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I was wondering if there was an extension of Euler's Identity to quaternions. Now, in two dimensions, there is only one direction of rotation. In three dimensions, there are two,"axes", of rotation. Therefore, the formula that I would like needs three different values of theta for rotation. Is there any such formula? Notice: The other formulas that were provided on this site are based on one angle of theta and a quaternion rotation on the angle theta. However, since four-dimensional space has three axes of rotation, the formula provided should have three values of theta.

Calcudev
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  • It does not. That only has one value for theta. – Calcudev Jan 22 '20 at 01:41
  • @Orb Can the exponential be extended to the quaternions? – Oliver Jones Jan 22 '20 at 01:45
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    I think you are misunderstanding quaternions. Every possible rotation in 3-space (not 4-space) is specified by a single (pure, unit) quaternion. I.e., a quaternion already corresponds to what you're thinking of as multiple angles. – Kimball Jan 22 '20 at 03:31
  • Every rotation is around an (oriented) axis by some angle. A unit quaternion specifies an axis and an angle. This is how every rotation can be specified by unit quaternion. You are simply wrong about needing three values for theta. It is true that there is such a thing as Euler angles, but that is a method that represents a rotation as a composition of three other kinds of rotations. If you want, you can represent a unit quaternion as a product of three other unit quaternions in an analogous, I guess. Is that what you want? What for? – anon Jan 25 '20 at 04:26
  • Also, your counting is weird. In 3D there are 3 coordinate axes to rotate around, not 2 (which is related to there being three Euler angles). In 4D it doesn't make sense to speak of rotation "axes," instead we speak of planes of rotation. And there are $\binom{4}{2}=6$ coordinate planes one may rotate around. I suppose you could say there are 2 coordinate plane rotations through a given axis in 3D, and 3 coordinate plane rotations through a given axis in 4D, but so what? – anon Jan 25 '20 at 04:29

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