Earlier in the week I did a short project on Anna Weltman’s loop-de-loops with my younger son:
He’s been doodling and coloring loop-de-loops since we did this project 🙂
His enthusiasm for the project made me want to revisit the idea again this morning since my older son missed it the first time because of an early morning school event.
We started by introducing my older son to the loop-de-loop idea and then we talked about some of the pictures that my younger son made during the week:
For our first exploration today, my younger son wanted to see if he could make a shape with really big “arms.” He tried to draw a 1,2,15 loop-de-loop. Unfortunately he accidentally made a little error while he was drawing the shape – that small mistake gave us an extra chance to talk through the ideas of turning left and right.
After revisiting the left / right rules at the end of the last video, we returned to draw a 1, 2, 15 loop-de-loop. He’d hope to make a shape that had big arms, but we got a surprise – this one had small arms. Why did that happen?
Next we decided to take a look at my older son’s idea: whether or not the arms ever “double split.” It seemed that the arms with 3 numbers never really double split, so he thought we’d need more numbers. He that we’d need 5 numbers for a really interesting reason – he thought ones with 4 numbers would always “spiral to infinity” rather than having arms that split. We decided to explore that idea with a 1, 3, 5, 7 loop-de-loop out of our Zometool set in the living room:
Here is that build as well as a few of their ideas about what they think is going to happen (and what is happening).
After they finished the loop-de-loop with 4 numbers, they built one with 5 numbers while I was publishing the first set of videos:
So, a super fun project – on so many levels – for kids, and an amazing way to get kids talking about and thinking about mathematical ideas!
One way to extend this project that I’m pondering is to talk about a relatively new (for math) idea – Langton’s Ant – that was invented / discovered in 1986. The concepts you need to talk about Langton’s Ant are pretty similar to the ones you talk about in these loop-de-loops. Here’s a short video by Katie Steckles and Numberphile about the ant which hopefully shows why I’m thinking of extending the project in this direction:
Mike's Math Page 
The loop-de-loops look like a lot of fun. We just started rainy season vacation (couple weeks break between terms) so we have a lot of time free for these types of explorations.
Also, thanks for flagging Langton’s Ant. Of course, I couldn’t resist creating a pencilcode program to animate it (basic version).
Then, I thought, why not give the little guy some company, so I made a version with two ants following the same rules. When they are both on the same cell, one of the “ants” acts first, changes the state of the cell and moves, then the other acts on the cell. I think the green one always acts first. This means that when they are on the same cell, at the end of the action, one will have turned left and the other right, while the state of the cell is left unchanged.
From looking at Wikipedia, this seems to be a non-standard implementation of multiple ants. Usually, the ants act simultaneously. In that case, when two ants land on the same cell, they both turn in the same direction and the state of the cell gets changed.
I guess one drawback of my implementation is that the ants can get “merged” for a step, while that won’t happen with simultaneous actors.
We finally got around the loop-de-loops and have been having a lot of fun. So many possible extensions, most of which seem non-obvious (to me, at least):
Loop-de-loop mash-up and extensions
Nice ideas – we played around with a different type of extension after seeing Frank Farris’s ‘Creating Symmetry” book.
I had fun figuring out when a sequence a,b,c and t,u,v,w,x make the exact same final result! It was a surprise to me that this could happen. There are so many things to explore here.