A certain type of system has progressive states as represented by
points on the XY coordinate plane. If totally random, then points
would range evenly over the plane, but these systems have strong
attractors that are often beautiful and /or interesting.

Check out a sample of 12 from a library of 512 :
https://postimg.cc/gallery/JB8TtTj

Note capture of special Windows program on system 93 :
https://postimg.cc/F7pdpQd1

Email casagiannoni@optimum.net to get the free special Windows
program.

* * * DETAILED DESCRIPTIVE PAPER * * *

SA18 Chaos Engine

( 1 ) Preliminary Notes :

SA18 software includes, the program executable SA18 Chaos Engine.exe,
and the library binary data file sa18lib.bin containing the required
information library where 88 bytes are used to store the unique
generating, locating and scaling information for each of the many
semi-chaotic systems. Also required are the five Visual Basic v6
auxiliary run time files, Asycfilt.dll, Msvbvm.dll, Oleaut32.dll,
Olepro32.dll and Stdole2.tlb . These five must be either with the
executable, or more properly at C:\Windows\System** with other DLLs.

The SA18 Chaos Engine is a 32 bit Windows application requiring an
appropriate version of Windows. Use a display at the maximum normal
resolution. Start the program SA18 Chaos Engine.exe by any of the
usual methods , e.g. - double clicking SA18 Chaos Engine.exe in the
Windows Explorer or permanently installing a shortcut with the program
icon (recommended).

The software is provided "as is" for free distribution, without any
warranty or condition of any kind, express or implied, and with the
firm understanding that the user assumes all responsibility for any
consequences of the use of the software.

( 2 ) Introduction and Background :

The Chaos Engine, has evolved from a study of a unique form of
mathematically defined systems of semi-chaos. Each state of these
systems is represented by a point on the XY coordinate plane.
Subsequent states or points, are mapped via application of 18 ordered
coefficients from two 9 element, 3x3 matrices, Aij and Bij, specific
to each unique system, according to the following algorithms
recursively :

X new ( X, Y ) = A00 + A01 Y + A02 Y^2
+ A10 X + A11 X Y + A12 X Y^2
+ A20 X^2 + A21 X^2 Y + A22 X ^2 Y^2

Y new ( X, Y ) = B00 + B01 Y + B02 Y^2
+ B10 X + B11 X Y + B12 X Y^2
+ B20 X^2 + B21 X^2 Y + B22 X^2 Y^2

Matrix coefficients are additively applied to every possible product
combination of the current X and Y state coordinates in powers 0, 1
and 2, thus defining each subsequent system state.

It was discovered that if the 18 matrix coefficients were chosen at
random from an interval a bit wider than -1 to +1, then about one in
every several hundred so defined systems would exhibit behavior that
was stable or bounded, non-degenerative and non-periodic. This semi
chaotic behavior would result in evolving points for each subsequent
state of the system, defining a progressive image where locations in
the image were clearly attractive of most system states ( i.e. - the
system, though fundamentally chaotic in nature, nevertheless "prefers"
certain states of attraction). Visually, it was observed that these
attractors tended to have pleasing and interesting qualities,
especially if the spectral colors are used to indicate orbital
accelerations in various image areas.

A computer developed a library of images by the random process
selection of sets of matrix coefficients and rejecting systems that
lacked the desired weak chaotic behavior. Each acceptable system was
stored as the 2 x (3 x 3) = 18 matrix coefficients together with 2
locating and 2 scaling parameters, requiring 88 bytes for each image
in the library file sa18lib.bin of images. The unique matrices can be
thought of as a kind of mathematical code for the corresponding
attractor images. The Chaos Engine enables the user to view the 2 x
(3x3) = 18 matrix coefficients while the image is evolving, and allows
for the dynamic "tweaking" of any selected coefficient and the
observed effect on the dynamic image. Given even the crude precision
of the chaos engine tweaking tools, there is still likely estimated to
be a vast number indeed of different "viable" possible images !

