By CLIFFORD W.ASHLEY, Published by Doubleday and Company, Inc. 1944
The most famous book on knots is the Ashley Book of Knots, which contains almost 4000 examples of knots, links, sinnets, hitches, bends and splices, with illustrations by the author of the way knots have been used.
So Ashley writes:
`To me the simple act of tying a knot is an adventure in unlimited space. A bit of string affords the dimensional lattitude that is unique among the entities. For an uncomplicated strand is a palpable object that, for all practical purposes, possesses one dimension only. If we move a single strand out of the plane, interlacing at will, actual objects of beauty result, in what is practically two dimensions; and if we choose to direct our strand out of this plane, another dimension is added which provides an opportunity that is limited only by the scope of our own imagery and the length of a ropemaker's coil.
What could be more wonderful than that? '
`But there is always another car ahead in a possible parking place. Here is a Mr. Klein asserting that knots cannot be tied in four dimensions.'
To see why knotted string can be untied in four dimensions, we work by analogy. The jump from 2 dimensions to 3 dimensions can be illustrated by an ant moving on a table. He can only get past a long obstacle by going over it, that is, into the third dimension.
To untie a piece of knotted string, you move one piece of string against another, and then past it by moving into the 4th dimension (in your imagination!).
In science and mathematics, there is always another question in any parked answering place. If knotted string can be untied in 4 dimensions, what then can be tied?
The answer to this question is a knotted balloon, or sphere.
So art and mathematics part company for a while, as mathematics tries to understand the structure and patterns of objects we cannot make, and can only perceive as through a glass, darkly, and for which the tools of study and certainty are not experiment but logic and the development of language and concepts to describe the previously indescribable.
Ashley also writes:
`From the earliest times to the present day, the joy incident upon occasional discovery has ever been considered sufficient reward in itself for any human effort or sacrifice.'
This is a motto for mathematics, and art.
© Mathematics and Knots/Edition Limitee 1996-2002
This material may be used freely for educational, artistic and scientific purposes, but may not be used for commercial purposes, for profit or in texts without the permission of the publishers.