Fractal is a new branch of mathematics and art. Perhaps this is the reason why most people recognize fractals only as pretty pictures useful as backgrounds on the computer screen or original postcard patterns. But what are they really?
With computers, we can generate beautiful art from complex numbers. These designs are called fractals. Fractals are produced using an iteration process. This is where we start with a number and then feed it into a formula. We get a result and feed this result back into the formula, getting another result. And so on and so on Fractals start with a complex number. Each complex number produced gives a value for each pixel on the screen. The higher the number of iterations, the better the quality of the image.
For the most part, when the word fractal is mentioned, you immediately think of the stunning pictures you have seen that were called fractals. But just what exactly is a fractal? Basically, it is a rough geometric figure that has two properties: First, most magnified images of fractals are essentially indistinguishable from the unmagnified version. This property of invariance under a change of scale if called self-similarity. Second, fractals have fractal dimensions.
Fractal geometry is an extension of classical geometry. It can be used to make precise models of physical structures from ferns to galaxies. Nature is rough, and until very recently this roughness was impossible to measure. The discovery of fractal geometry has made it possible to mathematically explore the kinds of rough irregularities that exist in nature.
Fractal geometry has permeated many area of science, such as astrophysics, biological sciences, and has become one of the most important techniques in computer graphics. For example, fractal patterns have appeared in almost all of the physiological processes within our bodies.
While Mandelbrot will always be known for his discovery of fractal geometry, he should also be recognized for bridging the gap between art and mathematics, and showing that these two worlds are not mutually exclusive.
In full agreement with the comment, I'd like to provide an additional perspective - as someone whose lifetime achievements are in geometrization of neuroscience and genomics. Fractals are beautiful, because sensory systems were built by nature to be the easiest to grasp and thus, enjoy fractals. Most people know that our visual system is very unlike the "scanning line-by-line" how e.g. the TV tube produces pictures. Our retina, and visual cortex (as well as our auditory, and even tactile sensory systems) were built to grasp "textures" ("roughness or smoothness", as they are patterned). We do not have to take a look at objects in their entirety - a glimpse (saccadic eye movement) directs our eyesight to e.g. a tree - and perceiving the pattern of branches and leaves we instantly can tell an apple tree from a spruce. Likewise, fractal dimension of music (depending on its "smoothness") varies from one classic to another. Our DNA (also a fractal, just as our organisms are, see FractoGene), can also be converted to "fractal music". It is interesting, that the fractal dimension of music of Bartok is the closest to the fractal dimension of our DNA. "Smoother" composers (e.g. Chopin) can be spotted not only according to the melody, but also of their repetitions and roughness - measurable by fractal dimension. - Pellionisz
What has fractals got to do with beauty? Actually, a lot. For examples our DNA sequences can be very similar to fractals in that they display qualities of Brownian self-similarity, which involve small random lines making up lines instead of small, more patterned lines making up lines. DNA can be modeled by fractal geometry.
Fractals may products beautiful shapes such as fern, Mandelbrout set , julia set, Lina image, and others.
My understanding of humans perception of beauty is that it has a degree of dependence on symmetry. Humans like symmetrical shapes, thus it seems logical that fractals would be appealing in that they display a complex symmetry or repetition of form. Fractals are beautiful, but I think mathematics is in fact closely related to art and the appeal of images in many other ways. For instance the 'rule of thirds' in art and photography guides the positioning of the focal point and critical components of the composition in pictures. Perceived beauty in faces is also dependent on their symmetricallity and this is considered to possibly be an evolutionary adaptation. Ie a more symmetrical face is less likely to have suffered mutation or deprivation and therefore make a better mate or produce stronger offspring. Mathematics is also strongly related to music and it has been observed that good musicians also often have strong mathematical skills.
The following paper focuses on beauty associated with fractals:
M. Johnson, Fractal Beauty, 2009:
http://www.a2u2.org/pictures/pdfs/sermon_20090301_fractalbeauty.pdf
Another interesting paper on fractal beauty is
G. Bennett, Chaos, self-symmetry, musical phrase and form
http://www.gdbennett.net/texts/chaos.pdf
Beauty and fractals? Fractal structures are all around us. There is a self similarity of a leafless branch with the whole tree, self similarity of a rock with the whole rocky mountain. This self similarity is not strict, but can we recognize a rock from a mountain on the photograph when the surroundings is filtered out and we have no signs of the scale? This kind of self similarity lies in a principle of the Japanese garden, simulating the Nature. Maybe fractals are beautiful to us, because we are naturally evolved to be perceptive to this kind of beauty : patterns on all scales disturbed by a slight noise.
