### extending Mandelbox fractals with shape inversions

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#### hobold

• Fractal Phenom
• Posts: 57

#### Re: extending Mandelbox fractals with shape inversions

« Reply #15 on: September 11, 2018, 03:44:55 PM »
Somewhat offtopic, but coincidentally related. This past weekend I saw an exhibition of works from the very beginning of computer generated art (in Bremen, Germany, in the "Kunsthalle").

Most of the pieces were from the late 1960s and early 1970s, and were made with technically primitive tools from today's point of view. Quite a few academics, researchers and engineers authored these early works, simply because pretty much no other playful minds had access to these priceless machines back in the day. Still, there were few academic publications back then. The scientists didn't think their dabbling was science, nor did they think of it as art; and artists were initially not involved with computers.

We (the community of people interested in fractal-related imagery as medium of expression) have progressed a lot further in that regard. There are a few papers of "pure math", there are a few papers from computer science perspective, and there are "engineering" publications about the industrial production of fractal imagery (e.g. for movies). Artists, mathematicians, computer scientists, are still dabbling in this field out of sheer enthusiasm, and thus hobbyists are still free to join even before they acquired expertise in any of the aspects. (And this forum here is maybe the best place for such chance encounters.)

The fascination with fractal imagery has already produced original mathematical research, original computer science research, and it has made big money on the silver screen. Seems the topic already got quite a bit of recognition. Beyond that, what we do is not an exception; a recently coined term is "citizen science". Throughout the history of science there were always "dilletants" who could not publish - the internet merely makes them more visible today than before.

BTW, of the several dozen works being on display in the Kunsthalle Bremen, only a single one is actually fractal based. A dragon curve from 1971, produced by a working group under physics professor Herbert W. Franke (I only knew him as author of hard science fiction and had been unaware of his involvement in the early computer art exploration). We were there right from the start - we just didn't know it yet. :-)

#### utak3r

• Fractal Phenom
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#### Re: extending Mandelbox fractals with shape inversions

« Reply #16 on: September 15, 2018, 11:31:56 PM »
just finding the time to look at improving those interfaces (subject for another thread)

BTW, take a look at Synthclipse, which is a generalized GLSL tool. It has many things similar to Fragmentarium, but what I love here, are animation tools.
Of course, not only that - I do use it, because I use it not only for fractals.

#### utak3r

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#### Re: extending Mandelbox fractals with shape inversions

« Reply #17 on: September 15, 2018, 11:36:14 PM »
templates for your favorite work processor (MS Word or Latex)

Wait... there are people using Word for preparing their papers?
The first time I was writing my own scholary paper, I was trying to use Word for this... yes, it was LONG time ago, but I don't think nowadays Word suddenly became a paper publishing tool   so, yeah, of course LaTeX is the only proper choice

• 3f
• Posts: 1536

#### Re: extending Mandelbox fractals with shape inversions

« Reply #18 on: September 16, 2018, 01:08:28 AM »
Wait... there are people using Word for preparing their papers?
The first time I was writing my own scholary paper, I was trying to use Word for this... yes, it was LONG time ago, but I don't think nowadays Word suddenly became a paper publishing tool   so, yeah, of course LaTeX is the only proper choice
It depends on the journal, but most I've encountered allow Word. All I know about Word is how to open a Word document, so I agree with you on that.

#### utak3r

• Fractal Phenom
• Posts: 58

#### Re: extending Mandelbox fractals with shape inversions

« Reply #19 on: September 16, 2018, 02:00:20 PM »
most I've encountered allow Word.
I don't mean they allow it or not - what I meant is, you can very easily go gray, while preparing your doc in Word    I switched to LaTeX right after my paper was destroyed by Word.

#### CozyG

• Fractal Freshman
• Posts: 7

#### Re: extending Mandelbox fractals with shape inversions

« Reply #20 on: October 04, 2018, 03:16:00 AM »
I've been making some videos using these shape inversion extensions to the Mandelbox. Just posted one on YouTube, a visualization of a 4D Mandelbox where the inversive shape is a linear blend of hypercube and hypersphere. Each frame is a 3D slice through the 4D geometry, with the parameterized fourth dimensional coordinate changing over time. I've been calling these HyperMandalas:

#### Alef

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• Posts: 52
• catalisator of fractals

#### Re: extending Mandelbox fractals with shape inversions

« Reply #21 on: October 04, 2018, 02:09:28 PM »
Offtop.

