So say you're me and you're in math class
and you're trying to ignore the teacher and
doodle fibbonachi spirals while simultaneously trying to
fend off the local greenery,
only if you become interested in
something the teacher said by accident.
And so you draw too many squares to start with.
So you cross them out, but you crossed out too many,
and then the teacher gets back on track and
the moment is over, so...
Oh well, might as well try to do the spiral from here.
So you make a 3 by 3 square,
and here's a 4 by 4
and then 7 and then 11...
This works, as in you got a spiral of squares,
so you write down the numbers.
1, 3, 4, 7, 11, 18.
is kind of like the Fibonacci series,
because 1 + 3 is 4
3 + 4 is 7 and so on
Or maybe you start as 2 + 1
or -1 + 2
Either way is a perfectly good series,
and it's got another similarity
with the Fibonacci series.
The ratios of consecutive numbers also approach phi.
Ok, so a lot of plants have Fibonacci numbers of spirals,
but to understand how they do it
we can learn from the exceptions.
This pinecone that has 7 spirals one way
and 11 the other, might be showing Lucas numbers.
And since Fibonacci numbers and Lucas numbers
are related, maybe that explains it.
One theory was that plants get Fibonacci numbers
by always growing new parts
a phi-th of a circle all around.
What angle will give Lucas numbers?
In this pinecone, each new pinecony thing
is about a 100 degrees around from the last.
We're going to need a Lucas angle-a-tron.
It's easy to get a 90 degree angle-a-tron,
and if I take a third of a third of that,
that's a ninth of 90 which is another 10 degrees.
There.
Now you can use it to get spiral patterns
like what you see on a Lucas number plants.
It's an easy way to grow Lucas spirals
if plants have an internal angle-a-tron.
Thing is, a hundred is pretty far from 137.5.
If plants were somehow meassuring angles,
you'd think the anomalous ones
would show angles close to a phi-th of a circle,
not jump all the way to 100.
Maybe I believe different species
use different angles,
but two pinecones from the same tree,
two spirals on the same cauliflower?
And that's not the only exception.
A lot of plants don't grow spirally at all.
Like this thing with leaves growing opposite from each other.
And some plants have alternating leaves,
180 degrees from each other,
which is far from both phi and Lucas angles.
And you could say that these don't count,
because they have fundamentally different growth pattern
and they are different in class of plant or something.
But wouldn't it be even better if there were
one simple reason for all of these things?
These variations are good clue that
maybe these plants get this angle
and Fibonacci number as a consequence of
some other process and not just because
it mathematically optimises sunlight exposure.
If this sun is right over head
which pretty much never is and
if the plants are perfectly facing straight up which they aren't.
So how do they do it?
Well you could try observing them,
that would be like science.
If you zoom in on the tip of a plant, the growing part,
there's this part called the meristem.
That's where new plant bits form.
The biggest plant bits were
the first to form of the meristem,
and the little ones around the center are newer.
As the plant grows
they get pushed away from the meristem
but they all started there.
The important part is that
the science observer would see
the plant bits pushing away
not just from the meristem
but from each other.
A couple physicists want to try this thing
where they drop drops of a magnetized liquid
in a dish of oil.
The drops repelled each other
kind of like plant bits do and
were attracted to the edge of the dish
just like a plant bits move away from the center.
The first couple drops would head
in opposite direction from each other,
but then the third was repelled by both,
but pushed farther by the more recent and closer drop.
It and each new drop would set off at a phi angle
relative to the drop before
and the drops ended up forming Fibonacci number spirals.
So all the plant would need to do
to get Fibonacci number spirals, is
to figure out how to make the plant bits repel each other.
We don't know all the details.
But here is what we do know.
There is the hormone that tells plant bits to grow.
A plant bit might use up the hormone around it.
But there is more further away,
so it will grow in that direction.
That makes plant bits move out from the meristem
after they form.
Meanwhile the meristem keeps forming new plant bits
and they're gonna grow in places that aren't too crowded
because that's where there's the most growth hormone.
This leaves them to move further out
into the space left by the other outward moving plant bits.
And once everything get locked into a pattern
it's hard to get out of it,
because there's no way for this plant bit
to wander off unless there were empty space
with a trail of plant hormone to lead it out of its spot,
but if there were
all the nearer plant bits would use up the hormone
in grow to fill out in space.
Mathematicians and programmers
was made their own simulations
and found the same thing.
The best way to fit new things in
with the most space
has some pop-up at that angle,
not because plant knows about the angle,
but because that's where
the most hormone has build up.
Once it gets started, it's the self-perpetuating cycle.
All that these flower bits are doing
is growing where there is most room for them.
The rest happens auto-math-ically.
It's not weird that all these plants
show Fibonacci numbers,
it would be weird if they didn't.
It had to be this way.
The best thing about that theory
is that it explains why Lucas pinecones would happen.
If something goes a bit differently in the very beginning
the meristem will settle into a different but stable pattern
of where there's the most room to add new plant bits.
That is 100 degrees away.
It even explains alternating leaf patterns.
If the leaves are far enough apart,
relative to how much growth hormone they like,
that these leaves don't have
any repelling force with each other,
and all these leaves care about is
being farthest away from the two above and below it,
which makes 180 degrees optimal.
And when leaves grow in pairs
that are opposite each other
the answer where there's most room
for both of those leaves
is at 90 degrees from the one below it.
And if you look hard
you can discover even more unusual patterns.
The dots on the neck of this whatever it is
come in spirals of 14 and 22
which may be as like doubled a Lucas numbers,
and this pinecone has 6 and 10 -
doubled Fibonacci numbers.
So how is the pineapple like a pinecone,
what do daisies and brussels
perhaps have in common?
Is not the numbers they show, it's how they grow.
This pattern is not just useful, not just beautiful.
It's inevitable.
This is why science and mathematics are still much fun.
You discover things that seem impossible to be true
and then get to figure out
why it's impossible for them not to be.
To get this far in our understanding of these things
it took the combined effort of mathematicians,
physicists, botanists and biochemists,
and we've certainly learned a lot,
but there's much more to be discovered.
May be you should keep doodling in math class?
You can help figure it out.