If real, irrational, rational, whole and complex numbers are the only numbers you’re familiar
with...boy have I got news for you.
Some special kinds of numbers, like quaternions and octonions, describe more than our three
dimensions.
Just let that sink in a little bit.
Can you even picture the kind of math involved for a reality that exists in more than 3D?
How would we even think about those shapes?
Let’s take it back to like, middle school for a sec.
If you want to travel in one dimension, you add and subtract real numbers.
Here you are, the dot, on your little line, and you want to move 7 units to the right?
Great, that’s plus seven.
Let’s take it into two dimensions.
To make moves like this you can use what’s called an imaginary number These can be combined
with real numbers to make what are called complex numbers.
Real numbers and complex numbers are the first two of four number systems in pure mathematics
that we call the normed division algebras over the real numbers.
We can think of complex numbers as two-dimensional numbers...because they help you move things
in 2D.
But something kinda funny happens when you use these division algebras to manipulate
an object in 3-dimensions.Say you’ve got a cube, and you need to be able to rotate
it around to see all its sides.
What kind of math are we looking at?
After the 1D real numbers and the 2D complex numbers… comes a division algebra with 4-dimensions,
not 3, like you’d expect!
And this 4-dimensional number system is known as the quaternions!
And they are awesome and SUPER crazy, so we got in touch with Cohl Furey, a postdoctoral
researcher at Cambridge who lives and breathes this stuff.
She helped us get a lot of this straight, and if you’re a math PhD, you can go watch
her series on quaternions and get real nitty-gritty.
But for the rest of us--quaternions are the number system we use to describe 3D shapes
in 4D space.
That sounds a little trippy, but that fourth dimension doesn’t necessarily represent
a spatial dimension, like up and down.
Sometimes we use extra dimensions to represent a different concept, like time.
So in your quaternion, you have a real number, nested inside your complex number, and two
new terms, j and k, which represent your two other dimensions--four components, hence the
‘quat’ in quaternion.
What makes quaternions extra hard to work with is that they’re noncommutative.
Stick with me here--that means the order of the calculations matters in a way it doesn’t
with real and complex numbers.
Think about it: If we go back to our 2D axis, it doesn’t matter if you rotate in the plane
by 10 degrees first, and then by 20 degrees, or 20 degrees first and then 10 degrees--either
way you end up in the same place.
But in 3D space, if we have our cube, it can matter which move you do first, partly because
you have more options.
And while they may sound insane, quaternions are super useful--they come in handy when
calculating the launch trajectory of an aerospace vehicle.
Your phone can use them to calculate your device’s position and orientation in space.
We use them all the time in generating computer graphics and for programming robots, which
move around in 3D space.
They are also especially important because a version of quaternions underpin Einstein’s
theory of special relativity--allowing us to calculate the movement of an object in
3 physical dimensions, plus the extra dimension, which can be defined as time.
But strap in--’cause here come the octonions.
That’s the next step up.
Now, because of some complex mathematical rules, these kinds of numbers come only in
dimensions 1, 2, 4, and 8, so after quaternions, we make the jump to octonions--numbers that
help us navigate eight-dimensions.
That’s the quaternion, which contains complex numbers, which contain real numbers, plus
4 new units.
E1, e2, and e4 are one way that i, j, and k from the quaternions are represented in
the octonion system.
These eight ‘dimensions’ again, don’t have to be spatial--they can represent characteristics
of particle behavior, like how certain particles behave under the strong force, making octonions
really useful for quantum calculations.
And there we have our nice neat set of the four normed division algebras [over the reals]
[over the real numbers] The thing is, with each next level, the math
of these division algebras gets even more tangled.
Quaternions are noncommutative, and octonions are too.
But they’re also nonassociative--which means not only does order matter, but it also matters
how you group the multiplication of octonions.
But the very structure that makes the math so difficult is one reason that the octonions
can be so useful in describing fundamental physics.
One recent paper revealed that octonion mathematics allows for the existence of sterile neutrinos--a
particle that scientists are actively looking for experimentally because...it might be dark
matter.
That’s just one of a whole host of applications--many of our models of particle physics are still
incomplete, but in the words of Dr. Furey “...there have been a number of very tantalizing
hints that the octonions might somehow underlie certain aspects of the structure of elementary
particle physics.”
So they’re famously difficult to work with.
But incredibly useful and underpin the forefront of mathematical exploration happening today!
They’re so crazy physicists even disagree on how to approach them, but researchers hope
octonions will help them peel back the layers of reality to explain why the universe behaves
the way we observe it to.
Wow.
We made it through that, pretty exciting.
Now we know more about extra-dimensional math!
For more mind-blowing close-up looks at the universe, check out this video here.
A huge thanks again to Dr. Cohl Furey for helping us make sure this was all correct,
and let us know in the comments if you have any questions or any other in-depth mathematics
that you’d like us to explore.
Thanks for watching!