When a new shoot starts, it depletes the
auxin in its immediate area. Subsequent
shoots, therefore, always start as far away
from the first shoot as possible, because
that’s where the most auxin is to be
found.
As the stem continues to create new
shoots, this simple behavior creates the
spirals. Mathematical models of this
process create the same types of spirals
that plants do, and like plants, the num-
ber of criss-crossing spirals they create is
usually a Fibonacci number.
If you want to explore Fibonacci
numbers in nature, go look at the sun-
flowers growing by the side of the road
or at a cactus.
Clearly the Fibonacci numbers can
describe much of what is beautiful in
nature. Who would have thought that
mathematics could be so “natural?”
This story is based on an episode of “Nature
Notes” produced by KRTS, Marfa Public
Radio, in cooperation with the Chihuahuan
Desert Research Institute in Fort Davis. “Nature
Notes” is heard throughout the region at 9:35
a.m. and again at 7:06 p.m. Thursdays on
KRTS, 93.5 FM.