A loop-de-loop mash-up and some extensions

The mash-up idea

We had an idea to combine Anna Weltman's loop-de-loops with one of Vi Hart's doodle games, making a knots/doodle braids:

Questions from our noticing and wondering about these:

1. Is it possible to make a VH doodle out of every loop-de-loop? If not, can we tell in advance which ones don't give us legitimate VH doodles?
2. Does this method always produce a knot distinct from the unknot?
3. Will any resulting knot/braid have a single string or could there be multiple separate strings? Can we tell in advance how many there will be?

Transcendentalism

Another set of ideas were inspired by John Grade & his daughter, the winners of Dan Meyer's loop-de-loop contest:

Their loop-de-loop was made using the digits in the decimal expansion of π as the step sizes. I think there were 49 digits used.

The immediate question was, why 49 digits? That led us to create a (not entirely successful) program that allows us to choose how many digits of π to use to generate a loop-de-loop. Of course, this quickly led to a second extension: what about other irrationals? For now, our code will let you choose whether to use π or e as the reference. The code is here.

Playing around with these a little, we came up with another extension.  Instead of a single loop-de-loop, consider a family of loops-de-loop {F_n(α)} where the n-th element of our family is the loop-de-loop generated by the 2n+1 terms of the decimal expansion of  α. Here are our questions about these families:

1. If A_n(α) is the area bounded by the loop-de-loop F_n(α), is the sequence A_n(α) bounded?
2. Does the sequence  A_n converge?
3. Does the answer to either 1 or 2 depend on α?
4. If there are α for which the area isn't bounded, does it grow with some power of n? Does that power depend on α?
5. We have similar questions about the perimeters of the family.

Take us to another world

One last idea was to think about drawing on a surface that isn't a flat plane. In order of interest:

1. sphere: my idea is to show the stereographic projection of the motion on a sphere. This doesn't seem too hard. Someone with stronger coding skills might have fun making a version that shows the loop appearing on the sphere itself.
2. disc model of hyperbolic geometry: our intuition is that this could look cool
3. torus: should be easy to write a program for this, but intuition is that it won't be all that interesting.
4. Klein bottle and Möbius strip: like the torus, not too hard, but we guess this won't look very exciting.

By the way

Go back and watch the Vi Hart video again (the link, for your convenience) and make sure to play those doodle games and think about all the connections. It has to be one of the coolest things ever and is well worth taking the time to appreciate it.