Mathematical Imagery by Jos Leys

Knots and dynamics

This page stems from a collaboration with Prof. Etienne Ghys of the Ecole Normale Supérieure de Lyon.
Prof.Ghys held a plenary lecture at the International Congress of Mathematicians in Madrid (August 2006).The professor and myself worked out the graphics for this lecture : I wrote custom algorithms in Ultrafractal to translate the math, supplied by the professor, into images and animations.

The lecture entitled "Knots and Dynamics" treats topics in fluid flow and knot theory. The graphics in these pages relate only to some aspects of these topics, as the subject matter is much broader.Some of the math used to produce these images is explained in this article.
See also this review.

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Seifert surface
See also the animations page

Seifert surface-tesselated

Multi Seifert

Seifert fibration

Integer matrix 1
The projection in R^3 of the image in C^2 of a matrix with integer coefficients and determinant=1.

Integer matrix 2
The projection in R^3 of the image in C^2 of a matrix with integer coefficients and determinant=1. Matrix: (162 127 , 287 225 )

Integer matrix 3
The projection in R^3 of the image in C^2 of two matrices with integer coefficients and determinant=1. Matrix: (4 1 , 7 2 ) and (2 7 , 1 4 )

Integer matrix 4
The projection in R^3 of the image in C^2 of a matrix with integer coefficients and determinant=1. Matrix: (5 2 , 747 299 )

Integer matrix 5
The projection in R^3 of the image in C^2 of two matrices with integer coefficients and determinant=1. Matrix: (6 5 , 403 336 ) and (336 403 , 5 6 )

Integer matrix 6
The projection in R^3 of the image in C^2 of two matrices with integer coefficients and determinant=1. Matrix: (5 3 , 333 200 ) and (200 333 , 3 5 )
(Rotated 90° in C^2)

Integer matrix 7
The projection in R^3 of the image in C^2 of two matrices with integer coefficients and determinant=1. Matrix: (4 1 , 7 2 ) and (2 7 , 1 4 )
(Rotated 90° in C^2)

Integer matrix 8
The projection in R^3 of the image in C^2 of two matrices with integer coefficients and determinant=1. Matrix: (8 13 , 3 5 ) and (5 3 , 13 5 )
(Rotated 90° in C^2)

Real matrix 1
The image of a collection of matrices with real coefficients.
Based on the matrix ( 6 5 , 403 336.6 )

Real matrix 2
The image of a collection of matrices with real coefficients.
Based on the matrix ( 11.1 13.1 , 16.1 19.1 )

Real matrix 3
The image of a collection of matrices with real coefficients.
Based on the matrix ( 11.1 13.1 , 16.1 19.1 )

Real matrix 4
The image of a collection of matrices with real coefficients.
Based on the matrix ( 52.2 71.12 , 41.1 56.1 )

Real matrix 5
The image of a collection of matrices with real coefficients.
Based on the matrix ( 5.12 7.1 , 7 10.1 )

Real matrix 6
The image of a collection of matrices with real coefficients.
Based on the matrix ( 19.1 31.1 , 11.1 18.1 )

Real matrix 7
The image of a collection of matrices with real coefficients.
Based on the matrix ( 5.1 7.1 , 7 10.1 )

Real matrix 8
The image of a collection of matrices with real coefficients.
Based on the matrix ( 3.2 5.1 , 10.1 17.1 )

Real matrix 9
The image of a collection of matrices with real coefficients.

Real matrix 10
The image of a collection of matrices with real coefficients.
Based on the matrix ( 2 1.2 , 1.2 1 )

Real matrix 11
The image of a collection of matrices with real coefficients.
Based on the matrix ( 11.1 13.1 , 16.1 19.1 )

Real matrix 12
The image of a collection of matrices with real coefficients.
Based on the matrix ( 11.1 13.0 , 16.0 19.1 )

Real matrix 13
The image of a collection of matrices with real coefficients.
Based on the matrix ( 11.1 13.1 , 16.1 19.1 )

Real matrix 14
The image of a collection of matrices with real coefficients.
Based on the matrix ( 3.2 5 , 106 177.2 )

Real matrix 15
The image of a collection of matrices with real coefficients.
Based on the matrix ( 17.1 21.0 , 21.0 26.1 )

Real matrix 16
The image of a collection of matrices with real coefficients.
Based on the matrix ( 5.4 7.9 , 2.9 5.4 )

Real matrix 17
The image of a collection of matrices with real coefficients.
Based on the matrix ( 10.1 11.0 , 12.0 13.14 )

Real matrix 18
The image of a collection of matrices with real coefficients.
Based on the matrix ( 5.2 3.11 , 13.11 8.26 )

Real matrix 19
The image of a collection of matrices with real coefficients.
Based on the matrix ( 13.1 20.779 , 8.04 13.1 )

Real matrix 20
The image of a collection of matrices with real coefficients.
Based on the matrix ( 13.1 20.779 , 8.04 13.1 )

Real matrix 21
The image of a collection of matrices with real coefficients.
Based on the matrix ( 5.12 7.1 , 7 10.1 )

Real matrix 22
The image of a collection of matrices with real coefficients.
Based on the matrix ( 97.2 115.1 , 70.1 83.1 )

Real matrix 23
The image of a collection of matrices with real coefficients.
Based on the matrix ( 13.0 20.779 , 8.04 13.6 )

Real matrix 24
The image of a collection of matrices with real coefficients.
Based on the matrix ( 5.1 7.1 , 7 10.1 )
See also the animations page

Horocycle 1
The horocycle dynamic.
See also the animations page

Horocycle 2

Horocycle 3

Horocycle 4

The Lorenz template
An object on which a certain kind of knot can be drawn. See also the animations page

The Lorenz template
Close-up.
The orange bar is called the 'Poincaré section'.

A variation of the Lorenz template

deformationc002a

The Lorenz attractor

Stable orbit 1
A stable orbit on the Lorenz attractor.
See also the animations page

Stable orbit 2
A stable orbit on the Lorenz attractor

Stable orbit 3
Multiple stable orbits

Knot deformation 1A
A knot, based on an integer matrix is deformed...

Knot deformation 1B
...and comes to rest on the template.

deformationc007

deformationc008
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Copyright 2014 Jos Leys