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contents Mathematics

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where is the order of deterministic chaos hiding

From the dimensions with coordinates to the dimensions of deformation

[start]    00   what is a dimension ?
The evolution of the notion of dimension, from antiquity to fractal dimensions of Mandelbrot.

[next]     01   the measurement of the deformation of a contrast
How we can measure a phenomenon without any notion of coordinates on an axis.
The fundamental difference between coordinate dimensions and deformation dimensions.

[next]       02   in theory the dimension of a contrast can be self-similar
The problem of dimensions which vary with the scale of the measurement.

[next]     03   the trap in the vectorial representation of the forces
Usually a force is summarized by a vector which applies on a point. This representation leads to a serious abnormality when we want to calculate the interference of several forces.
We suggest another type of representation that correctly uses the effects of their interference in all directions of space: this representation requires the measurement of infinity of vectors in every point.
The objet of the following text is precisely to show how it can be easy to measure infinity of vectors with a single number.

Counting in a different way
(why most of the time .5 is anything but between 0 and 1)

[next]      10   Cantor's absurd infinities
A reminder of the way for generating numbers according to set theory.
A reminder of the abnormalities caused by this way when we use infinities. The irrational decimal numbers such as 'Pi' or such as logarithms bear infinite numbers of decimals, therefore the abnormalities of infinite numbers have direct consequences in the measuring of physical phenomena.
A reminder of the demonstration by the "Cantor's diagonal."

[next]      11   let's go back from zero
We suggest another way for generating decimal numbers, a way without the abnormalities previously recalled regarding infinities.
This new conception of numbers considers that a decimal number has the value of a volume, therefore that it cannot be reduced in a single point on an axis without trouble, and that it needs a surface at least to be represented.

[next]      12   how to travel from one number toward another?
How to translate whole and decimal values on a diagram, without losing any information about the numbers.

About Mandelbrot fractal dimensions

[next]      20   Mandelbrot fractal dimensions to the rescue
Our conception for numbers gives to the fractal dimensions of Mandelbrot the status of 'normal and usual' dimensions.
The whole coordinates of space dimensions, usually considered as the very prototype of a dimension, would be only 'special cases' and non-representative sorts of decimal numbers. In fact, they would only be decimal dimensions with zero as a decimal value.

[next]      21   the difference between fractal and space dimensions
What the whole numbers before decimal value mean in fractal dimensions, such as digit '1' in the dimension Log 4/Log 3 ~ '1'.2618.
Why these whole numbers are not connected to the number of dimensions in space.

Birth of a new dimension in a natural phenomenon,
and simultaneous surge of chaos

[next]      30   continuity of deformations in space
Two points that are not linked together in space, do not necessarily work like separated points.

[next]      31   Ian Stewart drips his tap
A reminder of the theory of 'deterministic chaos'. For this we use a typical experiment.

[next]      32   birth of a dimension
How and why deterministic chaos inevitably surges in due circumstances.

[next]      33   to see what's going on in the 4th dimension
A 'strange attractor' is not a curve on which points erratically move: it's a surface and functions as a surface.

[end]      34   chains of dimensions
We suggest a table summerizing the different kinds of dimensions, their chain and their specific properties.
We hope this table will help to find the way to calculate and to predict with accuracy the behavior of the so called 'chaotic' phenomena, such as turbulence, without being limited in this calculation by any 'butterfly effect.'


This text is the almost complete translation of the last chapter of a French book whose résumé (in French) is in line:
résumé of the book   'l'adieu au big-bang'  (A farewell to big-bang)

This book can be bought directly through the Internet with "Fnac.com".
To see the offer, you just have to click on this image of the book:

Quatuor site has its own domain name: quatuor.org  -  Then, the page you are looking at, is available by: www.quatuor.org/english/mathematics.htm


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