We all know how three-dimensional oranges are stacked in grocery stores, but what do you think the best way to stack 100-dimensional oranges is?

I'm Kelsey Houston-Edwards, and this is "Infinite Series," a new PBS digital show about mathematics.

The study of math is old, really old.

But right now it's expanding faster than ever.

Each week, we'll explore a new corner of this puzzling universe that's being carved out by mathematicians right now.

For today, it's hyperspheres.

What the heck are those?

Well, it's a sphere in dimensions that are bigger than three.

And while stacking hyperspheres has surprising practical applications in error-correcting codes which help ensure accuracy when transmitting data, like through cell phones, satellites, and the internet, I want to talk about them mainly.

Because they're cool their properties are counter-intuitive, and even seemingly simple questions about hyperspheres are hard to answer.

But before we get there, let's start with spheres.

It's easy to define a sphere in any dimension.

It's all the points that are a fixed distance away from a central point.

On a two-dimensional plane, you can imagine holding a rod fixed at one end and spinning the other end to trace out a circle.

In three dimensions, the same trick works, but now the rod has more freedom to move and traces out a sphere.

In higher dimensions, this traces our a hypersphere.

So what does that look like?

Well, before we go there, let's talk about what mathematicians mean when they say higher dimensions.

We can visualize one dimension as a line.

But another way to think about it is all the points described by one coordinate, which we usually call x.

Two dimensions is just all the points described by two coordinates, usually called x and y.

Three dimensions is just three coordinates.

That means it takes three numbers to describe a point in three-dimensional space.

Two numbers alone isn't enough to give directions to a specific point in three-dimensional space.

Even though you can't really visualize it, four-dimensional space is just the point specified by four coordinate.

Eight-dimensional space is just eight coordinates and so on.

Mathematicians have wondered for centuries about the optimal way to fill space with equal-sized non-overlapping spheres, what's known as sphere packing.

You can try the two-dimensional version at home.

How should you arrange pennies on a table so that you have the least amount of table showing between the pennies?

At best, you can cover about 91% of the table.

What about in three dimensions?

Way back in 1611, famed astronomer Johannes Kepler guessed that the best way to pack three-dimensional spheres are this and this.

These 3D sphere arrangements are intuitive.

It's how tennis balls or oranges are often stacked in stores.

They waste the least amount of space between balls, filling about 74% of total space.

Kepler was right, but it took nearly 400 years for mathematician Thomas Hales to actually prove Kepler's conjecture.

But what about sphere packing in higher-dimensional space where spheres are less intuitive?

Mathematicians have struggled to prove anything about sphere packing in dimensions bigger than three.

However, just a few months ago, in March 2016, mathematician Maryna Viazovska proved the best way to pack spheres in eight and 24 dimensions.

The standard way to pack spheres in three dimensions has an analog in higher dimensions, but the spheres move further away from each other as the dimension increases.

Then something special happens in eight dimensions.

In eight dimensions, there's exactly enough room between the spheres to squeeze in new ones.

For a detailed explanation of Viazovska's ideas, check out the "Quanta" article in the description.

Now, here's the totally wild part.

Mathematicians don't know the best strategy for packing spheres in any other dimensions.

We only know the best arrangements in dimensions 2, 3, 8, and 24.

That's it.

All the other dimensions are pretty mysterious.

Part of why sphere packing in higher dimensions is so difficult is that hyperspheres are basically impossible to visualize.

People have used different visual representations of higher-dimensional spheres, particularly in four dimensions.

One method mathematicians find useful is to look at slices.

Think about it this way.

If you pushed a three-dimensional ball through a two-dimensional surface, like a table, the portion of the ball intersecting the surface would always look like a disk-- first a tiny disk, getting bigger, and then getting smaller again.

The disk is a 2D slice of the ball.

Similarly, if a crazy four-dimensional creature took a four-dimensional ball and pushed it through our three-dimensional world, we'd see it as a three-dimensional ball-- first a tiny ball, then bigger, and then smaller.

These are the 3D slices of a four-dimensional ball.

After four dimensions our visual imaginations are pretty much useless.

Here's two thought experiments to show how weird hyperspheres are.

First, what happens if you confine a sphere to a cube?

Start with a 2D square, and draw a circle that just barely touches the four edges.

Analogously, in three dimensions, draw a cube and inside it a sphere that touches all six sides.

The circle takes up 79% of the square, but the sphere only takes up 52% of the cube.

The pattern continues in higher dimensions.

Inside an n-dimensional cube, nestle a sphere that just barely touches all 2 times n sides.

As the dimension n goes up, the percentage of the cube that's occupied by the sphere gets smaller and smaller, approaching 0.

In 30 dimensions, the sphere is about 10 to the negative 13th the size of the cube inside it.

This is about how big a grain of sand is relative to a sports arena.

Except that in this analogy, the sand grain is touching each wall of the sports arena and is still round.

Yeah, I know that's weird, but just stay with me, and let's try one more experiment with cubes and spheres to see if we can understand what a hypersphere looks like.

Take a two-dimensional box, and cut it into quarters.

Draw a circle filling each one of these four boxes.

Now draw another circle in the center of the box that just barely touches the other four circles.

Notice that the interior black circle is far inside the box.

Now let's try the whole thing in 3D.

Start with a cube, and split it into eight parts, or octants.

Place a sphere so that it perfectly nestles inside each octant.

In the center of the whole cube, draw a sphere so that it touches each of the other eight spheres.

Again, this sphere is way inside the cube.

We can repeat the same thing in higher dimensions, nestling 2 to the n equal-size spheres into an n-dimensional cube and constructing a central sphere that touches the others.

The central sphere gets bigger as the dimension goes up.

But here's the totally mind-blowing part.

At nine dimensions, the central sphere touches the sides of the cube.

After nine dimensions, the sphere actually bursts through the sides of the box.

Why is this happening?

Roughly, because as the dimension goes up, the distance between opposing faces of the cube stays the same while the diagonal distance between opposite corners gets longer and longer.

.

See.

Hyperspheres are totally counter-intuitive.

But that's part of what makes them so awesome.

If you have a cool way of visualizing hyperspheres, let us know in the comments.

I'll see you next week on "Infinite Series."

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