Extra anti-twisters

The anti-twister is an interesting mechanism that gives a physical interpretation for a physical state that returns to the same state after the inner part turns 720 degrees.

This has connections with the quaternion sandwich product as the qvq^-1 is similar to the RTR^-1 transformation of the anti-twister.

One thing missing in the analogy is that the quaternion sandwich product can be applied in two ways. The quaternion is an isoclinic rotation, so it rotates by equal magnitude angles on two orthogonal planes in 4D. To create the double-cover of the 3D rotation one of these rotations is cancelled out in the sandwich product. The alternative sandwich product (using an alternative multiplicatino table, or possible using the complement of q) has the cancelled out rotation having the opposite sign.

The anti-twister's twist matrix T rotates around the y axis by an angle y in radius that depends on the radius x. The usual function is a smooth step from y=pi at x=0 down to y=0 at say x=2.

We can create two different anti-twisters if we make the object being rotated 720 degrees not the centre, but a unit sphere. Both the centre and the distance space is unrotated. The two types depend on which direction we rotate the inner space relative to the outer space. 

If we rotate in the same direction we get this anti-twister:

which corresponds to this y-axis rotation profile with respect to radius: 

If we rotate in the opposite direction we get this anti-twister:

which corresponds to this rotation profile:

Dual Anti-twister

The stereographic projection of the 4D quaternion multiply acts like nested Clifford tori, rotating along the fibres of the Hopf fibration. That is the fancy way of saying that it rotates in two orthogonal ways- in 3D: around an axis, and around the axis's orthogonal circle, but the same angle.

The sandwich product usually cancels out the rotation around the orthogonal circle, leaving the double cover of the standard rotation around an axis. But we can make a modified sandwich product that cancels out the axis rotation, leaving a double cover of the rotation around the unit circle.

Is there an anti-twister analogy to this alternative type of sandwich product? I think this is probably it:

It uses the same rotation profile as the second example above in its T matrix, but this time the sandwich product is CTC^-1 where C is the circle rotation. Shown above with the circle rotation's angle increasing from 0 to 720 degrees.

 

It therefore returns to the same configuration after two full rotations around the unit circle (shown in red). The blue and green surfaces are orthogonal two the circle.

Unlike the rotational case, this rotation is not local to the region around the unit sphere. This is not unlike the circle rotation itself, which can send points to unbounded locations.

Because this transformation is the orthogonal rotation, it could be combined with the rotational anti-twister above, with no tangling or self intersections of the space.

Weyl anti-twister

The Weyl spinor is like the Pauli spinor but has another parameter representing the rapidity in addition to rotational phase. We can represent this quite nicely as a Simple Mobius Transformation which bends the space downwards if the rotation axis is upwards. Here I show the up/down value varying as it is rotating:

Just as the rotational part doubles the angle at the central point, the boost part doubles the curvature at the central point. 

For the maths of these anti-twisters see here.