
Knots and dynamicsThis 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. Back/retour 
 Seifert surfacetesselated

 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 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
 Closeup.
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