( 3 ) About the Colors :

The color assigned to pixel points representing each system state, is
keyed to the acceleration at that point in the progressive development
of the attractor. It is the magnitude of the change in vector
displacements, between the vector of the preceding point to the
current point, and the vector from the current point to the subsequent
point. In a qualitative sense, it is the magnitude of the "jerk" felt
at each point if one was "riding" the points around the developing
image. Normal Spectral colors are used from Blue representing minimal
accelerations, increasing through Cyan, Green, Yellow, and up to Red
representing maximum accelerations. Excursions beyond either extreme
are represented by a progression to Magenta. The program samples the
early development of system states to define a mean and standard
deviation of accelerations. Normalized scaling from full Magenta below
Blue up to full Magenta above Red is indicative of from -2 to +2
standard deviations.

( 4 ) Periodic Random Orbit Perturbation :

On occasion, an otherwise well behaved attractor will suddenly fall
into a repeating sequence, sometimes only involving a limited number
of system states. Image number 275 from the original library is a good
example. Click the developed image to see the limited states. The
cause of this periodic degeneracy is not well understood, but the
round off error of the floating point math describing the system
states does impose a finite limit to the possible number of system
states within the domain of each attractor, and periodic degeneracy
can be the ultimate consequence. If the attractor is especially
"tight", as indeed is the case in some of the more interesting and
beautiful figures, then this periodic degeneracy can sometimes
overtake the attractor causing further development to cease. To offset
this tendency, code has been introduced to periodically perturb a
point (1 every 2^15 = 32768 points) in both the X and Y directions, by
random amounts selected from the interval form -.0025 to +.0025. This
is often just what such a figure needs to keep moving. This feature is
selectable in the chaos engine (click the label : ON shown green, or
OFF shown red).

( 5 ) Using the Chaos Engine :

On starting the Chaos Engine a semi-chaotic system is selected at
random from the library ( currently 512 ) and the attractor displayed.
The sizing and positioning buttons [Bigger], [Smaller], [Taller],
[Wider], [Up], [Down], [Left], [Right] all do what they say when
clicked. Left and Right Clicking are for Large and Small adjustments
respectively. [Taller] / [Wider] change the aspect ratio of the image
without changing the overall size. All of these controls do nothing to
the character of the images.

Images are selected from the library using the vertical scroll slider
and the selected image number is indicated above the top end of the
slider. Any of the 18 matrix coefficients as Aij (left) and Bij
(right) displayed at the top may be selected for "tweaking" by left
clicking the number. The selected coefficient will appear in a
different color than the rest. The coefficient will be rounded off to
six decimal places when tweaked up or down using the [Add] or [Sub.]
buttons respectively.

Six levels of additive or subtractive adjustments are possible
according to the following table :

Action Added or Subtracted Amount

Left Click 0.1
Right Click 0.01
Shift - Left Click 0.001
Shift - Right Click 0.0001
Ctrl - Left Click 0.00001
Ctrl - Right Click 0.000001

Immediately on tweaking a coefficient, the image clears and redraws
using the altered coefficient, allowing the user to observe the effect
on the image. On occasion, the tweaked coefficient will render the
system unstable or unbounded and the green "OK" indicator will
intermittently or continuously change to a red "OUT !". At this point
the user can recover to the previous stable state by reversing the
offending action using the [Add] or [Sub.] buttons appropriately. In
any case, clicking the image number will return all coefficients to
the library values and is therefore a sure way to recover.

An altered image can be stored, replacing the starting image in the
library by holding both the Ctrl and Shift keys while clicking [Save]
(wait for beep). All previous points in a developing image can be
deleted by clicking either the "OK" indicator or the display area.
This is often a good way to detect the previously mentioned periodic
degeneracy.

Exit the program by clicking [Exit].

Free Windows software on request.

Stephen G. Giannoni PE (retired)

casagiannoni@optonline.net

USA (631) 757-2793

What does it mean that the Mandelbrot set has to leave out certain overshooting quantities?
You can overshoot in the subtraction operations at their absolute no quantity limit.

Mitchell Raemsch

On 1/8/2022 3:31 PM, mitchr...@gmail.com wrote:
> What does it mean that the Mandelbrot set has to leave out certain overshooting quantities?

Afaict, the OP is showing strange attractors. Wrt overshooting in the
Mandelbrot set... Well, have you ever computed a Mandelbrot set in your
life? Think of escape time... For a moment of time... ;^) HINT, HINT!

> You can overshoot in the subtraction operations at their absolute no quantity limit.

