Topology is the mathematics of the shapes of bendy spaces. In topology two spaces are said to be homeomorphic if there is a continuous smooth transformation of one space to the other. However, “space” doesn’t need to follow your intuitive notions of space. A computer can take a space made up of zeros and ones and transform it into this performance by Ouchhh. (In fact that is what yours is doing when you watch this video.)
We live in a very interesting space. The space we live in has three dimensions which gives it a property that spaces of other dimensions cannot have. In our three dimensional world we can tie knots.
What is a knot? Well, like a lot of things in mathematics there is the definition everyone uses, and the definition mathematicians use. Webster’s Revised Unabridged Dictionary gives us the following definition:
1. (a) A fastening together of the pars or ends of one or more threads, cords, ropes, etc., by any one of various ways of tying or entangling. (b) A lump or loop formed in a thread, cord, rope. etc., as at the end, by tying or interweaving it upon itself. (c) An ornamental tie, as of a ribbon. &hand; The names of knots vary according to the manner of their making, or the use for which they are intended; as, dowknot, reef knot, stopper knot, diamond knot, etc.
Where WolframMathWorld says:
In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot). While in common usage, knots can be tied in string and rope such that one or more strands are left open on either side of the knot, the mathematical theory of knots terms an object of this type a “braid” rather than a knot. To a mathematician, an object is a knot only if its free ends are attached in some way so that the resulting structure consists of a single looped strand.
As already mentioned knots can only be made in three dimensions. In two dimensions there is no way for the string to “cross over” itself. In four or more dimensions the knots don’t stay tied as the extra dimensions allow them to unfold and fall open. If we lived with any other spacial dimensions knots would simply not be.
Knots are one of the oldest technologies we have. According to a 2010 article in Low-Tech Magazine,
The earliest fossilized fragments of ropes and knots date back 15,000 to 17,000 years, which makes the direct evidence of this technology much older than that of the axe (6000 BC) or the wheel (5000 BC). However, based on indirect evidence (perforated objects, wear marks on artefacts, bone needles, representations in art, etcetera), archaeologists believe that the use of ropes and knots dates between 250,000 and 2,500,000 years old.
Through out history knots have been used for hunting, sailing, climbing, recording information, helping to provide shelter and general decoration. The design on my wedding ring is a knot made of two links intertwined.
When do you use knots in your daily life? What is your favourite knot?