Fourier analysis is a branch of mathematics that uses the Fourier transform to analyse waves. When a violin and trumpet both play the same note, they sound different. The Fourier transformation of these sound waves let us see the difference. For this reason, Fourier analysis is sometimes called Harmonic analysis.

 

A Fourier analyst thinks of many things beyond sound and water as waves, even shapes. In this installation, we have presented a sequence of shapes, followed by their Fourier transforms in the same order. The sequences are accompanied by music created using the geometry of the Fourier transforms.

 

Since time immemorial, humans have used numbers to describe properties: how numerous are the fish in a pond? How much does an ox weigh? How wide is a river? How fast does a cheetah run? How vast is the universe, or is it unbounded?

 

One way Fourier analysts describe a shape is by determining whether its Fourier multiplier is bounded (finite) or unbounded. In 1972, L. Carleson and P. Sjölin discovered that the disks with smoothed edges that began our sequence, the Bochner-Riesz symbols, are bounded in this sense. In 1971, C. Fefferman proved the “unfortunate fact” that the Fourier multiplier of the disk is unbounded. Six years later, A. Córdoba determined sharp bounds for regular polygons like triangles. It is unknown if the last shape in our sequence, the Koch snowflake, is bounded or unbounded. The Koch snowflake is just one example of rough, geometrically rich shapes known as fractals, for which this question remains unresolved.

 

We are grateful to Professor Po Lam Yung, who inspired our interest in this topic.