Notes prepared by
Professor Marcus du Sautoy, University of Oxford

Spin the needle

Take a needle, dip it in ink and place it on a piece of paper. Now turn the needle on the page so it points in the opposite direction. As you turn the needle the ink prints out a shape on the page. What is the smallest area that you can get away with as the needle sweeps out the shape on the page?

Clearly I can fix one end of the needle and sweep out a semi-circle whose radius is the length of the circle. But I can do better by fixing the middle of the needle and spinning it. Now the area is a circle whose radius is half the length of the needle. But using a concave triangle I can do even better.

In 1917 the Japanese mathematician Soichi Kakeya challenged mathematicians whether they could find a way to spin the needle which used less area than the concave triangle. Amazingly ten years later Abram Besicovitch showed that there were ways to spin the needle which swept out as small an area as you wanted.

As an example of one ingredient in the proof, let me explain how one can translate the needle from one place on the paper to another sweeping out as small an area as you want. At first sight it looks like you will sweep out a huge slanted rectangle. The cunning move is to slide the needle along the line it is sitting on, then tilt the needle so that it points towards the new location you are trying to get to, then slide the needle along the new line and tilt the needle back again. The only area swept out on the page are the two tilts. By moving the needle further along the line we can gradually decrease the amount needed to tilt the needle so that the area is as small as we want.

To actually turn the needle rather than simply translating it, Besicovitch cuts up the triangle into pieces which he puts on top of each other. By piling the triangles on top of each other, the area covered by the triangles is smaller than the area covered by the original triangle. Besicovitch cooked up a way to turn the needle by using this spikey area made out of triangles together with the translation move above. Besicovitch proved that the more triangles you cut the first triangle into, the smaller the area used to turn the needle.

The challenge then became to understand these strange shapes that the needle was sweeping out. Carleson produced methods to be able to measure in some sense how big these sets are. His analysis implies that although the limit of these sets has zero area they are nonetheless big, having something called Hausdorff dimension 2.

This may look like nothing more than a fun puzzle. But it goes to the heart of the nature of the space we live in. In recent years the mathematics hiding behind this toy has impacted on questions in number theory, geometry and even the analysis of differential equations describing waves.

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