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Mathematical Background

How does the creation of the designs actually work?

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.

x(i) := f(x(i-1)|y(i-1))
y(i) := g(x(i-1)|y(i-1))

Clifford

x(i) = sin(a * y(i-1)) + c * cos(a * x(i-1))
y(i) = sin(b * x(i-1)) + d * cos(b * y(i-1))

Clifford-Designs

Play of colors on Clifford trajectories
Unequal pair of green twins in the wind
Streams of Heart

Bedhead

x(i) = sin(x(i-1)*y(i-1)/b)*y(i-1) + cos(a*x(i-1)-y(i-1))

y(i) = x(i-1) + sin(y(i-1))/b

Bedhead-Designs

Someone is in a hurry elegant Lady
Chaotic Ice Water Whirlwind
Happy play of colors

Gumowsky Mira

x(i) = y(i-1) + a*(1 - b*y(i-1)**2)*y(i-1) + G(x(i-1), mu)
y(i) = -x(i-1) + G(x(i), mu)


G(x, mu) = mu * x + 2 * (1 - mu) * x**2 / (1.0 + x**2)

Gumowsky Mira Designs

Rascal Starfish Sea Urchin Sea Animals
Bird of Paradise Icarus
green steering wheel green starfish

Symmetric Icon

zzbar(i) = x(i-1)*x(i-1) + y(i-1)*y(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 Designs

Colorful boomerang starfish sea animal
Star Christmas Star Mandala Lucky Star
Colorful seashells with thorns

Symmetric Icon SALI_1

Symmetric Icon SALI_1 Designs

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)

Spiral fan ammonite fossil
Starfish ammonite prehistoric animal
rotating fire of life
rotating fire of life

Jason Rampe Sali 1

x(i) = (cos(y(i-1)*b)+c*sin(x(i-1)*b))*e
y(i) = (cos(x(i-1)*a)+d*sin(y(i-1)*a))*f

Jason Rampe Sali 1 designs

Open fire campfire flickering fire
Color wind colorful chaos watercolor

Svensson

x(i) = d * sin(a * x(i-1)) - sin(b * y(i-1))
x(i) = c * cos(a * x(i-1)) + cos(b * y(i-1))

Svensson dessigns

Harmony in the Ocean of Elements
Winds of the Heart

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

Emerald weave spun from the chaos
cheerful blue diamond

Fractal Dream

x(i) = sin(y(i-1)*b)+c*sin(x(i-1)*b)
y(i) = sin(x(i-1)*a)+d*sin(y(i-1)*a)

Fractal Dream dessigns

Secret sign from chaos theory
Eye in the Inferno
glowing paths

other equations will follow ... under construction

About mathematical works of art

The idea of this WEB page is to show the visitors the beauty and charisma inherent in the magic of chaos theoretical fractals - which are otherwise mainly used for calculations such as turbulence, weather or for calculating economic cycles.

Welcome to my little mathematical art gallery !

Beauty in mathematics and physics can be seen in the fractals and strange attractors of chaos theory: each has its own charisma, fantasy and message from the universe of chaos and order.


The diversity of the mathematical works of art that emerges can be seen in the most varied of pictures, watercolors, mandalas or even in almost essential creations that sometimes look strikingly like marine animals, birds or plants.


Let yourself be surprised and look into the fantastic and imaginative world of visualized images from quantum theory, Heisenberg's uncertainty principle or dynamic systems.


The motifs shown form a (small) selection of what appears to appear almost randomly through selected parameters in physical or mathematical systems. However, a lot of attempts are usually necessary to come across really interesting and worth seeing results.


The idea of this WEB page is to make the elegance, grace and charisma tangible and visible to visitors, which is inherent in the magic of butterfly effects and various pattern formation processes - which are actually used for completely different things such as calculations such as turbulence, weather or also for calculation used by economic cycles.


For me as a mathematician, however, it was and is particularly exciting and surprising that the mostly so dry calculations that you have to carry out hundreds of millions of times (of course computer-aided) in order to (hopefully) find satisfactory results that seem like out of nowhere and how by chance such impressive and varied creations can conjure up.


So, I thought that these might also be worth seeing for many other people - and that mathematics and physics also hide a lot of beauty that one usually does not know or suspect. Of course, the beauty of every work of art is always in the eye of the beholder.


I would be happy if you too find one or the other of the resulting designs beautiful!

Our Team

DARIUS

Physics student


supports with ideas+ programming

DAGMAR

Creative Director


Creative Director of sali-math-arts

FRIEDRICH

Mathematician, Designer


mathematics support for design set up

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