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I start this thread to create a point of reference on the subject, because I think is interesting enough, time will tell.
     As I have been trying to show in different threads, recursive processes, by themselves, produce   accidental results at best, or mathematically sterile.
     There is something in nature that directs the processes to a particular order, as we can see in the picture, that something is the gnomon.
     In my view, we need a programmable gnomon that can be applied to fractal generating programs.
     If anyone wants to try, or just discuss it, this is the place.
   

At first glance, it seems odd that people generally do not value or appreciate many beautiful images generated fractally .
For those who do not understand this issue, it is a matter of nerds and actually is something that computers do , and for those who understand , it is clear that it is mathematically very simple, and artistically worthless .
    For a work of art is valuable must be unique , something done by thousands, is not art .
    Because for something to be art, an artist must do it.
    And you must have a content , an idea, a meaning, without it , there can be no art .
    And we all agree that penetrate to infinity in an M -set does not bring any meaning , because , as above so below , if at first it does not mean anything. the same at the end.
    The competition this year, we have seen several ways to create an axolot by fractal media, all valid, of course, I want to note is that each creator had to do their part , what programs do not reach to do , namely to introduce a specific purpose in the process .
    What is at issue, then, is to increase the control of the artist on the generation, allowing you to introduce your ideas to direct fractalisation .
    I have no idea how to implement this, but I can not believe that there is not an interesting challenge for matemathics  .
 This is the tool true artists will need if fractals  have to be artworks.

First of all: I find it a great idea to elaborate on the topic of gnomons that you brought up in other posts.
Very intersting and a crucial element of fractals, i think. It's the first time I heard a proper desciption of the phenomenon of recurring shapes in fractals like the seahorse or the 2-3-4-.. folded spirals.., the minibrots themselves
this seems very important.
and (as so often with such things) I'm surprised it hasn't been seen/discussed for such a long time.

but I need to make a little detour because of some of your statements:
until here, well said!

From here I disagree.
Not because I'm 'insulted' as an 'artist'. I see myself as a novice explorer. I find stuff that is already there.
The actual artist probably is the formula. Nature. Everything around us.
Some might call it god (I don't)

I think to bring out art, to show it, craftmanship is needed.
Take a stone block.
Every thinkable form is already in there. The artist 'just' has to set it free. Is Michelangelos David not art? The form (human body) existed long before.
Yet we clearly classifie this 'simple, accurate depiction of a man' as art.

ok, for fractals, the entry point is much easier, as all the beauty is obviously already there, calculated by some 'mindless' programm, a simple formula...
but still, it takes time and craftmanship to really show all the beauty that it contains in places that need to be discovered.

the images of paudelbrot are a perfect example. they are not simple snapshots anyone can do.
i tried.
i can't..

I'm a professional musician.
When I make music it often just happens. It's a flow. It starts with a small idea, the flow takes over and a few hours later I 'wake up' and the layout is done.
Sometimes when a song is finished and I hear it some months later I wonder if it was really me who did this.
It's like my concious mind wasn't involved, except for working out the details. Besides playing a minor and G major on a guitar. ;)
Maybe this is similar to zooming into the m-set.
I might have had a short musical idea "that sounds good" in the beginning (like "I'll zoom here in the m-set that looks interesting") but when the result is about to become really good, 'subconciousness' takes over. whatever that is..
the melody has already been there, in theory.

the picture also is already there. but hidden.


All this reminds me on the quote "We're all standing on the shoulders of giants" (by whom I forgot).



So now that I've got this off my mind I can proceed to your core idea:
To this I can strongly say YES! I agree!
I approached this from a different side, biomimicry-style.
I was wondering, do we know the exact formula for romanesco brocolli, or for mushroom gills ...natural fractals.
and could we invent new ones?!
this is where it becomes incredible interesting!
I think jesse did this on a large scale. and others did it(and keep doing it) with contributing new formulas.
combining these formulas into new stufff

I can only say, I wish I could do that(besides random strikes of luck).
I'd love to see an inspiring shape in nature, find the formula, vary it, explore it, do "my own" little evolution and then finalize it with a little personal touch.

or even come up with totally new formulas...


after all.. if new stuff comes up, I'm pretty shure we're at the best place to find out first.. ;)


------
sorry& back to gnomons!  ;)


edit: to avoid anymore offtopic, regarding great krams video in the following post:
vsauce, a great science channel also has a video about this:
youtube.com/watch?v=DAcjV60RnRw
"will we ever run out of new music"

thanks kram for your video! i never got into 12tone music, it's too far out for me.. but her little example at 12:30.. interesting, and more important, one can listen to it, without aching ears ;)
I should give that music a second chance.. thanks! :)

edit 2: wow! i love that sentence 14:50
"creative people are just skilled at navigating an exponential tree of possibilities..."

