Littlewood videos

by Greg Egan

Littlewood Dragons

View Littlewood Dragons in HD on the Vimeo site.

This video shows all the Littlewood polynomials of degree 25 with a positive constant term — that is, polynomials of the form:

L(z) = 1 ± z ± z2 ± z3 ... ± z25

— evaluated at a point that moves around in the complex plane. The inset shows the location of the point against a backdrop of the set of roots, within a disk of radius 0.8, of the same polynomials.

The small white cross in the centre of the main image marks zero in the complex plane for the evaluations. As various families of polynomials (those sharing the same low-order coefficients, indicated by the colour of the points) sweep over zero, the same polynomials, and some very similar geometry, can be seen in the roots set.

Littlewood Tour

View Littlewood Tour in HD on the Vimeo site.

This video tours the inner edge of the set of roots in the complex plane of the Littlewood polynomials of degree 36. Each point is coloured according to the coefficients of the particular polynomial of which it is a zero. The movie is best viewed in fullscreen mode (in the HD version, only available on the Vimeo page) to see the fine details!

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Science Notes / Littlewood videos / created Wednesday, 4 January 2012 / revised Saturday, 7 July 2012
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