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MATHS BITE: The Kolakoski Sequence

The Kolakoski sequence is an infinite sequence of symbols {1,2} that is its own “run-length encoding“. It is named after mathematician Willian Kolakoski who described it in 1965, but further research shows that it was first discussed by Rufus Oldenburger in 1939.

This self-describing sequence consists of blocks of single and double 1s and 2s. Each block contains digits that are different from the digit in the preceding block.

To construct the sequence, start with 1. This means that the next block is of length 1. So we require that the next block is 2, giving the sequence 1, 2. Continuing this infinitely gives us the Kolakoski sequence: 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, etc.

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Kakeya Needle Problem

The Kakeya needle problem asks whether there is a minimum area of a region in the plane in which a line segment of width 1 can be freely rotated through 360°, where translation of the segment is allowed.

This question was first posed for convex regions in 1917 by mathematician Sōichi Kakeya. It was shown by Gyula Pál that the minimum area for convex regions is achieved by an equilateral triangle of height 1 and area 1/√3.

Kakeya suggested that the minimum area, without the convexity restriction, would be a three-pointed deltoid shape. However, this is false.

Needle rotating inside a deltoid | Source: Wikipedia

Besicovitch Sets

Besicovitch was able to show that there is no lower bound >0 for the area of a region in which a needle of unit length can be turned around. The proof of this relies on the construction of what is now known as a Besicovitch set, which is a set of measure zero in the plane which contains a unit line segment in every direction.

One can construct a set in which a unit line segment can be rotated continuously through 180 degrees from a Besicovitch set consisting of Perron trees.

Kakeya Needle Set constructed from Perron trees | Source: Wikipedia

However, although there are Kakeya needle sets of arbitrarily small positive measure and Besicovich sets of measure 0, there are no Kakeya needle sets of measure 0.

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