The pattern of Mills Mess juggling and a link invariant in 3D?

Question

You can find the juggling pattern of the 3 balls called the Mills Mess juggling here.

And a [simplified animation is found here]

Observation and Attempt: - Notice that for each of the three balls, each ball circulates its own "infinity $\infty$"-shape path. The animation makes the paths less clear, but the above video demonstrates much more clearly.

Question:

• I wonder what would be the 3 closed curve patterns on the x-y plane, from the projection of the spacetime trajectory of 3 balls (3 ball dancing in the 3D space with 1D time) into the 2D plane such as the x-y plane shown on your computer screen?

• Can one identify the curves as the time-trajectory paths of 3 balls (allowing only a slightly deformation away from the x-y plane to the z-axis outside/into your screen direction) in 3D $\mathbb{R}^3$ space as some kind of a link invariant in 3D?

Apparently, the simplest candidates will be (1) a set of Hopf links or (2) the Borromean rings.

• BUT, is it really possible to allow such a link pattern from the 3 curves as the 3 time-trajectory paths of the above juggling 3 balls?

Answer 1

The Mills Mess pattern is very rich indeed. Several observations includes :

• We can see the trajectory as a snake formed by three balls (head, body, tail)
• These three balls can have fixed individual patterns (head makes a small $\cap$, body makes an $\infty$ and tail makes a big $\cup$
• The previous description is in the practical case, in your animation's case, the trajectory can be shown on the picture below( Averaged pixel value on a cycle)

What I did is coarsely track the balls and plot it in a 3D way. You have helixes, but it don't seem like baids (in your Borromean rings example). 3D-plot of three cycles of Mills Mess