Twist and Shout: Numbers, Rotations and All That
Imagine a universe of a different sort where everything is bound to a straight line. You can walk forward and backward along this line, but that is it. You can do nothing else. It would make for a pretty boring universe, wouldn't it? But you are a mathematician. And even though all you have ever experienced is just one dimension, you know how to make this boring universe a bit livelier: by cooking up more dimensions.
So, you conceive of this funny object living in two-dimensional space that you decide to call a circle — a round thing that closes on itself. And you give the movement of going around in a circular fashion a name as well: rotation. The difficult task, though, is to explain your ideas to the fellow inhabitants of your one-dimensional universe. But you have figured out a trick: even when all they have ever experienced is a monotonous one-dimensional world, you can help your friends visualize rotations by cutting up the circle and opening it up into a straight line. What were rotations in two-dimensional space are now simply movements along this line. (Technically, for there to be a proper correspondence between the two, you would need a straight line of infinite length. This trick of understanding an object with the help of another object of one lower dimension goes by the name of stereographic projection.)
In mathematics, another useful way to think about these rotations in two dimensions is through what we call complex numbers. Real numbers are one-dimensional. Any movement you make in your one-dimensional universe can be represented by a real number. So, how do we use complex numbers to think about movements in two dimensions? A complex number — say 3 + 4i — denotes going three units in one direction, taking a 90° turn and moving four units in the new direction. You can represent this entire motion as an arrow from the point where you started to the point where you finally reached. (Think of that cute, little i as something that always tags along with the second component and denotes one of the spatial dimensions.)
Let us try to make this a bit more magical. The magic lies in a simple multiplication rule: i x i = -1. Multiply 3 + 4i by i and you get -4 + 3i, and in effect what you have done is rotating your original arrow by 90°. Equipped with this rule, you can try multiplying 3 + 4i by any other complex number and what you have essentially achieved is a rotation in the two-dimensional space you are working in.
A complex number is a two-dimensional number — in the same manner in which real numbers are one-dimensional. The defining feature of this two-dimensional number system comes from the multiplication rule: i x i = -1. Once you know this simple rule, you know how to multiply any two complex numbers.
Having defined a two-dimensional number system that helps you understand rotations in two-dimensional space, it seems obvious that you can define a three-dimensional number system (something which has numbers like 3 + 4i + 5j, numbers with three components) to understand rotations in three dimensions.
Well, no. Turns out you need four components to have the freedom to do a rotation in three dimensions. Strange, right? If you are feeling perplexed, you are in good company. Give it some thought, you can possibly figure this out on your own. Or you can scroll down to find some great references that will help you figure it out.
For now, let us stick to talking about these new four-dimensional numbers. We call them quaternions. Like complex numbers, this number system — which includes numbers like 3 + 4i + 5j + 12k — comes with its own multiplication rule. Quaternions are supercool! However, visualizing anything that is four-dimensional is difficult. But remember how you could think about two-dimensional space even when you were restricted to a one-dimensional universe? You can take a similar approach to visualize quaternions, too.
Part of the coolness of quaternions also comes from how useful they are. Your favorite video games need quaternions all the time. That is how they keep track of the movements in the three-dimensional world of the game. From computer graphics to space-launching softwares, quaternions are always there to make your job easier.
Coming back to our number systems: there are one-dimensional real numbers, two-dimensional complex numbers and four-dimensional quaternions. Any guesses on how many components the number system next in line has?
The answer is eight. These eight-dimensional objects are called octonions. Like their predecessors, octonions come with their own multiplication rule, too. You can go even further and you will end up with sixteen-dimensional numbers. We call these higher-dimensional generalisations of complex numbers hypercomplex numbers. The way complex numbers help you understand rotations in two dimensions, hypercomplex numbers help you work with objects in higher dimensions. And quite often, things get really, really weird in these higher dimensions!
It's not that our three-dimensional space fails to stump us either. I will leave you with a small example. You will need a long strip of paper and a few of your favourite books for this. Choose one of the books that you will continue to keep in your hand and place one end of the strip inside its pages. Now, use the remaining pile of books to make the other end of the strip stay fixed. You can just place this end of the strip on a table and use the pile as a sturdy paperweight.
Now, rotate the book in your hand by 360°. In doing so, you have obviously given the strip of paper a twist as well. Now, here's the curious thing. Unlike other situations, where turning an object by 360° brings it back to its original position, there is no way you can get rid of the twist in the strip without further rotations of the book. Well, okay. Let's do that. Let us rotate the book again by 360° in the same direction. And what happens to the strip of paper? It appears to have become further twisted. But wait for the last piece of magic. Try moving the strip over the book and you have gotten rid of all the twists. Unlike most objects you see around, this tiny setup you had created needed a double rotation to get back to how it originally was. Pretty amazing, right? We will stop here, but the fun clearly does not. You can continue playing around with hypercomplex numbers and things you can find around to rotate and twist. Chances are you will discover something cool for yourself.
“Why, sometimes I've believed as many as six impossible things before breakfast.” Lewis Carroll, Alice's Adventures in Wonderland
I hope this post did its job of giving you a flavor of how amazing hypercomplex numbers are. If you want to dig in deeper and study all this more systematically, a great place to begin would be this great presentation by the Quanta Magazine: The (Imaginary) Numbers at the Edge of Reality. Also, here's an equally fascinating discussion on quaternions with Dr. James Grime on Numberphile: Fantastic Quaternions.
You may also want to understand stereographic projection better and delve into the ways one can visualise quaternions. In which case, check out this interactive portal by Grant Sanderson and Ben Eater: Visualizing Quaternions: An Explorable Video Series.
What we didn’t discuss in this post at all was the application of octonions, the eight-dimensional counterparts of quaternions. Here’s an interesting and insightful read that discusses the mathematics of octonions and the surprising ways they’ve made their presence felt in physics: The Peculiar Math That Could Underlie the Laws of Nature.
Also, while you're here, I will drop in a book recommendation for all things mathematics and physics: The Road to Reality by Prof. Roger Penrose. Whatever stage of learning you are at, you will find this magical book a good company to keep. It is also where the last example of our post comes from.
And finally the most important part. Heartfelt thanks are due to Charu Mehta for the very kind offer to write for this wonderful blog, and to Tejasi Bhatnagar for patiently reviewing the piece and sharing her very valuable feedback. And thanks to you, dear reader, for allowing me to share my silly love for the magical hypercomplex numbers with you. Hope you will find your own silly love in the beautiful world of mathematics.