I don't intend to keep up the diversion for longer than this but I want to add to Chillheimer's analysis of art this video, or at least a part of it:
http://youtu.be/4niz8TfY794?t=10m8s
From 10:08 onward but then especially from 13:39 to 15:30.
There are more sections relevant to this in that video. If you have time, watch the whole thing: It's amazing!

To start somewhere, I will try to define the type of interface.
   Basically, what is needed is something like the image.
   Above, we see the appearance of the gnomon in two configurations.
   Below, some of the changes that you need to do, namely, to transform a single line and let it spread its influence, or transform a number independently.
   Naturally, the ultimate goal is a gnomon in 3D, but we must start in 2D, I think.

 First of all, I agree with your latest sentence, gnomons are found in egyptian tombs.
  I must admit I´ve been a bit rude  with the old good M-set, but I wanted to make a clear cut.
  While I agree that the creative process overcome us and flows trough us , the fact remains that only true artists can reach this level, and only trough long effort and endurance.
   The beautiful images found in fractal programs can only be attributed to nature, I agree, the autor is nature and they are beautiful artforms, but you can say the same about everything around us, you still need an artist to create art.
   I agree that some skills are needed to bring the best of a scene, but these are artistic skills. :dink:

What exactly is a gnomon anyway? I'm assuming you are using Hero's definition as found here http://en.wikipedia.org/wiki/Gnomon ?
And how does that relate to the images you showed so far?

Edit: Oh, you're using the definition given by Midhat J. Gazalé in which case it's just a word for being self-similar.

If I understand stereoman correctly, I wouldn't say Gnomon is just another word for self-similar, but the gnomon is more like a word for 'the group of self-similar shapes of a specific kind'
The spiral is a gnomon I'd say.
or the branching pattern of rivers..

Do I understand you right stereoman?

     As you can see, there are many definitions of gnomon.
The Hero is the definition I used but just to have something to talk about.
I think the human head drawing   makes it clear what it is, you name it.
The fact is that recursion is a very simple process, such as a spinning wheel, no matter how long you continue to turn, never produce a thought.
The way we use fractals, is much like the proverbial bunch of monkeys typing away, hoping that by the sheer number of words Hamlet will be written at the end. .
   Nature uses the gnomons continually understand what they are is easier if they are observed, take an onion, cut it in half.
    No matter how you name it,it´s the mathematic and geometric tool each of us needs to work with fractals.
   

     Probably the Word gnomon can be used  in such many ways because  the idea behind is greater than what we can reach, after all, it´s one of the life´s most hidden secrets, ancient people think  gnomon was sacred.
     Maybe I should talk a bit about the horizon to make this a bit clear.
     Geometrically, horizon is just a line, very useful in perspective drawing, but for ancient egyptian people, it was a God,,Horus, the horizon line, separates the earth from the heavens, life, from death, light from shadows, and thus was sacred and was a concept far beyond it´s strictu sensu as we understand it.
      This was usual, Art, Science and Religion were one single thing, and all of our problems come from having separate these.

Another useful transform

This one is really complex, and probably must be the core for a 3D gnomon
  Note that the logarithmic spirals allow for an unlimited growing, while closed circles implies a limit.
  Edit to note that this design can be found at ancient petroglyphs

hm.. if my "the group of self-similar shapes of a specific kind" isn't the core of the concept of gnomons, then I don't get it at all and wish you could try to clarify it. to be honest, the pictures don't yet help, it hasn't made click ;)
I'd like to stay with this interesting topic. i can feel that there is lots of importance in it.


it sounds like that for you the gnomon is "larger" than the fractal. in my understanding fractals contain, include gnomons.
having different gnomons is a telltale sign for a fractal, but not the only one.