Again, think along the lines of escape time... ;^)

Funny we have Mitch who denies the existence of negative and imaginary
numbers, but knows what the Mandelbrot Set is...or does he?

On Saturday, January 8, 2022 at 4:27:55 PM UTC-8, Michael Moroney wrote:
> Funny we have Mitch who denies the existence of negative and imaginary
> numbers, but knows what the Mandelbrot Set is...or does he?

Zero math wins... yours is just imaginary...

Mitchell Raemsch

On Saturday, January 8, 2022 at 4:03:51 PM UTC-8, Chris M. Thomasson wrote:
> On 1/8/2022 3:31 PM, mitchr...@gmail.com wrote:
> > What does it mean that the Mandelbrot set has to leave out certain overshooting quantities?
> Afaict, the OP is showing strange attractors. Wrt overshooting in the
> Mandelbrot set... Well, have you ever computed a Mandelbrot set in your
> life? Think of escape time... For a moment of time... ;^) HINT, HINT!
> > You can overshoot in the subtraction operations at their absolute no quantity limit.
> Again, think along the lines of escape time... ;^)

There is absolute zero... with nothing below it...

On 1/9/2022 1:11 PM, mitchr...@gmail.com wrote:
> On Saturday, January 8, 2022 at 4:27:55 PM UTC-8, Michael Moroney wrote:
>> Funny we have Mitch who denies the existence of negative and imaginary
>> numbers, but knows what the Mandelbrot Set is...or does he?
>
> Zero math wins... yours is just imaginary...
>
Explain the Mandelbrot Set without any references to either imaginary or
negative numbers.

You may start with the way it's plotted.

Michael Moroney wrote:

> On 1/9/2022 1:11 PM, mitchr...@gmail.com wrote:
>> On Saturday, January 8, 2022 at 4:27:55 PM UTC-8, Michael Moroney
>> wrote:
>>> Funny we have Mitch who denies the existence of negative and imaginary
>>> numbers, but knows what the Mandelbrot Set is...or does he?
>>
>> Zero math wins... yours is just imaginary...
>>
> Explain the Mandelbrot Set without any references to either imaginary or
> negative numbers. You may start with the way it's plotted.

nothing imaginary plotting numbers in computers, all real. And the
negative are the registered data you subtract from the present, the
difference. But without a buffer, here memory, there are no negative
numbers, nor differences. You think like an ass.

On 1/9/2022 12:24 PM, Cody Alba wrote:
> Michael Moroney wrote:
>
>> On 1/9/2022 1:11 PM, mitchr...@gmail.com wrote:
>>> On Saturday, January 8, 2022 at 4:27:55 PM UTC-8, Michael Moroney
>>> wrote:
>>>> Funny we have Mitch who denies the existence of negative and imaginary
>>>> numbers, but knows what the Mandelbrot Set is...or does he?
>>>
>>> Zero math wins... yours is just imaginary...
>>>
>> Explain the Mandelbrot Set without any references to either imaginary or
>> negative numbers. You may start with the way it's plotted.
>
> nothing imaginary plotting numbers in computers, all real. And the
> negative are the registered data you subtract from the present, the
> difference. But without a buffer, here memory, there are no negative
> numbers, nor differences. You think like an ass.

Huh? What the heck are you talking about?

On 1/9/2022 11:49 AM, Michael Moroney wrote:
> On 1/9/2022 1:11 PM, mitchr...@gmail.com wrote:
>> On Saturday, January 8, 2022 at 4:27:55 PM UTC-8, Michael Moroney wrote:
>>> Funny we have Mitch who denies the existence of negative and imaginary
>>> numbers, but knows what the Mandelbrot Set is...or does he?
>>
>> Zero math wins... yours is just imaginary...
>>
> Explain the Mandelbrot Set without any references to either imaginary or
> negative numbers.
>
> You may start with the way it's plotted.

I have a feeling its going to be a very cold day in hell before he can
explain any aspect of the Mandelbrot set.

Chris M. Thomasson wrote:

>>>> Zero math wins... yours is just imaginary...
>>>>
>>> Explain the Mandelbrot Set without any references to either imaginary
>>> or negative numbers. You may start with the way it's plotted.
>>
>> nothing imaginary plotting numbers in computers, all real. And the
>> negative are the registered data you subtract from the present, the
>> difference. But without a buffer, here memory, there are no negative
>> numbers, nor differences. You think like an ass.
>
> Huh? What the heck are you talking about?

