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Issue 9 - June 2002

Get Knotted

Get Knotted A branch of mathematics thinks the twists and turns of knots might hold the key to the secret of the Universe. By looking at knots in a new way, scientists have found that knots may have their own quantum theory.

Until now mathematical knots were constructed out of lines which have no thickness, and theorists only looked at the way the knot was formed. Unlike many real life knots, mathematical knots have no loose ends. If you tie your shoelace, and then tape the two ends left over together, you have a mathematical knot. But your shoelace also has a thickness and is made out of certain materials, and these properties make a difference to the kinds of knots you can tie with them.

The question is not how long is a piece of string but what is the ratio of the length to the diameter? "The smaller this ratio becomes," explains Keith Devlin, Dean of Science at Saint Mary's College of California, "the harder it becomes to construct a given knot. Below a certain threshold in this ratio, you can't construct the knot at all".

All this may seem a lot of trouble to find the best way to tie your shoelaces, but it has important implications for molecular science. Knots are the stuff we are made of - the double helix strands of DNA knot themselves up to be able to fit into the cell nucleus. If a cell nucleus was the size of a basket ball then the 2 DNA strands could be compared to fishing lines 200km long, so it takes a lot of twisting and turning to fit inside.

A computer programme was made to construct the ideal configurations for different knots to minimise the length of string used, taking into account this ratio. The programme calculated the 'writhe' or over all 'intertwistedness' of the knots. This measures how much it crosses over itself from left to right (+1) and right to left (-1), and then measures again from different angles to get the 3D writhe. One scientist did this with ideal prime knots (knots which can't be split into 2 simpler knots) with up to 9 crossings, and plotted the writhe on a vertical and horizontal axes. The resulting pattern looked very like the evenly spaced energy levels of electrons in an atom.

Although it is still early days for knot theory, there appear to be important implications for understanding DNA and the construction of the universe. Knot theory began when Lord Kelvin put forward his idea that the whole world was made up of an invisible fluid of knots of particles that fill space. With knot theory now sending mathematicians, geneticists and quantum physicists round the bend his idea may not have just been money for old rope.

If this article has sent you round the twist find out more about knot theory and DNA at:
http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html
and
http://library.thinkquest.org

Source: The Guardian


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Microsoft fails maths

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Get Knotted

Maths Exams Are Getting Harder

Sum Fun




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