Fractals do contribute enormously to beauty in that they allow us to grasp a dynamic beauty - in contrast with previous assumptions of beauty such as the (Greek) Golden Ratio (1,618…).
Fractals have the merit of showing a sort of beauty in-process, a beauty in activity. Julia, Mandelbrot, and many others have produced a shift vis-à-vis beauty in this respect.
Fractals themselves are beautiful.They are piece of art and mathematical also. There are space filling curves which are fractals having integer dimension. Our nervous system is nothing but a space filling curve. In fact most of the natural objects around us are piece of art and indeed are fractals. Thanks to Mandelbrot to introduce such a beautiful geometry.
There is a basic paradox at work here. From one side we the physicists can claim, rightly so, that fractals are a fundamental concept of Nature and we can point to percolation theory (see Dietrich Stauffer's book "The theory of Percolation" as an excellent introduction) as the link between statistical dynamics and critical phenomena. On the microscopic side we look at the fractional diffusion equation as the truly governing equation of motion, as Nature is never perfect. From the other side the artist will point out that the nature of beauty cannot and should not be encased in an equation, no matter how elegant. It is an intrinsic part of the soul and cannot be contained by soulless mathematics.
So do we, by recognizing these fundamental patterns of nature, negate the very existence of beauty as traditionally defined? Or should we be looking for something a little deeper in our Philosophy of science?
There are many 'tensions' to be found in science and nature eg experimental vs theoretical physics; randomised clinical trials vs observational studies, naturopathic/holistic vs modern medicine. Do we need to say one is better than the other or that recognising one means excluding the other? I would have thought that by now we have seen enough paradigms in science fall to be humble enough to accept that there are many facets to reality, which can't be all known. And to recognise that for the most part, a combination of apparently antithetic concepts often paradoxically provides a better solution to a problem ie we need to use observational studies and RCT to understand complex medical problems.
I think traditional art or any art can and does exist in complete harmony with science. Many artists were great scientists eg Da Vinci, and many artists are intrinsically mathematical eg Bach. Mathematicians perceive equations as beautiful or ugly with similar emotional responses in the orbitofrontal cortex as associated with visual and musical beauty (http://www.scientificamerican.com/article/equations-are-art-inside-a-mathematicians-brain/).
Defining beauty, either mathematically or by understanding the evolutionary origins or unravelling the brains physiological response doesn't in any way diminish any metaphysical or spiritual connection humans may have to aesthetic creations. So no, I would certainly argue that scientifically defining beautiful patterns doesn't negate the existence of beauty as traditionally defined, but actually enhances the beauty by revealing another dimension and remaining open to this is what science (and art) should really be about.
cj Nev: What do you mean? Mandelbrot set is a beautiful piece mathematically and otherwise. A thing of beauty is a joy for ever.
RC, "Beauty is in the eye of the beholder." To me, the Mandelbrot set looks like a kidney bean with gaudy-looking Julia sets here and there. Although the mathematical formula of the Mandelbrot set is beautiful to me, its form is not. On the other hand, I find the E8 Lie Group strikingly beautiful, wouldn't you say?
Mandelbrot based his "mathematical science" on the Lorentz theorems, which contain signs asymptotics comparison. Present in them exponent alpha (determining the fractal dimension) is impossible to determine by computing.
In addition white noise (eg roughness without macroscopic abnormalities) has no reliably calculated spectrum, which determined only by probabilistic characteristics.
Trillions of dollars since the 1980s have been spent on insanity one dropout.
Myndelbrot - a liar and a mathematical ignoramus.
Its proximity to the art is determined by the unspeakable lie, which he cast as an experimental and theoretical physics.
Evidence of the beauty (intense self-symmetry) have been found in nano geometry. See, for example,
R.E. Dickerson, Higher order Mandelbrot fractals: Experiments in nanogeometry, 2003:
http://www.fractal.org/Bewustzijns-Besturings-Model/Higher-order-Mandelbrot-Fractals.pdf
A sample 4th order Mandelbrot image is given by Dickerson, p. 22 (see attached image).
Perhaps fractals are beautiful, but there is no reliable mathematical criteria confirming their compliance with the real physical or biological objects. All the criteria created to date - a mathematical lie.
@A. S. Kravchuk: … there is no reliable mathematical criteria confirming their compliance with the real physical or biological objects.