First I´ll try to define what I mean by fractal
   I think this is a very simple process, since nature always uses simple methods.
   In this case, recursion, but to start a process, you must have something to process.
   And here comes the gnomon, gnomon provides the information necessary for the recursion reaches a particular purpose, therefore the gnomon, is the heart, the engine and the reason for the recursion.
   Understand that recursion is just a process, no other entity, and can be applied anywhere, with or without sense, geometrically recursion is always an extended spiral over time.
    Let's separate then what is a process in time, recursion or fractalization, from what you apply it, and to apply the recursion to something, we need the gnomon, ie, a way to connect the recursion and the object to which we apply it.
    Not quite sure how to relate this with music but I´m sure it will make sense for you  if you can translate this into your own terms.
    Probably there´s no name for this thing since it does not exist, gnomon, or better gnomonic, it´s a word wich describes its behaviour.

I think this may help.
  This is the  gnomonic structure of the human brain , although it is too complex for it to start, it may serve to understand how nature Works, and what the aim is.
  The front section shows the  gnomonic development and the side view it´s logarithmic growth .
  The final result would be a flexible spherical gnomon wich can be edited at polygon, edge  or vertex level, both graphically and by numbers, then , you can apply recursions at different levels and see what happens.
   Of course, this allows to create transparent layers, , rigid or flexible, forms within forms, animations and a lot  of options to develop new ideas.
   I want to make it as clear as posible, I hope this way we will know if this thing can be done or not, so, I´m grateful to questions.
  
 

It's still very unclear to me what you are talking about.
Are you saying that fractals always are recursions while gnomons are more generic maps of similar underlying simplicity?

And what exactly are you showing in those pictures? What kinds of transforms are you talking about?

I have to disappoint you. I still don't get it, although I'm really trying..  :suspious:


the process I'd call iteration of the formula.

these are the numbers that you use to fill the variables of the formula.

you have a formula, start with a chosen number and iterate the formula to achive a result...

I don't understand whats new or different for gnomons or where the connection is and what they are 'needed' for..


obviously I'm not the only one having a problem understanding what your point is..
Is there an official definition of "gnomon", that describes exactly what you are talking about? the definitions I find don't really seem to fit. can you recommend a link/read to better understand it?
Or is it a new concept that you are thinking of and use the term gnomon in your own way?

The eye has a fairly simple gnomonic structure, and allows to understand the concept more clearly.
The lens is a special case, while forms, gets its feed through an artery, when it reaches its final state, the artery atrophy and disintegrates, the lens is formed by very thin wholly transparent layers, something like a squashed onion.
   Anyway, the curves of this lens are part of two overlapping and oppossed circles with different radius,thus this leads to further levels that are far beyond of our present aim.

Let me put it this way, suppose you can edit de M-set, or better, the A-box.
Suppose a 3D program where you can model anything using as a base a flexible sphere, this exists, Mudbox, or Zbrush.
 Now, apply recursive formulas to it, is so simple.
 The gnomon applies transforms trough time, this is the point, and here is where is the difference, because trough gnomons you can mimic any natural growth.
  The gnomon includes some nested layers wich develop trough time, each carrying its own info.

then we probably just have different standpoints.

I believe that fractals do exactly this perfectly.
I believe the difference between a pure mathematical fractal and nature is that these natural fractal forms react to influences from outside. this is why they don't look as perfect. a tree for example shows fractal branching. but the system is so complex, you have to think if wind direction, shade of other trees that will change its appearance to a more ragged, natural way.
I believe that we can explain this with 'multifractal systems'.
The tree (system 1) reacts to the weather (system 2), to other trees position-->shade (system 3) and so on..

Also, I see evolution, the tree of life as a grand fractal. where over time the formula changes and grows in a fractal, self similar way.
which is why it is soo complex after billions of years that we just can't (yet) reverse engineer the formula..

thats my view..

but I'll keep reading this thread, as different approaches to a topic don't have to contradict.. :)

cheers!

Ah, ANY transformation?
So what you are saying is that, unlike fractals where you tend to apply the same limited set of transformations over and over again, in a gnomon you could at any point in time switch to any other transformation? Is that the difference?