Realy so called spherefold of mandelbox consist of spherefold and sphere inversion. It's beautifull shapes is mostly a sphere inversion, then less of a boxfold. And it is just slightly modified by spherefold.

If I remember correctly:

//So called spherefold:
if magnitude of z < 0.5:
z := z * 4   /// spherefold
else if magnitude of z < 1:
z := z / (magnitude of z)^2 // sphere inversion!
catalisator of fractals

#### Alef

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• catalisator of fractals

#### Re: extending Mandelbox fractals with shape inversions

« Reply #22 on: October 06, 2018, 02:53:08 PM »
Most of what is decribed in paper I had done in JIT formulas for Mandelbulb3D. Exept in higher dimensions.
It is https://en.wikipedia.org/wiki/Lp_space

Code: [Select]
//===modulus==in==Lp========()==================================/// calculate radius in Lp space where circle is circle (2), square (1) or smartphone shape (4)// Using just int values - faster.intPow:=round(v_Lp_space); if intPow = 1 then  // choice 1. If curvatures is 1 then use fastest formula begin modulus  := abs(x) + abs(y) + abs(z); end   else if intPow = 2 then  // choice2. If curvatures is 2 then use standart formula   begin modulus  := sqrt( x*x + y*y + z*z); end else  //choice 3. else use heavy formulas begin invPower:= 1/ intPow; temp:= intPower(abs(x),intPow) + intPower(abs(y),intPow) + intPower(abs(z),intPow); modulus  := power( temp ,invPower); end;
And use this as sphere inversion radius.
It just = abs(x) + abs(y) + abs(z) or = sqrt( x*x + y*y + z*z);   or = square root of 4( x^4 + y^4 + z^4);

But sphere inversion in 4D hypersphere must be different and 4D hypercube inversion alsou is something different. But then it could be done with hypersupersphere.

But radius can be calculated with different power in each direction. I used integer powers only to speed up calculation:

Code: [Select]
intpowerX:=round(sCurvatureX);intpowerY:=round(sCurvatureY);intpowerZ:=round(sCurvatureZ);//inverse of geometric mean, calculates necesary root value invpower:= power(intPowerX *intPowerY *intPowerZ,-0.33333333333333333333); temp:=  intPower( abs(x),intPowerX) + intPower( abs(y),intPowerY) +intPower( abs(z),intPowerZ); radius  := power( temp ,invpower);
I alredy posted this in fractalforums.com. Here it have different p powers in each axis. 1,2,2 Stretched eye corners are where p=1.

Sorry if I 'm repeating myself.

Alsou boxfold can be made with different lenght in each dimension - more control over its gabarites and if FOldZ=1 a possibility to make something like Amazing Surface formula.
« Last Edit: October 08, 2018, 02:00:25 PM by Alef »

#### DarkBeam

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• Formula crazy engineer
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• 2 plus 2 is 4 minus 1 is 3 quick maffs

#### Re: extending Mandelbox fractals with shape inversions

« Reply #23 on: October 24, 2018, 01:02:37 AM »
Seven years  ago I created ABoxVSShapes for Mandelbulb3D...
http://www.fractalforums.com/mandelbulb-3d/custom-formulas-and-transforms-release-t17106/
it cannot be considered your invention.The concept of tweaking inversion shape was shown to me by Aexion I think

#### CozyG

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#### Re: extending Mandelbox fractals with shape inversions

« Reply #24 on: October 24, 2018, 04:10:16 AM »
Seven years  ago I created ABoxVSShapes for Mandelbulb3D...
http://www.fractalforums.com/mandelbulb-3d/custom-formulas-and-transforms-release-t17106/
it cannot be considered your invention.The concept of tweaking inversion shape was shown to me by Aexion I think

I'm certainly not claiming I was the first to come up with the idea of modifying the "SphereFold" part of the Mandelbox formula. Nor the idea of "shape inversion" in general (see my paper for references on earlier work by others, and I can list more that couldn't fit into the paper if anyone is interested). But I do think that the idea of replacing spherical inversion in the SphereFold with shape inversion is new. I could be mistaken, but I pored over a lot of code from Fragmentarium, Mandelbulber, and MB3D, and a lot of old fractalforums posts before coming to the conclusion that this hadn't been done before. I posted some of my initial work on this last year on the old fractalforums site: Generalized shape inversion applied to the Mandelbox, asked if anyone knew of similar pre-existing modifications, and mentioned MB3D's ABoxVaryShapes formula(s) as the closest I had found but pointed out what I think is the difference:

