Mathematical Imagery by Jos Leys

Knots and dynamics animations.

This page is a collection of animations that where made made in collaboration with Prof. Etienne Ghys of the Ecole Normale Supérieure de Lyon.
Professor Ghys used some of them during a plenary lecture at the International Congress of Mathematicians in Madrid, August 2006.

See also this page for a series of images and this article for an insight in some of the math.

All the animations require Quicktime. For best results, make sure you have version 7.1.


All animations copyright Jos Leys / Etienne Ghys.
(all made in Ultrafractal )

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The 'S3' animation shows the trefoil knot in the three-sphere ( obviously projected onto R^3). The knot is rotated in four dimensions (clip 1), and the Seifert surface is built up. (clip 2). Next (clip 3), the Seifert surface rotates around the trefoil knot. Finally (clip 4), several partial Seifert surfaces rotate around the trefoil knot, and the Seifert fibration is shown.
S3 Full length movie 4 Mb
S3 Clip 1 (0.2 Mb) S3 Clip 2 (0.4 Mb) S3 Clip 3 (0.2 Mb) S3 Clip 4 (1.1 Mb)

The 'Knots' animation shows a series of knots, associated with a series of matrices with integer coefficients, and determinant equal to 1, in the complement of the trefoil knot in C^2, projected in R^3.
Knots Full length movie (0.7 Mb)

The 'Knots-Seifert' animation shows a series of knots, in the complement of the trefoil knot in C^2, projected in R^3, and their relative positions and crossings of the Seifert surface.
Knots-Seifert Full length movie (1.3 Mb)

The 'Cross' animation shows a series of knots associated with matrices with real coefficients, in the complement of the trefoil knot in C^2, projected in R^3.
The knots are chosen so that they form a cross shape at their start position.
Cross Full length movie (0.6 Mb)

The 'Horocycle'animation shows the gradual buildup of a horocycle knot around the trefoil. (the "spaghetti factory")
Horocycle Full length movie (4 Mb)

The 'Lorenz' animation shows the Lorenz attractor, and a couple of stable orbits on the attractor. The attractor is built up in clip 1.Clip 2 and clip 3 show different stable orbits: 'knots' on the Lorenz attractor.
Lorenz Clip 1 (0.3 Mb) Lorenz Clip 2 (0.4 Mb) Lorenz Clip 3 (0.4 Mb)

The 'Lorenz template' is an object on which certain types of knots can be drawn.The animation shows how.
Lorenz template Full length movie (1 Mb)

The 'Deformation' animation shows a number of knots in C^2, projected into R^3, based on matrices with integer coefficients and determinant=1.
A deformation procedure is applied which sends the knot onto a template equivalent to the Lorenz template.
Clip 1 shows the template, and the other clips show the procedure for different knots.
Deformation Full length movie 6 Mb

All of the above has strong links to fluid flow.
Although the colors don't match, we call it the 'Dame Blanche' animation. (you know, chocolate and ice cream)
Now imagine this is an ideal fluid, and you manage to put in a cup so that the perfectly fluid ice cream and chocolate do not mix.
The animation shows what happens when four vortices are let loose .....
Dame Blanche Full length movie (25 Mb)


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Copyright 2014 Jos Leys