If that's the case, I'd think that gnomons really are just recursions, as found in fractals, in which you extend the parameter space of a single transformation to infinitely many dimensions.

And the exact way in which a transformation happens is based on all those parameters. And the choice of parameters can be partially random, partially fixed and partially determined by an arbitrary feedback loop.
(roughly corresponding to Noise/Errors, genetic adaption and learned adaption respectively in a living organism)

Is that about right?

 This thing does not exists yet, so has no name, gnomonic it´s the only name that fits with it´s purpose because of their expected behaviour.
 It´s   something that mimics life trough electronic pulses. sinchronized trough three levels, a kind of very sophisticated pencil, if you ask me, but this is too far for now.
   I need this tool, let me say it this way, I´m now planning some scenes, trough standard programs I can achieve a decent integration between  3D objects and fractal backgrounds, but there is no way to generate a fractal human head, whatever this be.
    But I have in mind a wonderful head made by small and countless iterations of the same head with all its details, hairdo, etc, moreover, I want different hues and shades for them, moreover, its outer cover is transparent and let see some organic inner structures made of different materials.
    It´s nearly imposible to achieve such an image with standard programs, or if it can be done it will cost a lot of time and effort, but this would be nothing for this tool .
    In your terms, it´s a kind of universal and editable formula, if I can say it.

I am with Chillheimer on this one. It's really just a vastly expanded rule-set in interplay.
If you will, the fractals we are typically rendering here are just what theoretical physicists or biologists would probably call "toy systems" - very simple cases of much more general possible forms.
They have little direct application in the real world, as they are too simple / make too many assumptions, but they are very helpful to highlight specific intricate details about these kinds of structures in general.

Though even if you throw in more and more of the rules that would actually be in interplay in real life, it's still a highly self-recursive, iterated feedback-loop that ultimately very often ends up defining a fractal of some sort (a strange attractor, usually)

You get it I think

All viewpoinst are welcome so we can discuss.
 But, to me, any tree has a function that not depends on any evolution, and is the function what defines the shapes and sizes of the tree, in the same way, is the function what defines the shape and size of everything.
     So we try to go as near of the core function as posible .
    I agree that  fractals show some kind of growing, but I´ve never seen a decent fractal generated tree, nor a fractal generated terrain that can be used together with 3D models without further edit.
    

 No problema with this, since , as above, so below.
  The interesting part for us is the level we can reach and work with.
  I agree,our present fractals are toy systems if compared with natural true systems ( and are impressive), this is the best we can approach them, and that´s why I look for a new and more powerful tool.

Really, the amazing complexity of natural systems partially stems from the effective starting conditions and effective rules not only having interesting attractors but also from being dynamically changed in a fractal manner.

Note that I said "effective". - if you were to follow each individual particle and all the forces that play between all those particles exactly, you'd end up with a very simple (relatively speaking) rule set that underlies it all.
It's just that the sheer number of particles we are typically dealing with makes such a detail view beyond impractical. We have to coarse-grain and approximate the globally valid rules with local ones that are more circumstantial, hide more of the details but yield the right results (within the tolerances we allow through the approximations we choose) more directly. This abstracted, local rule-set is what actually changes over time. And since it is a zoomed-out version of the underlying much simpler but much less practical rules below, this local rule-set effectively is a fractal in itself.

This way, nature as naively defined by our own perception, is a fractal of fractals.

I should clarify: The base-rules are "simpler" in that they consist of primitive notions (individual particles, only the four fundamental forces of physics - at least under our current description). They, however, end up giving much more complicated behavior, because you do not approximate away corner cases that give interesting scenarios that, however, are ultimately really hard to deal with.

This can especially be seen in thermodynamics where it's fairly straight forward to describe a gas of homogeneous, constant temperature, pressure, volume, particle number or chemical potential. But once you introduce spacial or temporal inhomogeneous regions or allow phase transitions, as found in so called non-equilibrium-thermodynamics, you very soon hit the very edge of what can currently be described. Our models just aren't good enough for that for now.

So, in your view, it´s not  posible to do it?
I assume it´s very difficult to implement  the full concept, but that´s why I think it must be done with small steps starting with some simplified 2D model, Is that a right approach?

