Alice in Wonderland wonders (I guess that’s her job?) what kind of knots we’re tying in Second Life.
She’s not alone. So let me explain. They are “ideal” or “tight” knots. If you imagine a perfect piece of string – one you can bend until it touches itself and that always has a perfectly circular cross section, always the same diameter – and you try to tie a knot with this string, then the shortest possible version, given a fixed diameter piece of string, of that knot is the ideal version.
Ideal knots are interesting not only because they are pretty
but because they predict some of the properties of knotted DNA. Why, no-one knows.
BTW, when I do pictures of ideal knots, I usually show them at half their real diameter. This is because when they’re full sized you can’t see the interesting details in the middle. Pictures like
solve that problem to some extent – and I could do better ones now. The one above doesn’t quite touch itself because when it was drawn we were using points and straight lines to approximate the knot, and those don’t really work, whereas now we use arcs of circles.
Anyway, I’ve been working on these things on and off for around ten years with various collaborators, and I still am, as you can see from the Second Life foolishness. By the way, watching these things render is so much fun its almost worth creating a Second Life character for. I did!
I’ve been working with Matt Biddulph of Second Nature to render ideal knots in Second Life. Here’s an image of one being fabricated.
A bit ropey currently, but we’re getting there.
One of my co-authors tells me that “Physical and Numerical Models in Knot Theory” is out. I did the cover, as well as working on chapter 5. Since their picture is rather teensy, you can see the artwork for the cover here.
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Earlier I wrote about an ideal link as if anyone had any clue what they were. So, here’s a brief explanation.
An ideal link is the form of the link that uses the least possible string. More formally, it is the shape that minimises L/r, where L is the total length of the centrelines and r is the radius of the thickest tube that can be put around that centreline without any overlapping (note that the same thickness of tube is used on all components of the link). Or, you can think of it as fixing the diameter of the tube and minimising L – intuitively, this is like tightening a piece of perfect string – which is why ideal links are also known as tight links.
So, why are they interesting? Well, apart from being very pretty, they predict the properties of knotted (or linked) DNA. The length of the ideal knot predicts very accurately the distance which the corresponding piece of knotted DNA travels when subjected to gel electrophoresis. They also predict the average writhe of a piece of knotted DNA that’s wriggling about in solution.
They also have some intrinsic properties that interest me. For example, they exhibit spontaneous symmetry breaking (the two components of that link are the same, topologically, but they end up different in the ideal link). Another is that, contrary to intuition, they don’t necessarily touch themselves everywhere – sometimes the shortest path is achieved by curving tighter than you can by wrapping around some other part of the knot, which means you have to break contact to do so.
Other people have other interests – for instance, my friend and collaborator John Maddocks is obsessed with the contact surface. Which, incidentally, I was recently given in 3D – aren’t 3D printers wonderful? I wish I had one.
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The magic incantation was whispered to me…
mencoder mf:///path/to/some-*.png -mf fps=25:type=png -ovc lavc -o spinning.avi
Which takes your PNGs and makes an AVI of them. Here’s 7.4 again. Make sure you have your player set to repeat. And don’t use Winamp. At least, not the version I was using. It was like my childhood: no vertical hold.
One of my interests is these things called ideal knots (and links) about which I’ll write more later, no doubt. Anyway, I recently slammed together some code and scripts to produce raytraces of them spinning. Here’s the two component link 7.4, spinning. The colours indicate curvature, by the way.
Incidentally, this is an animated GIF, which sucks somewhat, since GIFs are limited to 256 colours, as well as being politically incorrect. Anyone out there know of command-line UNIX tools that will produce moving images with 24-bit colours from a set of frames?