I think I talk with two braindead imbeciles, here Kibo Parry Morony and
Chris M. Thomasson.

On 1/9/2022 1:43 PM, Cody Alba wrote:
> Chris M. Thomasson wrote:
>
>>>>> Zero math wins... yours is just imaginary...
>>>>>
>>>> Explain the Mandelbrot Set without any references to either imaginary
>>>> or negative numbers. You may start with the way it's plotted.
>>>
>>> nothing imaginary plotting numbers in computers, all real. And the
>>> negative are the registered data you subtract from the present, the
>>> difference. But without a buffer, here memory, there are no negative
>>> numbers, nor differences. You think like an ass.
>>
>> Huh? What the heck are you talking about?
>
> I think I talk with two braindead imbeciles, here Kibo Parry Morony and
> Chris M. Thomasson.

You mention buffer, memory, there are no negative numbers, ect... What
are you talking about? The question was to explain the Mandelbrot set.
Got it?

Chris M. Thomasson wrote:

>>>>> Explain the Mandelbrot Set without any references to either
>>>>> imaginary or negative numbers. You may start with the way it's
>>>>> plotted.
>>>>
>>>> nothing imaginary plotting numbers in computers, all real. And the
>>>> negative are the registered data you subtract from the present, the
>>>> difference. But without a buffer, here memory, there are no negative
>>>> numbers, nor differences. You think like an ass.
>>>
>>> Huh? What the heck are you talking about?
>>
>> I think I talk with two braindead imbeciles, here Kibo Parry Morony and
>> Chris M. Thomasson.
>
> You mention buffer, memory, there are no negative numbers, ect... What
> are you talking about? The question was to explain the Mandelbrot set.
> Got it?

was about negative numbers and imaginary in computers. There is nothing
about that in 0s and 1s in computers. Just like moving a zero on scale,
hence interpretation. Go read about computers. You guys are idiots.

On 1/9/2022 2:04 PM, Cody Alba wrote:
> Chris M. Thomasson wrote:
>
>>>>>> Explain the Mandelbrot Set without any references to either
>>>>>> imaginary or negative numbers. You may start with the way it's
>>>>>> plotted.
>>>>>
>>>>> nothing imaginary plotting numbers in computers, all real. And the
>>>>> negative are the registered data you subtract from the present, the
>>>>> difference. But without a buffer, here memory, there are no negative
>>>>> numbers, nor differences. You think like an ass.
>>>>
>>>> Huh? What the heck are you talking about?
>>>
>>> I think I talk with two braindead imbeciles, here Kibo Parry Morony and
>>> Chris M. Thomasson.
>>
>> You mention buffer, memory, there are no negative numbers, ect... What
>> are you talking about? The question was to explain the Mandelbrot set.
>> Got it?
>
> was about negative numbers and imaginary in computers. There is nothing
> about that in 0s and 1s in computers. Just like moving a zero on scale,
> hence interpretation. Go read about computers. You guys are idiots.

Humm... I am a programmer, c/c++ mostly... By the way, you are
completely missing the point. How can you create a source/sink vector
field without using negative numbers?

https://youtu.be/1SXnsexrHvA

http://fractallife247.com/test/webgl/

;^)

On 1/9/2022 2:04 PM, Cody Alba wrote:
> Chris M. Thomasson wrote:
>
>>>>>> Explain the Mandelbrot Set without any references to either
>>>>>> imaginary or negative numbers. You may start with the way it's
>>>>>> plotted.
>>>>>
>>>>> nothing imaginary plotting numbers in computers, all real. And the
>>>>> negative are the registered data you subtract from the present, the
>>>>> difference. But without a buffer, here memory, there are no negative
>>>>> numbers, nor differences. You think like an ass.
>>>>
>>>> Huh? What the heck are you talking about?
>>>
>>> I think I talk with two braindead imbeciles, here Kibo Parry Morony and
>>> Chris M. Thomasson.
>>
>> You mention buffer, memory, there are no negative numbers, ect... What
>> are you talking about? The question was to explain the Mandelbrot set.
>> Got it?
>
> was about negative numbers and imaginary in computers. There is nothing
> about that in 0s and 1s in computers. Just like moving a zero on scale,
> hence interpretation. Go read about computers. You guys are idiots.

lol. Of course you know that negative real numbers are easily
represented in binary, wrt, say IEEE 754. Binary is just an encoding
technique. I can encode in quinary, so what. You are missing the point.

