The Mind-Blowing Mathematics of Sunflowers – Instant Egghead #59

How fast does a colony of rabbits grow,
assuming they breed like rabbits? If mature rabbits produce
a new litter every month, first you’ll have just the one pair… …then two, then three, and then 5 pair as the offspring start
having litters of their own. Soon the colony is growing fast. Eight, thirteen, twenty-one, thirty-four,
and then fifty-five pairs of rabbits. These are the numbers
of the Fibonacci Sequence Named after 13th Century Italian mathematician Leonardo of Pisa
(better known as Fibonacci). He introduced the rabbit problem
about 800 years ago. The sum of two consecutive numbers
in the Fibonacci Sequence gives you the next number. Of course, real rabbits
don’t breed so predictably. But surprisingly, the Fibonacci Sequence
is common in nature. Look at the head of the sunflower.
It typically contains two types of spirals. Thirty-four spirals in one direction
and fifty-five in the other. 34 and 55 appear back to back
in the Fibonacci Sequence. Some sunflowers have even larger
Fibonacci numbers, such as 89 and 144. Pine cones often display similar patterns
with 8 and 13 spirals. Again, consecutive Fibonacci numbers. This can’t be coincidence.
And in fact, there is more math at play. For growing sunflower, it’s beneficial
to push out each new floret as far as possible
from the existing florets. That gives each floret
the most space to grow. Studies have shown that
under many growth conditions each floret should emerge
at an angle 137.5 degrees from the one that came before. Amazingly, 137.5 degrees is a well-known
angle called, “The Golden Angle”. It’s famous for being derived from a number
called the Golden Ratio or Phi. The Golden Ratio is closely related
to the Fibonacci Sequence. Take any number in the sequence, divide it
by the one that came before, and voila! A good approximation of the Golden Ratio. So the Golden Angle produces
the most efficient use of the sunflowers limited space. And the relationship between
the Golden Angle, the Golden Ratio, and the Fibonacci Sequence
is what causes the sunflower spirals to appear in numbers
straight out of Fibonacci. As beautiful as the sunflower is, isn’t even lovelier knowing there is
a deep mathematical order to it? For Scientific American’s
Instant Egghead, I’m John Matson.