### What sort of beast is a choreography?

Imagine that you and a couple of friends get bored one
day, so you decide to play a game. Each of you hops
in a spaceship and grabs a star. You drag them out to
a quiet corner of the universe, set them up in some
configuration, and then let them go and see what
gravity does.

How should you set up your three stars?

You might instead try making a regular triangle
and, again, starting the stars off standing still. They
would still end up colliding.

### A simple choreography

Another possibility is to put the stars in the same
triangle, but this time start them off moving.
If you pick the velocities just right, then the triangle
will rotate but maintain its shape. The inward pull of
gravity balances out the outward speed, much like how
a satellite can stay in a circular orbit around the Earth.

If you follow the positions of the stars over time,
you see that they trace out a circle over and over again.
That is, not only does each star individually trace
out the same curve over and over (the orbit is
*periodic*), but each star traces out
*the same curve* as each other.

A configuration of stars that does that is called a
*choreography*. The stars in a choreography never
collide; they just majestically dance in the sky. The
circular orbit we just found is the simplest choreography.

### A more interesting choreography

It is perhaps not surprising that stars can orbit each
other in a circle. But can they orbit in more interesting
shapes? Are there choreographies other than circles?

In 1993
Cris Moore
found
some.
In particular, he found that there is a choreography
with three stars tracing out a *figure-eight* curve.

In 2000 the Moore figure-eight orbit was rediscovered by
Alain Chenciner
and
Richard Montgomery.
Whereas Moore only had numerical evidence that there is a
figure-eight choreography, Chenciner and Montgomery mathematically
*proved* its existence. It was kind of a
big deal.

### Where do we stand now?

After the (re)discovery of the figure-eight choreography,
many,
many
more
examples were found, especially by
Carles Simó.
Almost all of these examples were found numerically,
without mathematical proof.

There are two components to
my work
on choreographies. First, I argued that it is
even easier than previously appreciated to find new examples;
you can even
try your hand
at finding a new choreography
in your browser.
Second, I developed a practical system for proving
existence of choreographies that the computer finds.
Combining these two, I have been able to vastly
increase the number of specific choreographies which
have been proven to exist.