Why would you write about this?

My handle is the Dancing Geek.  The geek part is not just for show – I have a degree in this stuff, and sometimes I actually enjoy solving unnecessary puzzles just for fun.

The theory behind Level 3 as a mathematics puzzle

I’ve made up some notation to keep this shorthand.  H is horizontal, V is vertical and the number relates to the position.  So H1:V1 is left arm in Horizontal 1 position, right arm in vertical 1 position.  Clear?

1 complete round (i.e. you get back to the starting position and are ready to go on the opposite side) is 16 moves at Level 3.  It breaks down into 4 repetitions of the same 4 moves (i.e. the instructions are the same).  Looking at where your arms are after each of these 4-move combos for the first starting position gives us the pattern:

H1:V1 to V1:H3 to H3:V3 to V3:H1 to H1:V1

This can be generalised (using algebra – yippee!) to:

Hx -> Vx

Vy -> Hy+2

In words:

A horizontal starting position will end with that arm in the same position on the vertical plane.

A vertical starting position will end with the arm in the transquartered position on the horizontal plane.  (Hy+2 uses modulo 4 arithmetic, in case you weren’t clear).

How does it work?

Another way of looking at it would be to say that the movement swaps the positions of the arms, moves the right arm 2 positions on, and swaps the planes of the arms.

Either way, the most important aspect of the pattern is that it has symmetry 4.  That is after 4 repeats it returns to where it started.  Simple example of this would be a rectangle with a dot in the corner being turned 90 degrees clockwise:

Example of 4 symmetry

A brand new level 3 pattern!

Yes, I have created a new pattern which I am calling Level 3 because it works in a similar fashion.  Since I don’t know about anything higher except for a bit of Level 4, I could be talking complete and utter bull.  Blame @Sari_O for getting the idea stuck in my head.

It goes:

• Transquarters
• Mirror reflection
• One forward, one back
• Mirror reflection

Or, you could do:

• One forward, one back
• Transquarters
• One back, one forward
• Both forward

Not sure if I’ll try these or not.

Still, I got through Level 3 with basic legs without too much hassle (hurrah, and yet, oh no!) so after whizzing through Level 2 for fun (ha, it sounds crazy still but the pattern’s in my arms and legs now, not much for me to do) I decided to have a bash at Level 4 arms!  (Thanks to @Sari_O for saving me from bugging @havi as to the details).

So I’m working through (what I understand to be) the Level 4 arms and it strikes me that it’s so pretty!  I’m not talking about watching the movements either, though if I do another video we can all judge that for ourselves, but the patterns that came up in my mind as I worked through the positions.  Now I love patterns, it’s part of why I get on so well with mathematics, so I could start rambling on about oscillations and standing waves, but even that isn’t quite what I was at.

Whilst standing there moving my arms around with one of our bunnies by my feet (so cute!) I could feel the loops and swirls and colours and cycles that it formed and it was really quite awe inspiring.  If you’ve tried even just Level 1 arms then go from starting position 3:1 and tell me you don’t find it just the most beautiful movement to do.

I thought I’d do another, slightly longer (5 minutes) video since I enjoyed doing the last one so much and you guys seemed to like it!

This one is of me doing the level 1 horizontal arms, but with a twist!  Totally unedited, so you get to watch me struggle and gurn and even jump when I get interrupted (Warning: British guy swearing at 2:55)!

Feel free to laugh and/or comment.  Thanks.