Complex numbers, quaternions and octonions can be equated with three types of rotations: general rotations SO(n), simple rotations (rotations in a single plane), and isoclinic rotations Is(n). I also showed how SO(n) and simple rotations are built from Is(n). What's more, isoclinic rotations can also be used to generate reflections, e.g. qvq reflects v around 3D normal n when q is the isoclinic rotation defined (0,n). All of this suggests that isoclinic rotations are really the fundamental rotation type.
It also appears that these isoclinic rotations Is(2), Is(4) and Is(8) have a representation in 3D, giving a generalisation of the standard idea of rotations:
The complex rotations are in a single plane, which is also our standard idea of rotations in 3D. The quaternion Is(4) rotations can be represented in a couple of ways in 3D. The first is as a 'Hopf vortex' consisting of a rotation around one axis and an equal angled rotation around the orthogonal unit circle. The second is the so-called anti-twister mechanism, which is a rotation around a S-shaped curve which itself is rotating at the same rate. In both cases the rotation is around a curve which itself is rotating; a sort of vortex.
It is tempting therefore to consider whether the 8D isoclinic rotations associated with Octonions have a representation as a sort of vortex. I can think of two possibilities, which are all close I think:
- Two anti-twister mechanisms that orbit each other at the same speed that they do their rotation. This can be configured to be equivalent to two spinorial balls rotating around a ball 3 times larger, which is equivalent to one spinorial ball rotating on a projective ball 3 times larger. This unusual sounding system describes the algebra of split-octonions, more specifically imaginary split octonions with normalised temporal and spatial components. The vortex is rotating by an angle in 3 different ways, unlike the 4 planes that Is(8) rotates in, on the other hand the split-octonions remove one of those planes of rotation, replacing it with the Lorentzian transformation. So its a well-founded vortex, but only represents a sort of split-octonion, which is close to but not the same as standard octonions.
- It is possible to make an anti-twister mechanism in 4D. If I is an isoclinic rotation with increasing angle theta, and T is a a rotation around one plane in 4D, modulated by radius (180 degrees at r=0 and 0 degrees at r=1) then ITI^-1 is the anti-twister transformation of 4D space. For theta 0 to 2pi the centre is an isoclinic rotation of 4pi, so it has spin 1/2, and theta is basically applied 4 times, since it double covers an isoclinic rotation. That is like the 4-angle isoclinic rotation of octonions.
If we use the axes of a radial tetrahedron as the four spatial axes, then there are two fixed planes (equivalent to the curved line pair in the 3D anti-twister), drawing these planes with lots of radial lines gives this motion:
We know these are fixed planes because we can watch their behaviour when the camera is following the rotation I, this is equivalent to yawing the camera by theta in the 3D anti-twister, and seeing the curved line pair as fixed. So we're viewing TI^-1:
So if the camera follows the motion of I, all points in the space rotate around these fixed planes. In 4D you rotate around a plane. In a fixed camera this is a rotation around a rotating plane, giving a similar mechanism to two rolling balls, which is also a rotation around an orbiting thing (ball).
If you take a 3D cross-section of the 4D space, again using the camera that follows I, you get this:
The inner and outer circles are the cross section of the fixed planes, they are the equivalent of the two fixed points that space circles around in the 3D anti-twister. Here it shows that the nearby points rotate about these circles, and the circles themselves are rotating. So these fixed curves are going a sort of isoclinic rotation as they also rotate with I. Suggestive of the 'fermionic ball' around another ball used to describe unit imaginary split-octonions. More info about 4D anti-twisters here, and a summary of Octonion multiplications is here.
