The Scientific Book

This is the mother page for things that I think might lead to an actual science of Fractal Society.  The posting Is This a Theory or  a Vision might shed some light on this section.

When FSATFOTI came to me, it came as a flash, a glimpse of something.  The first thing had to do was figure out what that something was, and the menu tabs on overview, Fractal Society, and Fetish of the Individual are attempts to flesh out that initial glimpse.

In fleshing out the initial glimpse, and perhaps finding clothes to decorate it would be a better metaphor, i realized that it was way beyond me to put these ideas into the mathematical language which is the basis of our understanding of fractals.  To be clear, i never took Calculus in high school, and while i have an idea of what quadratic equations are, i have no idea of what they are really all about.

So i came up with (or found) the ideas of s-fracts and s-functions in an attempt to move towards formalizing this entire endeavor.

Fractal Images and the Equations that create them

I first got into the whole idea of fractals from seeing some of the incredible images available online.  Then I got the idea of FSATFOTI, and then thought I should try to figure out the “science” of “Fractal Society.”  Actually suceeding in this endevour is way beyond me, as the following will clarify.

This image of the Mandelbrot set is probably the most famous fractal picture around, and I think it actually captures the idea of Fractal Society.  Think of each part of the image below as an individual, or a part of an individual, or a collection of individuals. (AKA S-fractals.)

When you look at that image, and zoom in on any part of it, you can see what fractal means.

But as for the equations used to create that image; Wikipedia says

The Mandelbrot set M is defined by a family of complex quadratic polynomials

P_c:\mathbb C\to\mathbb C

given by

P_c: z\mapsto z^2 + c,

where c is a complex parameter. For each c, one considers the behavior of the sequence

(0, P_c(0), P_c(P_c(0)), P_c(P_c(P_c(0))), \ldots)

obtained by iterating P_c(z) starting at critical point z = 0, which either escapes to infinity or stays within a disk of some finite radius. The Mandelbrot set is defined as the set of all points c such that the above sequence does not escape to infinity.

A mathematician’s depiction of the Mandelbrot set M. A point cis coloured black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of c, respectively.

More formally, if P_c^n(z) denotes the nth iterate of P_c(z) (i.e. P_c(z)composed with itself n times), the Mandelbrot set is the subset of thecomplex plane given by

M = \left\{c\in \mathbb C : \exists s\in \mathbb R, \forall n\in \mathbb N, |P_c^n(0)| \le s \right\}.

As explained below, it is in fact possible to simplify this definition by takings=2.

Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number c either belongs to M or it does not. A picture of the Mandelbrot set can be made by colouring all the points c that belong to Mblack, and all other points white. The more colourful pictures usually seen are generated by colouring points not in the set according to how quickly or slowly the sequence |P_c^n(0)| diverges to infinity. See the section oncomputer drawings below for more details.

The Mandelbrot set can also be defined as the connectedness locus of the family of polynomials P_c(z). That is, it is the subset of the complex plane consisting of those parameters c for which the Julia set of P_c isconnected.

That is a bit beyond me

If Society as a Fractal is the image, what is the ‘math’ beneath the image?

That is what all the posting gathered underneath the heading “Scientific Book” explore.

Help me out

If you have any ideas about how to “concretize” the ideas of Fractal Society and the Fetish of the individual, I would love for you to share them in the comments.

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