PBS Nova: The great math mystery (The Great Math Mystery)

Roger, copy mission.

We live in an age of astonishing advances.

Descending at about .75 meters per second.

Engineers can land a car-size rover on Mars.

Touchdown confirmed.

Physicists probe the essence of all matter,

while we communicate wirelessly on a vast worldwide network.

But underlying all of these modern wonders

is something deep and mysteriously powerful.

It's been called the language of the universe,

and perhaps it's civilization's greatest achievement.

Its name?

Mathematics.

But where does math come from?

And why in science does it work so well?

Albert Einstein wondered,

"How is it possible that mathematics

does so well in explaining the universe as we see it?"

Is mathematics even human?

There doesn't really seem to be an upper limit

to the numerical abilities of animals.

And is it the key to the cosmos?

Our physical world

doesn't just have some mathematical properties,

but it has only mathematical properties.

"The Great Math Mystery," next on NOVA!

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And the George D. Smith Fund.

Human beings have always looked at nature

and searched for patterns.

Eons ago, we gazed at the stars

and discovered patterns we call constellations,

even coming to believe they might control our destiny.

We've watched the days turn to night and back to day,

and seasons as they come and go,

and called that pattern "time."

We see symmetrical patterns in the human body

and the tiger's stripes

and build those patterns into what we create,

from art to our cities.

But what do patterns tell us?

Why should the spiral shape of the nautilus shell

be so similar to the spiral of a galaxy?

Or the spiral found in a sliced open head of cabbage?

When scientists seek to understand

the patterns of our world,

they often turn to a powerful tool: mathematics.

They quantify their observations

and use mathematical techniques to examine them,

hoping to discover the underlying causes

of nature's rhythms and regularities.

And it's worked, revealing the secrets

behind the elliptical orbits of the planets

to the electromagnetic waves that connect our cell phones.

Mathematics has even guided the way,

leading us right down

to the sub-atomic building blocks of matter.

Which raises the question: why does it work at all?

Is there an inherent mathematical nature to reality?

Or is mathematics all in our heads?

Mario Livio is an astrophysicist

who wrestles with these questions.

He's fascinated by the deep and often mysterious connection

between mathematics and the world.

If you look at nature, there are numbers all around us.

You know, look at flowers, for example.

So there are many flowers

that have three petals like this, or five like this.

Some of them may have 34 or 55.

These numbers occur very often.

These may sound like random numbers,

but they're all part of what is known as the Fibonacci sequence,

a series of numbers developed by a 13th century mathematician.

You start with the numbers one and one,

and from that point on,

you keep adding up the last two numbers.

So one plus one is two,

now one plus two is three,

two plus three is five,

three plus five is eight, and you keep going like this.

Today, hundreds of years later,

this seemingly arbitrary progression of numbers

fascinates many, who see in it clues

to everything from human beauty to the stock market.

While most of those claims remain unproven,

it is curious how evolution seems to favor these numbers.

And as it turns out,

this sequence appears quite frequently in nature.

Fibonacci numbers show up in petal counts,

especially of daisies, but that's just a start.

Statistically, the Fibonacci numbers

do appear a lot in botany.

For instance, if you look at theottom of a pine cone,

you will see often spirals in their scales.

You end up counting those spirals,

you'll usually find a Fibonacci number,

and then you will count the spirals

going in the other direction

and you will find an adjacent Fibonacci number.

The same is true of the seeds on a sunflower head--

two sets of spirals.

And if you count the spirals in each direction,

both are Fibonacci numbers.

While there are some theories

explaining the Fibonacci-botany connection,

it still raises some intriguing questions.

So do plants know math?

The short answer to that is "No."

They don't need to know math.

In a very simple, geometric way, they set up a little machine

that creates the Fibonacci sequence in many cases.

The mysterious connections

between the physical world and mathematics run deep.

We all know the number pi from geometry--

the ratio between the circumference of a circle

and its diameter-- and that its decimal digits

go on forever without a repeating pattern.

As of 2013,

it had been calculated out to 12.1 trillion digits.

But somehow, pi is a whole lot more.

Pi appears in a whole host of other phenomena

which have, at least on the face of it,

nothing to do with circles or anything.

In particular, it appears in probability theory quite a bit.

Suppose I take this needle.

So the length of the needle

is equal to the distance between two lines

on this piece of paper.

And suppose I drop this needle now on the paper.

Sometimes when you drop the needle, it will cut a line,

and sometimes it drops between the lines.

It turns out the probability

that the needle lands so it cuts a line

is exactly two over pi, or about...

...64%.

Now, what that means is that, in principle,

I could drop this needle millions of times.

I could count the times when it crosses a line

and when it doesn't cross a line,

and I could actually even calculate pi

even though there are no circles here,

no diameters of a circle, nothing like that.

It's really amazing.

Since pi relates a round object, a circle,

with a straight one, its diameter,

it can show up in the strangest of places.

Some see it in the meandering path of rivers.

A river's actual length

as it winds its way from its source to its mouth

compared to the direct distance on average seems to be about pi.

Models for just about anything involving waves

will have pi in them, like those for light and sound.

Pi tells us which colors should appear in a rainbow,

and how middle C should sound on a piano.

Pi shows up in apples,

in the way cells grow into spherical shapes,

or in the brightness of a supernova.

One writer has suggested

it's like seeing pi on a series of mountain peaks,

poking out of a fog-shrouded valley.

We know there's a way they're all connected,

but it's not always obvious how.

Pi is but one example

of a vast interconnected web of mathematics

that seems to reveal

an often hidden and deep order to our world.

Physicist Max Tegmark from MIT thinks he knows why.

He sees similarities between our world

and that of a computer game.

If I were a character in a computer game

that were so advanced that I were actually conscious

and I started exploring my video game world,

it would actually feel to me like it was made

of real solid objects made of physical stuff.

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Yet, if I started studying, as the curious physicist that I am,

the properties of this stuff,

the equations by which things move

and the equations that give stuff its properties,

I would discover eventually

that all these properties were mathematical:

the mathematical properties

that the programmer had actually put into the software

that describes everything.

The laws of physics in a game--

like how an object floats, bounces, or crashes--

are only mathematical rules created by a programmer.

Ultimately, the entire "universe" of a computer game

is just numbers and equations.

That's exactly what I perceive in this reality, too,

as a physicist,

that the closer I look at things that seem non-mathematical,

like my arm here and my hand,