Quaternion introduction: Part I

I generally really like the content produced at 3blue1brown, but their recent video on quaternions was just downright terrible. It entirely lacked Grant Sanderson's signature "discover it for yourself" approach, i.e. motivating the idea from the ground-up, and focused too much on an arbitrary formalism (stereographic projections aren't necessary for visualising anything).

The right way to motivate quaternions is to start by thinking about generalising complex numbers to higher dimensions. Complex numbers are a remarkable and elegant idea -- if you don't understand why I'm saying this, you could either get off the grid and spend the rest of your life as a circus monkey, or you could read my posts "Null and row spaces, transpose and the dot product" and "Making sense of Euler's formula".

The key idea behind complex numbers is that they are an alternate, simple representation of a specific set of linear transformations, namely: two-dimensional spirals (scaling and rotations). Note, similarly, that the real numbers can also be considered an alternate representation of e.g. scaling in one dimension.

The natural way to generalise complex numbers to more than two dimensions may seem to be to have an imaginary unit for each possible rotation (or more precisely, each "basis rotation"). In three dimensions, the basis has three planes of rotation, and could be e.g. rotations in the xy-plane, rotations in the yz-plane and rotations in the zx plane (you may have heard these as rotations "around" the z, x and y axes respectively, referring to the axes that remain invariant during the rotation -- however, as it turns out, in a greater number of dimensions $n$, the number of dimensions held invariant is $n-2$, which is only equal to 1 -- i.e. a single axis -- in 3 dimensions. e.g. in 4 dimensions, an $xy$-rotation would leave the $zw$ plane invariant.)

So let's try out this formalism, because it seems promising. We could write, e.g. i for the yz rotation, j for the zx rotation and k for the xy rotation. Try to work out some of the algebra here for yourself. What does $ij=?$ equal? What does $jk = ?$ What does $i^2=?$ equal?

As it turns out, none of these transformations result in anything very interesting. It would have certainly been elegant if you'd gotten nice results, like $ij=k$, or something, but you don't. One of the neat things about the complex number system is that not only do all complex numbers together, or all unit complex numbers together, form a group -- even $\{1,i,-1,-i\}$ forms a group under multiplication. But $\{1,-1,,i,j,k,-i,-j,-k\}$ do not form a group.

How would one solve this problem? Well, the reason $i^2$ doesn't equal minus 1 is that it only offers a reflection across the $x$-axis. The matrix representing $i^2$ is:

$${\left[ {\begin{array}{*{20}{c}}1&0&0\\0&0&{ - 1}\\0&1&0\end{array}} \right]^2} = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&{ - 1}&0\\0&0&{ - 1}\end{array}} \right]$$
(If you can't come up with the matrix for $i$, you should review the linear algebra series -- or the circus monkey thing.) What if you reflected across all three axes, in some order? You'd have:

$${i^2}{j^2}{k^2} = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&{ - 1}&0\\0&0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 1}&0&0\\0&1&0\\0&0&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 1}&0&0\\0&{ - 1}&0\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]$$
In other words, ${i^2}{j^2}{k^2} = 1$. Additionally you may have observed while crunching the numbers above that ${i^2}{j^2} = {k^2}$.

This may give you an idea*. Here's another thing that may give you an idea: the reason you had $i^2=-1$ with complex numbers was that $i$ rotated all the axes in the plane. By contrast, $i,j,k$ only each rotate two of the three axes in 3-dimensional space.

*the idea being that perhaps combinations of two rotations can give us more interesting results

Well, how do you solve this problem? How do you create a rotation that "rotates all the axes"? Seemingly, you can't. Sure, you can define a rotation that rotates all three of the x, y and z axes, but that would still leave some other axis invariant, which we call "the axis of rotation". Can we define a rotation that leaves no axis invariant?

In three dimensions, the answer is no. Any rotation leaves one axis invariant, and trying to rotate this axis requires rotating it with another axis, and the resulting product rotation still leaves some, calculable axis invariant.

Calculate this axis.

The key is to extend our thinking to four dimensions. Here, you can have pairs of rotations acting simultaneously on two different pairs of axes. Since there are only four dimensions in four dimensions, all four axes are transformed.

Now, the obvious thing to do here may be to define an imaginary number for each pair of rotations in four dimensions -- there are $\left( {\begin{array}{*{20}{c}}4\\2\end{array}} \right)=6$ rotations, and $\left( {\begin{array}{*{20}{c}}6\\2\end{array}} \right) = 15$ such pairs. But this would be too many "basis rotations", and the rotations would not be independent of each other, since rotations in 4 dimensions can be described with only 6 basis rotations.

So how could we make use of our idea of using pairs of rotations as our basis for describing rotations?

The key is to make one of our four axes "special" -- call this axis $t$, and the other three axes $x, y, z$. Instead of considering all 15 rotation-pairs, we only consider the following three:

$$\begin{array}{l}i = (tx,yz)\\j = (ty,\overline{xz})\\k = (tz,xy)\end{array}$$
This is not the only possible representation of the quaternions, of course. Even among complex numbers, you have two possible representations -- you could make $i$ a counter-clockwise rotation, as is conventional, or a clockwise one, i.e. there is a symmetry between $i$ and $-i$. For quaternions, it turns out there are 48 different possible representations -- prove this.

Where $tx$ represents a rotation that sends $t$ to $x$ (i.e. a counter-clockwise rotation on a plane where $t$ is the x-axis and $x$ is the y-axis) and $\overline{xz}$ represents a rotation that sends $z$ to $x$, i.e. the clockwise rotation on a plane where $x$ is the x-axis and $z$ is the y-axis.

It turns out that these pairs -- called quaternions -- in fact allow the representation of 3-dimensional rotations, since you need only a $\left( {\begin{array}{*{20}{c}}3\\2\end{array}} \right)=3$-dimensional basis to represent rotations in 3 dimensions.

Think: Are there any other dimensions that allow such a system to be defined? Can you have, e.g. "hexternions"?

Note that however tempting it may seem, there is no known natural description of special relativity in terms of quaternions. Sorry.

One may work through the algebra of these new quaternions by tracking the position of each axis through the multiplication, and as it turns out, it is indeed much more elegant than the more obvious representation detailed earlier:

$$\begin{array}{l}j = k,jk = i,ki = j\\{i^2} = {j^2} = {k^2} =  - 1\\ijk =  - 1\end{array}$$
In the next several articles, we will look at exactly how 3 dimensional rotations can be represented with quaternions, the relation between quaternions and the dot and cross products through the commutative and anti-commutative parts, and further extensions of the quaternions to higher dimensions.

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