How long is a piece of string?

Once again we have the dynamic duo of Alan Davies and Marcus du Sautoy explore the realms of mathematics.

This time Alan wants to know how we measure things. For this purpose, he wants to start with a very simple question: how long is a piece of string? Since the whole infrastructure of science is based on achieving increasingly accurate measurement, this question is pertinent to the very soul of science itself.

Alan buys a piece of string from a hardware store and measures it with a ruler. It comes out as 32 cm. He then takes it to Marcus Du Sautoy's office, wanting to know if the measurement was correct.

Alan sums up the situation rather neatly: to me it's a short piece of string, to Marcus, it's a major mathematical problem.

Marcus tells Alan that measurement of something so apparently trivial as a piece of string has consumed mathematicians for thousands of years. He decides to take Alan to the National Physical Laboratories (NPL) where they would give the piece of string the proper measurement. During the journey on train, he expanded on the ancient Egyptian method of length measurement in cubits- which is the length between your elbow to the tip of your finger. Since different people would have different lengths in cubits, the problem, according to Marcus, is to be able to come up with a measuring unit that is universal.

At the NPL, the duo are taken to a tour of measurement histories. The original meter bar, a prototype cubit, a two hundred year old yardstick and an exact measurement of the kilogram is shown to the visitors. Then they are taken to the Length Bar Interferometry where they measure length by the speed of light. Using a laser tracking robot, they come up with a pretty concise measurement of Alan's string: 319 mm and 442 micrometers.

Marcus then demonstrates the difficulty of measuring lengths of coastlines, both in maps and in actual scale. He gives Alan a demonstration of a whole new area of mathematics: fractals. By drawing a score of triangles on the beach, Marcus demonstrates to Alan the possibility that the actual length of his piece of string could be infinite. The incredulous look Alan gives Marcus is enough to leave you stitches!

Not satisfied with infinity, Alan goes off solo in search of more measurement. He meets the most hyper physics teacher you are ever likely to meet, Becky Parker, with whom he embarks on a journey into the world of atoms. A most disturbing fact comes to surface when Alan finds out that most atoms are composed of empty space and all the matter composing human mass can be fitted onto a sugar cube. Yes, I am still pondering over that myself. But I must confess, I am jealous of the boys of Simon Langton School. If I had a physics teacher like Ms Parker, I probably would have devoted my life to physics.

Alan continues on his exploration of atoms and quantum mechanics as a means to measuring his string. Soon he comes up with the problem of particles existing in multiple different places at the same time. He is given a thorough lesson on Schrödinger's cat problem by Professor Seth Lloyd, who takes into another journey. He explains the quantum weirdness of photosynthesis and the human sense of smell. Alan becomes well schooled in the fact that quantum mechanics is the underlying fact of reality in the world. Much to Alan's bemusement, Seth postulates that the most accurate measurement of hos piece of string could result in a black hole!

The entire concept of measurement of length can lead you to a philosophical journey where you can learn about the fundamental building blocks of reality itself. In end, the answer to the question: how long is this piece of string? The answer is a qualified: we don't know.