The question was about plotting the Mandelbrot set.

Chris M. Thomasson wrote:

>>>> I think I talk with two braindead imbeciles, here Kibo Parry Morony
>>>> and Chris M. Thomasson.
>>>
>>> You mention buffer, memory, there are no negative numbers, ect... What
>>> are you talking about? The question was to explain the Mandelbrot set.
>>> Got it?
>>
>> was about negative numbers and imaginary in computers. There is nothing
>> about that in 0s and 1s in computers. Just like moving a zero on scale,
>> hence interpretation. Go read about computers. You guys are idiots.
>
> lol. Of course you know that negative real numbers are easily
> represented in binary, wrt, say IEEE 754. Binary is just an encoding
> technique. I can encode in quinary, so what. You are missing the point.
> The question was about plotting the Mandelbrot set.

yes, you fucking imbecile, I was saying it above. There are no negative
or imaginary numbers in computers, only higher level of interpretation. Ie
the MSB, most significant bit, a 1, gives you a "negative" number,
encoded i 1s and 0s. Fucking stupid. The question is about the existence
of negative numbers in physics, here computers. How can you be this
stupid, you idiot.

On 1/9/2022 5:24 PM, Chris M. Thomasson wrote:

> Huh? What the heck are you talking about?

You got suckered by the nymshifting troll. Again.
It's only purpose in life is to mess up usenet discussions and create noise.

Chris M. Thomasson wrote:

>>>> Chris M. Thomasson.
>>>
>>> You mention buffer, memory, there are no negative numbers, ect... What
>>> are you talking about? The question was to explain the Mandelbrot set.
>>> Got it?
>>
>> was about negative numbers and imaginary in computers. There is nothing
>> about that in 0s and 1s in computers. Just like moving a zero on scale,
>> hence interpretation. Go read about computers. You guys are idiots.
>
> Humm... I am a programmer, c/c++ mostly... By the way, you are
> completely missing the point. How can you create a source/sink vector
> field without using negative numbers?

you are an imbecile, not a programmer. The higher level crap you use, in
"programming" is hiding the stuff you actually are doing. You are an
idiot.

On 1/9/2022 2:35 PM, Cody Alba wrote:
> Chris M. Thomasson wrote:
>
>>>>> I think I talk with two braindead imbeciles, here Kibo Parry Morony
>>>>> and Chris M. Thomasson.
>>>>
>>>> You mention buffer, memory, there are no negative numbers, ect... What
>>>> are you talking about? The question was to explain the Mandelbrot set.
>>>> Got it?
>>>
>>> was about negative numbers and imaginary in computers. There is nothing
>>> about that in 0s and 1s in computers. Just like moving a zero on scale,
>>> hence interpretation. Go read about computers. You guys are idiots.
>>
>> lol. Of course you know that negative real numbers are easily
>> represented in binary, wrt, say IEEE 754. Binary is just an encoding
>> technique. I can encode in quinary, so what. You are missing the point.
>> The question was about plotting the Mandelbrot set.
>
> yes, you fucking imbecile, I was saying it above. There are no negative
> or imaginary numbers in computers, only higher level of interpretation. Ie
> the MSB, most significant bit, a 1, gives you a "negative" number,
> encoded i 1s and 0s. Fucking stupid. The question is about the existence
> of negative numbers in physics, here computers. How can you be this
> stupid, you idiot.

You are missing still missing the whole point! Wow. Negative numbers
exist, and can be encoded in a lot of different schemes, binary is one
of them. Still, I don't think you can explain how to create a Mandelbrot
set or a source/sink vector field. Remember, damn it, this was the main
question! I don't even think you could create a strange attractor, let
alone explain how to create one. You are great at missing the main
point. Binary, blah blah.

Michael Moroney wrote:

> On 1/9/2022 5:24 PM, Chris M. Thomasson wrote:
>
>> Huh? What the heck are you talking about?
>
> You got suckered by the nymshifting troll. Again.
> It's only purpose in life is to mess up usenet discussions and create
> noise.

not sure. As said, you are both brainless uneducated idiots, not knowing
what you do and what you say. Are you implying "plotting" without
computers, fucking stupid. What an idiot.

