Mandelbrot on real numbers (new post)

[Kali]

First of all, I'm so embarrased because I messed up things in a incredible way in my last post.
A mistake in my formula file for UF5 made a graph that wasn't what I wanted, and also I posted a formula that didn't work as I presented it.

So I deleted the previous topic and now I start again...

I was wondering about combining two 1D Mandelbrot sets (the formula applied only to real numbers), in order to obtain a 2D map of values, without using complex numbers at all.

This is the formula I made:

f1=f1^2+x
f2=f2^2+y

Z=abs(f1-f2)

bailout value for Z: 4

So if the difference between the result of the two functions i'm iterating at the same time, becomes larger than 4, the point it's outside the set, and leads to a value for painting the pixel just like the standard mandelbrot.

I use abs function, and not the modulus to obtain the distance of the two points, because I'm measuring the linear distance in the same 1D real line.
Off course the result is a 2D mapping of the values, but the formula itself works in 1D.

Here's the result:


The whole set




The interesting area




Deeper zooms











Did you see that there are 'minisquaredbrots'?

The first thing I tried was adding f1+f2 instead of obtaining it's difference and it works, but I like more the patterns obtained with this method.



Well, I'm not a mathematician, just a very curious guy... but I think there's something here that can tell us more about the m-set.

I really don't know much about mathematical theory of the m-brot, only the basic stuff, so perhaps any of you can tell me more about it.

I'll be waiting for your comments...

P.S.: sorry about my mistake in the other post.

[Kali]

I tested the Julia version and it works great!

initializing values:
f1=x
f2=y
c=(real constant)

and then iterating:
f1=f1^2+c
f2=f2^2+c
z=abs(f1-f2)

The constant c defines the pattern, and the bailout value the density of it.

This is the result for values c=-1.7134806 and bailout=6.5  (I used orbit traps for coloring it)

[Kali]

Ok, I thought it was kind of interesting... perhaps not.

Here it's another pic anyway, with another method.

[sonofthort]

It looks like you might be onto something.  I wonder if anybody else has come across this before. 

Did the previous image use the simple escape time algorithm for coloring?  I think it's cool that it has a 3d look to it.

[Fractal Ken]

Kali, I'm very interested in what you've done. I'm shocked that such a simple method can produce chaotic images. I haven't been able to reproduce your results, but with some experimenting I can make pictures with a fractal appearance.



I intend to examine your excellent idea some more as I have time.

[Fractal Ken]

Kali, I'm getting close to reproducing your first image. I'm using a bailout threshold of 2, rather than 4, on Z = abs(f1 - f2). I'm pretty sure of why I misunderstood: This new criterion is the same as bailing out when Z squared is greater than 4.

[trafassel]

The 3d variant is also interesting

[trafassel]

A zoom into the last one

[msltoe]

Kali,

 By chance, yesterday, I noticed a new formula in Chaospro 4.0 called "Mandelbrotnew" which apparently is the normal Mandelbrot but where one gets to define i^2 as something besides -1. If I set i^2 = 0.5, I get an object very similar to yours except rotated by 45 degrees. So perhaps your formula can be cast as redefined complex multiplication? Very nice.

-mike

[Kali]

It looks like you might be onto something.  I wonder if anybody else has come across this before. 

Did the previous image use the simple escape time algorithm for coloring?  I think it's cool that it has a 3d look to it.

Hi sonofthort, I used basic coloring method in UF5 with setting on "Real part", instead of "Iteration"... However don't know really how it works

[Kali]

Kali, I'm getting close to reproducing your first image. I'm using a bailout threshold of 2, rather than 4, on Z = abs(f1 - f2). I'm pretty sure of why I misunderstood: This new criterion is the same as bailing out when Z squared is greater than 4.

Ken, my mistake (again). I use to evaluate Z in UF5, the syntax |Z|, because if not, the program gives an annoying warning message that "the imaginary part is being ignored". So, as the imaginary part for this formula is always 0, I leave as it is, but I thought that |Z| means the modulus (the modulus should be =Z if the imaginary part is 0), but no, |Z| means "modulus squared", just like you said. Sorry for the inconvenience and thanks for your interest!
I'm not really very experienced in all of this, but since I know about the Mandelbrot set and how's generated, I'm kinda obsessed with learning, researching, and perhaps discovering something new about how it works and the implications of it...

[Kali]

@trafassel, thanks for the 3D images... I was thinking of asking Jesse to make a formula for Mandelbulb 3D to see how it looks...
I'd like to know how you implemented the 3D version of it

@msltoe, interesting... it looks similar to what I get if I evaluate the bailout with the sum of the two functions (I used this method in the last image I posted). Don't know about the implications of this yet... let me do some math about what you are telling.

Regards,

[Kali]

Please note that I answered to all in three different posts
I say this because perhaps you go to the last post I wrote and didn't notice the previous... (I know, I must answer to all in one post)

Now I want to show you something:

I managed to implement in UF5 what Mike said about changing the result of i². First I just made a standard Mandelbrot using real variables, and not the built-in complex variables that UF5 has. Then, in the part where the square of the imaginary part is calculated, I changed the - sign with + (this implies that i²=+1).
I got this:



Nothing but a boring black diamond shape

But with a bailout of 1.5, I got a small area of patterns:



I decided, intuitively, to change the way of evaluating the formula. I put a small bailout value of .1, and evaluated if the modulus were smaller than this.

And then...



A closer look:



The same patterns again. Or at least very similar...

 I will try to make some study of this patterns when I have some more time.

Meanwhile I'd love to hear any idea or conclusion.

[Fractal Ken]

I'm going to take a shot at expanding on Mike's suggestion: presenting general formulas which encompass both the Mandelbrot set and Kali's ideas.

     Iterate (x, y) = (f(x, y), g(x, y)) + c
     Stop if h(x, y) > b

x and y are real variables [(x, y) is complex; typical notation would be z = (x, y)]
f, g, and h are real-valued functions
c is a complex constant [pixel dependent in the Mandelbrot case; pixel independent for Julia's]
b is a real constant [the bailout threshold]

For Kalibrots : f(x, y) = x², g(x, y) = y², and h(x, y) = |x - y|.

For the Mandelbrot set: f(x, y) = x² - y², g(x, y) = 2xy, and h(x, y) = sqrt(x² + y²) [the modulus].
Note: You get f and g by using the definition of complex multiplication to compute z², where z = (x, y).

Please excuse my mistakes and imprecisions; I'm a rusty mathematician trying to balance simplicity against generality.

Ken

[msltoe]

trafassel: the 3-d julia versions might look more ordered than the m-set. Also, the inverse set (like you did with the mandelbox) might show a lattice of blobs.

kali & fractalken: thanks for looking into my suggestion. Whoever invented "Mandelbrotnew" in ChaosPro must've seen this first When people were looking for the 3-D mandelbrot, they explored various definitions of quaternion multiplication, but I don't remember seeing the 2-D version.

-mike