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Symmetric Tangles

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Tangles

Stage: 3 and 4
Article by Mike Pearson

You can begin to explore Conway's tangles with the NRICH Problem "Twisting and Turning". The article "Symmetric Tangles" explores the symmetry of tangles.

 

The 4/3 tangle in symmetrical pillowcase form
This is a personal story of discovery that began in February 2007 when a colleague of mine, Ian Short, showed me an interesting trick he'd seen in a lecture given by the mathematician John Conway in Cambridge in 1998.

At the very start of that lecture, he said this, "What I like doing is taking something that other people thought was complicated and difficult to understand, and finding a simple idea so that any fool and in this case, you can understand the complicated thing."

He's telling us there are lots of really difficult ideas out there that really aren't that difficult at all if only we can see them from a better angle. Well, after reading the lecture transcript it still took quite a long time to find the better angle to see his little rope trick, but he had captured my interest and I ended up having a lot of fun investigating it, and learnt quite a lot of mathematics on the way...

Read the whole article...(pdf download)


Appendix

Here are some interactive resources that were difficult to fit in the printable article:

There is a real player video stream of a similar lecture delivered by Conway to an audience of mathematicians at http://www.msri.org/publications/ln/msri/1997/ldt/conway/1/index.html .

I'm working on a 3D tangle environment.
You can use this environment to explore tangles without having the ropes to hand, but you'll find that the tangle looks ever more and more tangled, and never simplifies. I'm trying to use tangle theory to help us program this environment to do the disentangling too. It's not so easy, but tangle theory should make it possible!

Here's the explanation of why a drain pipe is handy...

This text is usually replaced by the Flash movie.


You can also use the Geomlab environment to explore tangles. The NRICH version of Geomlab has some useful tangle functions built into it.

ZERO - draws the zero tangle.
ONE - draws the tangle with one twist.
turn(x) - turns the tangle x
twist(x) -twists the tangle x
untwist(x) - untwists the tangle x
twists(n, x) - adds n twists to tangle x
twil(list) - draws a tangle from a list of twists. Twil([2,3]) will twist twice, then turn, then twist thrice.
twirl(list) - is simlilar to twil, but the list is reversed first. The result is easier to use.

Finally, the most interesting function is rat(p,q):
rat(p,q) - draws the tangle for p/q. It uses Euclid's Algorithm to do this:
define rat(x,0) = turn(rat(0,x))
| rat(0,x) = ZERO
| rat(x,x) = ONE
| rat(p,q) = rat(-p,-q) when q < 0
| rat(p,q) = twist(rat(p-q,q)) when p > 0
| rat(p,q) = turn(rat(q,-p));