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Wyrd Smythe's avatar

Not bad! I'm a little surprised you didn't mention Hamilton's famous equation: i²=j²=k²=ijk=-1.

FWIW, there's an easy way to remember the multiplication rules: i ⇔ j ⇔ k ⇔ i (best represented by a three-part circle combining the two instances of i). When you go right-to-left (or clockwise around the circle) the result is positive. When you go left-to-right (or counterclockwise), the result is negative.

So, j×k=+i but k×j=-i.

BTW: On my browser, the spiral symbol you used for the 4D axis just shows as a box; the browser doesn't support it. I looked in the app, and it appears okay there. For some authors, the canonical 4D axis is w (and the 5D axis is u), which is an easier typography. I think of them as standing for Weird and Unusual. For more than five dimensions, I've seen authors use xₙ, usually starting at x₁.

Are you next going to enter the deeper waters of octonions? 😁

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Ryan's avatar

> There’s just one real reason for it: in the quaternion space, there is an easily-memorized formula for rotating a point by an angle around a line of our choice.

Isn't the primary reason not the ease of the formula, but to avoid gimbal lock? Euler angles are a lot more intuitive for people otherwise. (I suppose there are also some computational benefits but I don't think that was the primary driver historically?)

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