Simple math formulas ...

In principle, very simple mathematical formulas were used to generate the designs.

These are simple equations that begin with a pair of starting values x (0) | y (0).

The next pair of values x (1) | y (1) is then calculated from this, and from this the next pair of values x (2) | x (2).

This continues hundreds of millions of times.

For example, one calculates one billion x | y points - so we get:

x(0)|y(0)

x(1)|y(1)

x(2)|y(2)

x(3)|y(3)

...

x(1.000.000.000)|y(1.000.000.000) 

Counting and coloring ...

The calculated x | y points are then more or less densely scattered on an x-y plane. This level is then divided into a grid of, for example, 1000 x 1000 points. You get a grid with 1,000,000 squares or pixels.

And now comes the really first simple ingenious trick that leads to the wonderful fractals: you COUNT how many of the calculated x|y points are in a grid point or pixel (you count the "hits", so to speak). In this way one then obtains pixels (grid locations on the x|y plane) in which no, few, many or very many calculated points (hits) lie.

This then results in a scale from 0 to, for example, a maximum of 100,000 values found in a grid square or pixel. The second nice trick is to give the scale values colors: e.g. You color a pixel with 0 to 10 hits with white, with 11 to 1000 hits red, with 1001-10000 hits green, with 10001-100000 hits blue.

Well chosen starting values and parameters + good coloring strategy ...

With a little "luck" you will have a good pair of starting values and, above all, the right parameters. With a good “coloring strategy”, you will get graphics that look like the designs in sali-math-arts.

Lots of attempts to find the right parameters ...

By the way, it can happen that you need hundreds of billions of attempts just to find suitable parameters that should lead to beautiful designs. Some of the calculated designs took a few days or weeks of uninterrupted calculations.

Which designs are "good" ...

But how do you determine whether parameters are actually candidates for a design to be "good"? The simplest criterion for whether a design is worth seeing was initially to determine whether the parameters and starting values used "generate" enough hits for pixels. The "most" attempts namely scatter the hits very far into individual pixels of the x|y plane, so that "apart from a fog with individual points one cannot see anything" - they are therefore useless.

The "art" therefore also and above all consists in actually initially finding parameters that lead to "visible" results. A further step then consists in the admittedly subjective optical selection of "beautiful designs" from the visible candidates.

New equations for new design types ...

In the literature you can find a multitude of well-known equations from chaos theory and from the world of strange attractors and fractals, the skillful parameterization of which - as described above - already leads to a certain range of different graphic characteristics for each type of equation. I also used such equations and - as mentioned - determined suitable parameters with sometimes considerable computing effort in order to obtain different designs.

However, in order to find completely new design types, parameter variations alone are not enough. I therefore modified existing types of equations on the one hand, and set up new types of equations on the other in order to find new design types and designs.

Equations that come from my own pen or that I have modified or expanded from existing ones always contain the abbreviation SALI in the name. All the others come from well-known sources that I found one after the other, for example on relevant Python sites, in Wikipedia or in books on chaos, fractals or non-linear dynamic systems.

Chic designs with a mathematical background from chaos theory give fashion and gifts their very own special charm!

Symmetric Icon

Symmetric Icon SALI_1

zzbar(i) = x(i-1)*x(i-1)+cos(y(i-1))*cos(y(i-1)) + y(i-1)*y(i-1) + cos(x(i-1))*cos(x(i-1))
p(i) = a*zzbar(i) + l
zreal(i) = x(i)
zimag(i) = y(i)

for k in range(1, d-1):
za(i) = zreal(i) * x(i) - zimag(i) * y(i)
zb(i) = zimag(i) * x(i) + zreal(i) * y(i)
zreal(i) = za(i)
zimag(i) = zb(i)

zn(i) = x(i)*zreal(i) - y(i)*zimag(i)
p(i) = p(i) + b*zn(i)

x(i) = p(i)*x(i-1) + g*zreal(i) - om*y(i-1)
y(i) = p(i)*y(i-1) - g*zimag(i) + om*x(i-1)

Symmetric Icon SALI_1 Designs

Hopalong 2

x(i) = y(i-1) - 1.0 - sqrt(fabs(b * x(i-1) - 1.0 - c)) * sign(x(i-1) - 1.0)
y(i) = a - x(i-1) - 1.0

Hopalong 2 dessigns

more equations will follow ... in progress