A hidden pattern is popping up in seemingly unrelated places,
A hidden pattern is popping up in seemingly unrelated places,

from a bus system in Mexico to chicken eyes
from a bus system in Mexico to chicken eyes

to number theory and quantum physics.
to number theory and quantum physics.

This phenomenon, known as universality,
This phenomenon, known as universality,

continues to surprise mathematicians and reveal
continues to surprise mathematicians and reveal

a deeper understanding of our world.
a deeper understanding of our world.

Imagine arriving at an empty bus stop in New York City.
Imagine arriving at an empty bus stop in New York City.

The last bus must have just left.
The last bus must have just left.

But the sign says a bus comes every 10 minutes on average.
But the sign says a bus comes every 10 minutes on average.

What's the probability that the next bus will
What's the probability that the next bus will

arrive within five minutes?
arrive within five minutes?

The probability of a random event happening
The probability of a random event happening

within some interval, like a bus coming within the next five
within some interval, like a bus coming within the next five

minutes, is given by a curve called a Poisson distribution.
minutes, is given by a curve called a Poisson distribution.

But what if bus arrival times are not independent?
But what if bus arrival times are not independent?

In the 1970s in Cuernavaca, Mexico,
In the 1970s in Cuernavaca, Mexico,

bus drivers would hire spies to sit along their route.
bus drivers would hire spies to sit along their route.

And drivers either speed up or wait at a stop
And drivers either speed up or wait at a stop

depending on how long ago their spy said the previous bus let.
depending on how long ago their spy said the previous bus let.

This spaced out the buses and maximized their profits.
This spaced out the buses and maximized their profits.

In that case, the spacing between buses
In that case, the spacing between buses

is defined by a very different probability distribution.
is defined by a very different probability distribution.

The Cuernavaca bus system, the Riemann zeta function
The Cuernavaca bus system, the Riemann zeta function

related to prime numbers, chicken retinas,
related to prime numbers, chicken retinas,

and atomic nuclei are all examples
and atomic nuclei are all examples

of complex correlated systems.
of complex correlated systems.

The components of these systems aren't independent.
The components of these systems aren't independent.

They interact and repel one another.
They interact and repel one another.

And this leads to a statistical distribution
And this leads to a statistical distribution

in between randomness and order.
in between randomness and order.

The same distribution or pattern arises,
The same distribution or pattern arises,

even though the components of these various systems
even though the components of these various systems

are very different.
are very different.

They are said to exhibit universality.
They are said to exhibit universality.

Mathematicians model these complex
Mathematicians model these complex

correlated systems using random matrices.
correlated systems using random matrices.

The numbers in random matrices are drawn randomly
The numbers in random matrices are drawn randomly

from probability distributions.
from probability distributions.

The matrix might be randomly filled
The matrix might be randomly filled

with zeros and ones or any set of numbers,
with zeros and ones or any set of numbers,

like the integers between 1 and 100.
like the integers between 1 and 100.

You can characterize a matrix by its eigenvalues,
You can characterize a matrix by its eigenvalues,

a series of numbers that can be calculated
a series of numbers that can be calculated

by multiplying components of the matrix
by multiplying components of the matrix

together in a certain way.
together in a certain way.

Eigenvalues of certain random matrices
Eigenvalues of certain random matrices

are always spaced along a number line
are always spaced along a number line

in a characteristic pattern, with consecutive eigenvalues
in a characteristic pattern, with consecutive eigenvalues

never too close together or too far apart.
never too close together or too far apart.

The same pattern of eigenvalue spacings
The same pattern of eigenvalue spacings

arises no matter how you fill the matrix with random numbers.
arises no matter how you fill the matrix with random numbers.

If you plot the distance between consecutive eigenvalues
If you plot the distance between consecutive eigenvalues

on the x-axis and the probability of getting
on the x-axis and the probability of getting

a particular spacing on the y-axis,
a particular spacing on the y-axis,

the familiar lopsided curve begins to appear.
the familiar lopsided curve begins to appear.

Researchers are still looking for a general answer
Researchers are still looking for a general answer

to where this universal pattern comes from,
to where this universal pattern comes from,

but clues continue to emerge.
but clues continue to emerge.

The idea of random matrix universality
The idea of random matrix universality

goes back to Eugene Wigner, a Nobel Prize winning
goes back to Eugene Wigner, a Nobel Prize winning

theoretical physicist who worked on the Manhattan Project.
theoretical physicist who worked on the Manhattan Project.

Wigner was attempting to calculate the energy
Wigner was attempting to calculate the energy

levels of a uranium nucleus, which
levels of a uranium nucleus, which

has more than 200 protons and neutrons
has more than 200 protons and neutrons

that can arrange themselves in all different configurations.
that can arrange themselves in all different configurations.

The associated energy levels of the system
The associated energy levels of the system

were far too complex to calculate.
were far too complex to calculate.

Wigner used random matrices instead
Wigner used random matrices instead

and plotted the statistical distribution of eigenvalues.
and plotted the statistical distribution of eigenvalues.

He found that the spacing of these numbers
He found that the spacing of these numbers

matched the spacing of energy levels of uranium
matched the spacing of energy levels of uranium

and other heavy atomic nuclei.
and other heavy atomic nuclei.

Two decades later, the pattern was
Two decades later, the pattern was

seen in gaps between consecutive numbers called zeros
seen in gaps between consecutive numbers called zeros

of the Riemann zeta function.
of the Riemann zeta function.

These zeroes are thought to control how
These zeroes are thought to control how

prime numbers are distributed.
prime numbers are distributed.

Since then, the pattern has been seen
Since then, the pattern has been seen

in many different settings, like in human bones
in many different settings, like in human bones

and social networks.
and social networks.

Just recently, it showed up in yet another unlikely place,
Just recently, it showed up in yet another unlikely place,

the eyes of chickens.
the eyes of chickens.

It was the first instance of the pattern seen in biology.
It was the first instance of the pattern seen in biology.

While the number line exhibits a pattern of universality
While the number line exhibits a pattern of universality

in one dimension, the chicken retina cells
in one dimension, the chicken retina cells

reveals it in two dimensions.
reveals it in two dimensions.

The color-sensitive cone cells of a chicken's retina
The color-sensitive cone cells of a chicken's retina

seem haphazardly distributed, but with a remarkably uniform
seem haphazardly distributed, but with a remarkably uniform

density.
density.

Looking closer, the cells appear to be surrounded
Looking closer, the cells appear to be surrounded

by what's called an exclusion region,
by what's called an exclusion region,

a space where cones with different color sensitivity can
a space where cones with different color sensitivity can

be found, but cones of the same kind can cannot.
be found, but cones of the same kind can cannot.

Just how these cones cells create the exclusion
Just how these cones cells create the exclusion

zones remains a mystery, but it's
zones remains a mystery, but it's

similar to the repulsion between consecutive random matrix
similar to the repulsion between consecutive random matrix

eigenvalues on the number line.
eigenvalues on the number line.

Researchers say we're just at the tip
Researchers say we're just at the tip

of the iceberg in understanding universality in math, physics,
of the iceberg in understanding universality in math, physics,

and even biology.
and even biology.