Quote from: CozyG
I think this may be a new technique to add to the Mandelboxen toolkit. But it's also possible that I've been exploring ground that's been covered before. I've only been deep diving into the Mandelbox for the last few months, so I've been playing catchup on many years of great brainstorming sessions on the forum! And looking at lots of Fragmentarium fragments and MB3D formulas for similar ideas. And there is a set of formulas, probably best represented by MB3D's ABoxVaryShapes, that on first glance may seem similar. However, as best I can tell these all modify the calculated distance from the origin of inversion to the iterated point --
what I've labelled above as d(OP). Whereas the shape inversion that I'm implementing modifies the distance from the origin of inversion (and following the ray OP) to the boundary of the inversion -- d(OS). Does anyone know of similar techniques applied to the Mandelbox?

The two techniques (modifying distance to iterated point, and modifying distance to shape boundary) can also be combined, one modifying the numerator and one modifying the denominator of the inversion scaling. This has been done in other contexts that use spherical inversion, for example see Pseudoinversion Fractals (Gdawiec, 2016) or the Inversion variation that I added to JWildfire last year. I will try to implement this technique combo for a Mandelbox formula in Fragmentarium and post again soon, to illustrate similarities and differences in code and images.

#### lkmitch

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#### Re: extending Mandelbox fractals with shape inversions

« Reply #25 on: October 25, 2018, 05:00:22 PM »
The Bridges conference, to which the original paper was submitted, is a very nice mixture of accessible and rigorous. Anyone can submit a paper and the formatting guidelines are provided in Word and Latex. The audience is wide, from mathematicians to artists (of all stripes), so a wealth of technical knowledge is not assumed. In fact, overly technical papers are sometimes rejected for not being sufficiently accessible. But all submissions are peer reviewed, leading to (if not guaranteeing) high-quality work. I encourage anyone with an interest in math & art and with something interesting to say to consider submitting a paper.

#### Softology

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#### Re: extending Mandelbox fractals with shape inversions

« Reply #26 on: November 29, 2018, 10:32:21 PM »
Where must you be to access those earlier papers without paying through the nose?

Sci-Hub
https://en.wikipedia.org/wiki/Sci-Hub

• Fractal Phenom
• Posts: 56

#### Re: extending Mandelbox fractals with shape inversions

« Reply #27 on: November 30, 2018, 12:11:14 AM »
Quote
I wonder about all that amazing work from previous years -- do any of you involved back then ever think nowadays about publishing some of your work as a more traditional journal article or conference paper? There is still very little published (in the traditional peer-reviewed sense) about the Mandelbox and other fractal graphics developments that have first appeared in fractalforums.

This might not be a published paper but it is a very well formalised analysis of the Mandelbox, by Rudy Chen at the University of Waterloo: http://digitalfreepen.com/mandelbox370/

I published one paper about using fractal dimensions in image enhancement: http://digital-library.theiet.org/content/journals/10.1049/iet-ipr.2014.0642 (paywall)

A lot of people who don't have the time to push through a peer-reviewed journal (it is quite an effort) use arxiv or researchgate. Here's one I did on variable dimension surfaces:
https://www.researchgate.net/publication/309391846_Three_Variable_Dimension_Surfaces

I agree that publishing is a worthwhile exercise, if nothing else because no-one is going to cite you and extend research on your topic if they can't cite a paper (hopefully this will change).

On the other hand I agree with Softology and others, why should I spend months pushing through the slow publication process only for it to be behind a paywall. More than this, forums and websites allow loads of images, videos, interactive webGL examples etc. I do find most journals to be really behind the times... some still don't even allow colour images!

• 3f
• Posts: 1536

#### Re: extending Mandelbox fractals with shape inversions

« Reply #28 on: November 30, 2018, 01:59:00 AM »
This might not be a published paper but it is a very well formalised analysis of the Mandelbox, by Rudy Chen at the University of Waterloo: http://digitalfreepen.com/mandelbox370/
Thanks for that link. I've been searching occasionally for something like that about the M-box for years. Search engines are not everything...

#### mclarekin

• Fractal Fluff
• Posts: 378

#### Re: extending Mandelbox fractals with shape inversions

« Reply #29 on: November 30, 2018, 09:47:25 AM »
Quote
This might not be a published paper but it is a very well formalised analysis of the Mandelbox, by Rudy Chen at the University of Waterloo: http://digitalfreepen.com/mandelbox370/

that is good, ( i skipped the maths, far too hard for this old man), the paper explains the concepts very well.

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