Here is the deal:

There are two interlinked circles, like in a chain. You move around one of the circles for half of eternity almost without any change. Almost. Because one day you reach a threshold, there is an action potential, a quantum jump. And you move to the second circle, where you will continue for the second half of eternity.

This is the archetype, the essence, the kernel, of the Dzhanibekov effect that my series of posts is about.

The essential math is summarized in my previous post. Here comes the animation.

Click on the image to open gif animation. It will take time to load as it is 2.5 MB.

The upper image shows the unit quaternion representing the rigid body rotation. It moves first on one circle, then makes the jump to the second circle.

The lower image shows the rotating rigid body. Time on both images is synchronized. The quantum jump is the flip of the rotating rigid body.

**Update**:

**My Mathematica animation (almost the same as above) code that can be analyzed and should run with free**Mathematica CDF Player (probably needs a recent one) is here.

WOW! That Path wins the beauty contest! So the object flips through 180deg. The unit quaternion which uses 1/2 the angle is fliped/rotated through 90deg and displaced upward. It is fascinating to me thet the circles are offset from the origin. Can you offer an explanitation as to why?

Eq. (4) from previous post is suspected wrong.

I think that the square root in denominators should be extended (to make sum of the squares to be equal to 1).

Thanks. All fixed.