Depends on what precisely "it" is.
Also, I do not think any simulation would be impossible. Just... hard.
The more details you want, the harder and slower it gets.
More correctly, "it" might be impractical given our current understanding of the basic rules as well as our technological capabilities. It might become very feasible one day.
And yeah, step-by-step better approximation is a good way to get there and it's what scientists have been doing since at the very least the invention of Newton's / Leibniz' calculus.

This thread was an interesting read, and then I saw this and had to reply:


Well, I must say that I have created just such a human head as you have described, in that everywhere you look you see smaller and smaller versions of the same head! (without the transparent surface or hairdo though). I don't have anything to post just yet because I am preparing a high-quality render for the contest. But I can tell you a little about how it was made, and I think it relates to your Gnomon idea. The concept I used was a literal interpretation of a Multiple Reduction Copy Machine (MRCM). This is similar to an IFS, in that you define transforms, but IMO is much more intuitive and flexible. The transforms were defined using the collage theorem. In other words, the first iteration is hand-placed. I made the head in a standard 3d package with just a few lines of scripting, and while the fractal rules are certainly simple, defining the transforms took quite a bit of artistic (not mathematic!) input to achieve a human likeness. It was akin to doing any other artistic 3d modelling. I was able to intuitively "tweak" the details until I was satisfied with the result. Here's a render of some broccoli I made with the same method.  Please note that this is polygon based, so is easily amenable to the techniques for lighting/texturing/rendering available in general 3d packages. In this case I chose a simple Ambient Occlusion look
(http://i58.tinypic.com/2yla5bp.jpg)
I have also implemented 2d versions of an MRCM in Photoshop without a single line of scripting, and without a single calculation on my part. It doesn't get much more artistic than that! The process involves recording an Action to define all the transformations, then playing back the action as desired. To define each transformation, the base layer is copied and then manipulated. The final step in the Action is to collapse all the layers into a single base layer again. This process gives you easy access to powerful (non-affine) transformations as well as color modulation (and any other effect recordable in an Action) that will be repeated automatically at ever iteration.
This simple smiley fractal I whipped up real quick shows what I mean. The layer copies were deformed (transformations involving more than just scale/rotate/translate/skew), and the outer "border" smileys were darkened every time. This is not possible with standard IFS approaches. The absolutely amazing thing is that now that the action is defined, I can start with literally any image and it will converge on these smileys after a few iterations!
(http://i62.tinypic.com/33vjmn7.jpg)

here is my take on a fractal human head, enjoy!
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16034

Lol, that's great :D
Now make something you can actually look at without getting nightmares :P
Alternatively, make a version where each and every pair of eyes stares at you.

thanks!
theoretically, it would definitely be possible to do the eyes, I actually considered this at one point but decided it would just take too much time to get right. One problem was that the "eye sockets" weren't actually very round and each pair of eyes would require manual adjustments.  But with proper planning from the beginning, yeah it would be cool, each eye could be rigged prior to iteration and then they would even update and follow a camera as it moves.
The nightmare part though, I don't know if I can do anything about it LOL, that stuff just comes out without me trying, most of my art winds up having that sort of feel... :evil1: 

Man, just imagine that with an entire body as a fractal. Where it all is fully rigged too.
Infinitely recursive rigging. Whoa. Fractals in motion.

yeah totally, I've been rolling the idea around in my head for how to do the whole body, but it's far from being realized.  Here's my quick "sketch" of a fractal stickman... I'm not too crazy about how it turned out, I felt like I fighting with the shape the whole time.
(http://i62.tinypic.com/116m788.jpg)
I think to make the body properly it might require a multi-fractal, like one fractal for the head, one for the arm, one for the leg, etc.  One thing I was considering for the arms/legs would be to form it out of its own muscles.  Meaning you'd take the whole arm and scale it down to form the bicep, use another copy to form the tricep, and so on.  I haven't tried it yet though so not sure if that's the way to go.

Yeah, while you're at it, make it a full volume fractal, considering all the internal organs, each being built from the whole body.  :smileysmileys:

and you all know its "resemblance" time :D

  HI Billtavis, Sorry, I was really busy  to enter FF the last months.
  Really glad  to read your post, I haven´t entered the core of your explanation, but in a first look, it seems we are approaching the same idea,
  maybe we can finally  make this approach feasible.
  keep us updated