A Breakthrough in Higher Dimensional Spheres | Infinite Series | PBS Digital Studios


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.” [MUSIC PLAYING]