Cicadas In Their Prime
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2,300 years ago, the
Greek scientific writer, poet and astronomer, Eratosthenes of Cyrene,
became the first person to calculate the circumference of the Earth. He
also dabbled in a bit of maths, and invented the famous Sieve of
Eratosthenes, which is an easy method for finding prime numbers. But
now it seems that some recent research into cicadas gives us another
way of finding prime numbers.
Prime numbers are numbers that can
be divided only by themselves and one. So 2, 3, 5, 7, 11, 13 and 17 are
all prime numbers  but 18 is not a prime number, because you can
divide it by both 2 and 9. To work the Sieve of Eratosthenes you write
down all the numbers, and then simply strike out every second number
that comes after 2, every third number following the number 3, and so
forth. All the numbers that you have left will be prime numbers. It's
called a Sieve because all the numbers that are not prime just "fall
through".
You also find prime numbers in the life cycles of
cicadas. There are about 1,500 species of cicadas known. There are
those that appear yearly in midsummer, and there are also the socalled
"periodic" cicadas. They appear at prime number intervals  7 years, 13
years and 17 years. The cicadas are part of the insect order
Homoptera. These are all sucking insects, which pierce plants with
their pointy mouthparts and suck out the juices. The breeding cycle
begins when huge numbers of adult cicadas emerge in the spring. They
mate within a week, and a few days later, the female lays her eggs. She
drills into the wood of trees, and inserts up to some 400to600 eggs.
These eggs hatch up after two to six weeks. The little babies make
their way down to the ground (by crawling down, or just dropping), dig
their way into the soil with their claws and begin the next phase of
their life, feeding on the roots of shrubs and trees for the next 6, 12
or 16 years. The 17year cicadas are almost fully grown into nymphs by
8 years, but they continue to feed underground until the 17th year when
they come out of the soil, and attach themselves to any nearby tree or
post. Their shell splits open, the adults emerge and live only for a
few weeks before dying.
Now biologists have asked for a long
time whether it's just a coincidence that the emergence period of the
three species of periodic cicadas (7, 13 and 17 years) are all prime
numbers.
One previous theory was that if the cicadas are running
on different cycles, and if these cycles are prime numbers, they'll
cross over only very rarely. For example, a 13year cycle and 17year
cycle will meet only every 221 years. That means that both species of
cicadas would come out in huge numbers and all have to compete for the
same amount of food only once every 221 years. The rest of the time,
there would be enough food.
This is a nice theory, but Mario
Markus, a physicist from the Max Planck Institute for Molecular
Physiology in Germany has come up with a new theory. It's related to
periodic predators. Suppose there are some predators (like birds, and
the Cicada Killer Wasp) that attack cicadas, and that the cicadas
emerge every 12 years. Then the predators that come out every two years
will attack them, and so will the predators that come out every 3
years, 4 years and 6 years. But according Mario Markus, "if the cicadas
mutate to 13year cycles, they will survive."
So Markus and his
colleagues created a mathematical model. In this mathematical model, if
a prey happens to be met by a predator, then it loses. According to
this mathematical model, as the years roll by, the length of the cycle
increases until the cicadas hit a prime number, and then it stays
there.
This model has an unexpected and delightful side
effect. It turns out to be a machine, like the Sieve of Eratosthenes
2,300 years ago, that can generate prime numbers. Large prime numbers
are rare, and they're difficult to find, but a biological mathematical
model like this, based on cicadas, will click through the nonprime
numbers, and land on the primes  and that will leave the
mathematicians chirping.
A letter from Paul NorrisHi Dr Karl I just finished your article on Cicadas and prime numbers. I don't have anything to add, but just for your interest, here is a sequence of photographs I took of a Green Grocer making the change from nymph to adult.
I
was surprised by the amount of volition shown by the cicada. I would
have thought that bursting out of your skin would be something that
just happens regardless of whether you're ready. I found the nymph
above the ground early on a Sunday morning. I brought it home to show
the kids and possibly treat them to the experience of it emerging. I
put it into a tall jar with some moist soil in the bottom (to prevent
dehydration), some leaves, and a stick for it to climb up.
Sunday night came and the nymph climbed the stick but just hung there.
Monday morning and no cicada in sight.
Monday night and nymphonastick again.
Tuesday morning and no cicada.
By
this time I was very concerned that the nymph would die and I wondered
if perhaps it wasn't ready to emerge. On Tuesday night I came home late
from work (about 10pm) only to see the nymphonastick trick again. I
decided to release it. I loosened some soil in the ground and emptied
the nymph onto it. To my surprise, rather than digging, the nymph
seemed more interested in climbing. I quickly planted the stick from
the jar and put the nymph at the base of it. The nymph climbed to the
top of the stick and started reaching upward. Clearly the nymph wanted
to be higher.
I put the nymph at the base of our little lemon
tree and rushed inside for the photographic equipment. By the time I
had set up, the nymph was still climbing so I dashed inside, threw some
food onto a plate and raced back out. I watched as I ate my dinner and
in turn fed the local mosquitoes. From here, the photos tell the story.
I learned two things in particular from this experience:
Firstly
the cicada seems to have absolute choice over whether it emerges or
not. This one had waited at least three days and then hatched as soon
as it was happy. I have in the past lifted paving slabs to discover the
tunnels and bodies of cicada nymphs which were clearly ready to hatch.
Clearly the nymph had chosen to die rather than hatch. If hatching were
involuntary then I would have found the body of a dead cicada with the
empty shell nearby. On one occasion I released a nymph from under the
paving but it had waited too long and died during the day.
Secondly,
it is clear that at some stage the nymph ceases to be a nymph and
becomes a cicada wearing a nymph shell. Clearly this must cause
complications for breathing, digging your way out of the ground,
climbing a tree, feeding, and seeing
Hope you found this as interesting as I did.
Cheers, Paul Norris cicada images © Paul Norris
© Karl S. Kruszelnicki Pty Ltd 2003.
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