On 1/9/2022 2:36 PM, Michael Moroney wrote:
> On 1/9/2022 5:24 PM, Chris M. Thomasson wrote:
>
>> Huh? What the heck are you talking about?
>
> You got suckered by the nymshifting troll.  Again.
> It's only purpose in life is to mess up usenet discussions and create
> noise.

Yikes! Damn. Need to work on that. This nymshift seems to be dumber than
the others. Humm...

Chris M. Thomasson wrote:

>>> lol. Of course you know that negative real numbers are easily
>>> represented in binary, wrt, say IEEE 754. Binary is just an encoding
>>> technique. I can encode in quinary, so what. You are missing the
>>> point.
>>> The question was about plotting the Mandelbrot set.
>>
>> yes, you fucking imbecile, I was saying it above. There are no negative
>> or imaginary numbers in computers, only higher level of interpretation.
>> Ie the MSB, most significant bit, a 1, gives you a "negative" number,
>> encoded i 1s and 0s. Fucking stupid. The question is about the
>> existence of negative numbers in physics, here computers. How can you
>> be this stupid, you idiot.
>
> You are missing still missing the whole point! Wow. Negative numbers
> exist, and can be encoded in a lot of different schemes, binary is one
> of

not in physics, here computers, idiot. You encode them, which related to
moving a zero on a scale. But there are negative numbers in this
universe. What a fucking idiot we have here, ladies and gentlemen.

Cody Alba wrote:

>>> yes, you fucking imbecile, I was saying it above. There are no
>>> negative or imaginary numbers in computers, only higher level of
>>> interpretation.
>>> Ie the MSB, most significant bit, a 1, gives you a "negative" number,
>>> encoded i 1s and 0s. Fucking stupid. The question is about the
>>> existence of negative numbers in physics, here computers. How can you
>>> be this stupid, you idiot.
>>
>> You are missing still missing the whole point! Wow. Negative numbers
>> exist, and can be encoded in a lot of different schemes, binary is one
>> of
>
> not in physics, here computers, idiot. You encode them, which related to
> moving a zero on a scale. But there are negative numbers in this
> universe. What a fucking idiot we have here, ladies and gentlemen.

"But there are *NO* negative numbers in this universe"

On 1/9/2022 2:43 PM, Cody Alba wrote:
> Chris M. Thomasson wrote:
>
>>>> lol. Of course you know that negative real numbers are easily
>>>> represented in binary, wrt, say IEEE 754. Binary is just an encoding
>>>> technique. I can encode in quinary, so what. You are missing the
>>>> point.
>>>> The question was about plotting the Mandelbrot set.
>>>
>>> yes, you fucking imbecile, I was saying it above. There are no negative
>>> or imaginary numbers in computers, only higher level of interpretation.
>>> Ie the MSB, most significant bit, a 1, gives you a "negative" number,
>>> encoded i 1s and 0s. Fucking stupid. The question is about the
>>> existence of negative numbers in physics, here computers. How can you
>>> be this stupid, you idiot.
>>
>> You are missing still missing the whole point! Wow. Negative numbers
>> exist, and can be encoded in a lot of different schemes, binary is one
>> of
>
> not in physics, here computers, idiot. You encode them, which related to
> moving a zero on a scale. But there are negative numbers in this
> universe. What a fucking idiot we have here, ladies and gentlemen.

Of course there are negative numbers, computers even need to be able to
encode them. Blah! You cannot answer the main question, and basically
hijacked the thread. Go away. Plonk.

Chris M. Thomasson wrote:

> On 1/9/2022 2:36 PM, Michael Moroney wrote:
>> On 1/9/2022 5:24 PM, Chris M. Thomasson wrote:
>>
>>> Huh? What the heck are you talking about?
>>
>> You got suckered by the nymshifting troll.  Again.
>> It's only purpose in life is to mess up usenet discussions and create
>> noise.
>
> Yikes! Damn. Need to work on that. This nymshift seems to be dumber than
> the others. Humm...

the dumbest uneducated fucks are you both, as proved above, of which you
are in a hurry to delete. You disgusting subhuman excrement.