$\text{Visualizing}$ $\text{quaternions}$

How do you multiply two complex numbers? Well, you start by using the distributive law.

$\begin{aligned} & (2 + 3i)(4 + 5i) \\ =\ & 2\cdot4 + 2\cdot5i + 3\cdot4i + 3\cdot5 i^2 \end{aligned}$

Then use the rule that $i^2 = -1$ to simplify further.

$\begin{aligned} & 2\cdot4+ \color{#29ABCA}{2 \cdot 5i + 3 \cdot 4i} \color{#494949}{{} + {}} \color{#FC6255}{3 \cdot 5i^2} \\ =\ & (2\cdot4 \color{#E65A4C}{{} - 3\cdot5} \color{#494949}{) + (} \color{#58C4DD}{2\cdot5 + 3\cdot4} \color{#494949}{)i} \\ =\ & -7 + 22i \end{aligned}$

Multiplying two quaternions is more complicated, but in principle not too different. There are some rules for how $i$, $j$ and $k$ multiply amongst themselves,

$\begin{aligned} ij &=-ji =k \\ jk &=-kj =i \\ ki &=-ik =j, \end{aligned}$

and multiplication is carried out by distributing and simplifying.

This is what computers do, and it’s what your computer is doing in the visualizations you will see below. But knowing how to compute products is just one way of understanding multiplication. In much the same way that multiplication by real numbers is made more understandable envisioning a process of scaling or squishing, and that understanding complex multiplication is made easier when understanding how they rotate points on the plane, quaternion multiplication can be understood by looking at how it transforms four-dimensional space.

This is not a simple idea or a simple visual, but with some patience, understanding quaternion multiplication in this way can make clear why quaternion multiplication describes 3d rotation the way that it does. We’ll limit our view of this action to the unit hypersphere, as viewed under a stereographic projection filling the entirety of our 3d space.

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4:43

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Credits and thanks

Many thanks to Andy Matuschak, Cameron Christensen, Michael Nielsen, and Kitt Hirasaki for useful conversations and feedback on this experience.

Each of our YouTube channels is supported on Patreon (Ben Eater, 3blue1brown). Funding for this collaboration came from the 3blue1brown patreon community, with special thanks to the following individuals:

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