Post #1815112

The following is in analogy to i²=-1 from complex numbers. I know that imaginary quaternions can be written using any orthonormal basis. Which means they are spherically symmetric. In fact, the symmetries cover the unit sphere twice! So we can take any solution and rotate it continuously. Since I still felt surprised I wanted to get my hand on some of them: Take any point (a,b) on the unit cirle. It means a²+b²=1. Expanding (ai+bj)² we get a²i²+abij+abji+b²k². Now, also by definition, i²=j²=-1 and ij=-ji. So we can simplify to get -a²-b², which is what we wanted: Using a subspace of just two imaginary dimensions we get a circle of solutions! I think we can simply use general imaginary unit quaternions and do the algebra. Appealing to symmetry seems odd at this point, because symmetry alone already solves the problem. @hallasurvivor