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12 agosto 2006

The Knot Round Dance

Lorenz attractor
From Wikipedia, the free encyclopedia

A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3

The Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator.

The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.

A trajectory of Lorenz's equations, rendered as a metal wire to show direction and 3D structure

The attractor itself, and the equations from which it is derived, were introduced in 1963 by Edward Lorenz, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere.

In addition to its interest to the field of non-linear mathematics, the Lorenz model has important implications for climate and weather prediction. The model is an explicit statement that planetary and stellar atmospheres may exhibit a variety of quasi-periodic regimes that are, although fully deterministic, subject to abrupt and seemingly random change.

From a technical standpoint, the Lorenz oscillator is nonlinear, three-dimensional and deterministic. For a certain set of parameters, the system exhibits chaotic behavior and displays what is today called a strange attractor. The strange attractor in this case is a fractal of Hausdorff dimension between 2 and 3. Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension to be 2.05 ± 0.01.

The system also arises in simplified models for lasers (Haken 1975) and dynamos (Knobloch 1981).
[edit]EquationsSensitive dependence on the initial condition

Time t=1 (Enlarge) Time t=2 (Enlarge) Time t=3 (Enlarge)

These figures — made using ρ=28, σ = 10 and β = 8/3 — show three time segments of the 3-D evolution of 2 trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10-5 in the x-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.

Java animation of the Lorenz attractor shows the continuous evolution.

Rayleigh number

The Lorenz attractor for different values of ρ

ρ=14, σ=10, β=8/3 (Enlarge) ρ=13, σ=10, β=8/3 (Enlarge)

ρ=15, σ=10, β=8/3 (Enlarge) ρ=28, σ=10, β=8/3 (Enlarge)
For small values of ρ, the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.28, the fixed points become repulsors and the trajectory is repelled by them in a very complex way, evolving without ever crossing itself.

Java animation showing evolution for different values of ρ


Growth and Unfurlment of 3 Pairs (RGB) of First Hand(tm)Models

Displayed as Three Loops

THE 3,10 KNOT Growing, Blossoming, and Fading
Continuous Creation: A Fruit Tree Yielding Fruit whose Seed is In Itself-Genesis 1.11

Displaying Six First Hand(tm) Models in Rainbow Order


Displaying 3 pairs (RGB) of First Hand(tm) Models
This animation is based on original designs by Stan Tenen. This figure is actually three loops, and not a true knot, but this animation was so beautiful that we have decided to post it here in addition to the true 3,10 knot version. An alternate animation of this three-loop figure can be seen at:

Displaying 3 pairs (RGB) of First Hand(tm) Models

Displaying 3 pairs (RGB) of First Hand(tm) Models

This animation is based on original designs by Stan Tenen.

The 3,10 Torus Knot, Ring, Sphere, Tetrahelix and Hand

Poster and explanatory text originally published in TORUS, the Journal of the Meru Foundation, Vol. 2 #2, December 1992

©1992, 1996 S. Tenen

This illustration shows: 
How the standard "Ring" form of the 3,10 torus knot can be transformed to fit on the surface of a dimpled-sphere torus.
How the 3,10 torus knot is defined by a "touch-pad magic square" whose diagonals, central row, and column add to 15).
How the dimpled sphere form of the 3,10 torus knot defines 6 hand-shaped regions wound around a (6-thumb)tetrahelical central column.
How the central column of the dimpled-sphere form of the 3,10 torus knot is composed of and defined by a column of 99-tetrahedra.
How each hand is defined by a central colunn (wound on the thumb and extended over the palm and 4-fingers) of a "jubilee" of 49-tetrahedra; and
How the 99-tetrahedra tetrahedral column consists of 3-ribbons of 3x22=66 triangular faces, with one triangular face for each of the 3 sets of 22-letters of a string of 3-Hebrew alphabets.
A Model of Continuous Creation

For a more details on the upper portion of this poster, see the poster:

3,10 Torus Knot, Ring, Sphere, Tetrahelix, and Hand

To view animations of the 3,10 Torus knot displaying 3 pairs of First Hand(tm) models, see:
3,10 Torus Knots