Current time:0:00Total duration:6:07

0 energy points

Studying for a test? Prepare with these 9 lessons on Doodling in Math and more.

See 9 lessons

# Doodling in math: Spirals, Fibonacci, and being a plant [3 of 3]

Video transcript

So say you're me and
you're in math class and you're trying to
ignore the teacher and doodle Fibonacci
spirals, while simultaneously trying to fend off
the local greenery. Only, you become
interested in something that the teacher
says by accident, and so you draw too many
squares to start with. So you cross some out,
but cross out too many, and then the teacher
gets back on track, and the moment is over. So oh well, might
as well try and just do the spiral from here. So you make a three
by three square. And here's a four by four, and
then seven, and then 11 This works because then you've
got a spiral of squares. So you write down the
numbers, 1, 3, 4, 7, 11, 18. It's kind of like the Fibonacci
series, because 1 plus 3 is 4, 3 plus 4 is 7, and so on. Or maybe it starts at 2 plus
1, or negative 1 plus 2. Either way, it's 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 pine cone that has
seven spirals one way, and eleven 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 around. What angle would
give Lucas numbers? In this pine cone, each
new pine cone-y thing is about 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
the Lucas number plants. It's an easy way to
grow Lucas spirals, if plants have an
internal angle-a-tron. Thing is, 100 is
pretty far from 137.5. If plants were somehow
measuring angle, 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'd believe different
species use different angles, but two pine cones
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 a fundamentally
different growth pattern, and are in a different
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 a good
clue that maybe these plants get this angle, and
Fibonacci numbers, as a consequence of some other
process and not just because it mathematically optimizes
sunlight exposure if the sun is right overhead, which
it pretty much never is, and if the plan 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
where the first to form off the meristem, and the little
ones around the center are newer. As the plants grow, they get
pushed away from the meristem, but they all started there. The important part is
that a science observer would see the plant
bits pushing away, not just from the meristem,
but from each other. A couple physicists
once tried this thing where they dropped 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 how plant bits
move away from the center. The first couple jobs would
head in opposite directions 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 numbers spirals. So all a plant would
need to do to get Fibonacci numbers
spirals is figure out how to make the plant
bits repel each other. We don't know all the details,
but here's what we do know. There's a hormone that
tells plant bits to grow. A plant bit might use up
the hormone around it, but there's more
further away, so it'll 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 going
to grow in places that aren't too
crowded, because that's where there's the
most growth hormone. This leads them to
move further out into the space left by the
other outward moving plant bits. And once everything gets
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 an 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
grow to fill in the space. Mathematicians and programmers
made their own simulations, and found the same thing. The best way to fit new
things in with the most space, has them pop up at that angle,
not because the plant knows about the angle,
but because that's where the most
hormone has built up. Once it gets started, it's
a self perpetuating cycle. All these flower
bits are doing is growing where there's
the most room for them. The rest happens
auto-mathically. 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
pine cones 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, then
all this leaf cares 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 of where there's the 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 maybe is like double the Lucas numbers. And this pine cone has 6 and
10, double Fibonacci numbers. So how is a pineapple
like a pine cone? What do daisies and Brussels
sprouts have in common? It's 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 so 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. Maybe if you keep
doodling in math class, you can help figure it out.