Fractals are defined in terms of the self-symmetry along the axes of the fractals. That brings to the question of beauty in fractals and the contention that fractals mirror the architecture of natural phenomena. I think you will agree that self-symmetry is commonly found the architectural of physical structures such as a tree leaves or the bilateral symmetry of the bodies of animals. In addition, it is common to conclude that symmetries in natural phenomena evoke a sense of beauty.
Hence, we can conclude that fractals have close ties (mirror) natural phenomena. It is not so much that fractal "comply" with physical objects but rather that fractals make it possible for us to focus on self-symmetrical structures. Do you agree?
Science (particularly Physics) is not Art.
Times of Alchemy long gone.
Now the adequacy of the models are estimated by penetration depth of Mathematics (in particular existing of criteria of eligibility).
An example is the treatment of statistics. In this case, the forecast made by one researcher in accordance with approved model completely coincides with the forecast of another person.
In the application of fractal theory, I know that forecasts for the same objects by different researchers are entirely different. They coincide only if they are adjusted.
Therefore fractals have no connection with Physics in principle.
@A. S. Kravchuk: ...fractals have no connection with Physics in principle.
I have found evidence that fractals are useful in Physics.
Consider, for example,
1. V.K. Horvath, H.J. Herrmann, The fractal dimension of stress corrosion cracks, 1991:
http://www.comphys.ethz.ch/hans/p/118.pdf
2. S.S. Dalziel, J.M. Redondo, New visualization methods and self-similar analysis in experimental turbulence studies, Models, Experiments and Computation in Turbulence, 2007:
http://www.damtp.cam.ac.uk/lab/people/sd/papers/2007/ModelsExperimentsAndComputationInTurbulence_DalzielRedondo.pdf
3. See Section 2.5.5.8, starting on page 50, in
A. Khalil, Analyse Structurelle de L'Hydrogene Neutre dans la Voi Lactee, Ph.D. Thesis, Laval University, 2004:
www.theses.ulaval.ca/2004/22165/22165.pdf
All authors listed publications either absolutely have no understanding about mathematical grounds of authentication fractality proposed by Mandelbrot (just like parrots repeating someone nonsense), or liars. Western Physics completely turned into magic or infomercial with beautiful pictures. It (Physics) deliberately yielded by Mandelbrot “hypnosis” because in 80 years of the last century, it had no idea how else to take money from the state, because have all been investigated.
It's something like a corporate contract between physicists and the state: someone pretend that something investigated and the state pay for it. Because of American States deeply still whether it's true or lies - most importantly to be a leader. In order to look the truth lies must be infinite.
This can only cause charity
Let me remind readers that 500 years ago through simple observation, it was obvious that the sun revolves around the earth. It's time to reject western Physics principles of simple observation (and admiration) and to create mathematically acceptable basis for experimental confirmation of the fractal properties of real objects. I am sure that in this work it will found that it is not possible. I am an example of passing this stage.
@A. S. Kravchuk: ... It's time to reject western Physics principles of simple observation (and admiration) and to create mathematically acceptable basis for experimental confirmation of the fractal properties of real objects.
Contrary to what you have written, observe that there is considerable interest in fractals in recent articles in Physics journals. Further, It is self-similarity and underlying fractal structure that interests some physicists.
For evidence of this see page 83 in
M.K. Ghosh, P.K. Haldar, S.K. Manna, A. Mukhopadhyay, G. Singh, Ring and jet-like
Structures and two-dimensional intermittency in nucleus-nucleus collisions at
200 A G3V/c, Nuclear Physics A 858, 2011, 65-85.
See also
T. Kalaydzhyan, Chiral superfluidity of the quark-gluon plasma, Nuclear Physics A
913, 2013, 243-263.
See page 245: In addition,thelatticedata[2229,31,32] suggest that the topological chargedensity itself (for uncooled configurations) is localized on low dimensional defects with fractaldimension between 2 and 3, i.e., presumably on (percolating) centralvortices (seealso[33,34] for a similar result based on the scalar density distribution and[35,36] for a localization on vortex intersections).
Unfortunately, the above sample articles in the Nuclear Physics journal available in my university library but not available as open source articles. I do admit that if one were to make a judgment about fractals in physics based on what is freely available on the web, one would probably arrive at the conclusion that fractals do not carry much weight in physics. There are a few exceptions. Here is an interesting article that can be downloaded from the internet:
E.A. Timofeev, A. Plotkin, Statistical estimation of measure invariants,
St. Petersburg Math. Journal 17 (3), 2005, 527-551:
http://www.ams.org/journals/spmj/2006-17-03/S1061-0022-06-00919-8/home.html
See Sect. 2.4, p. 533
Without going into a long discussion about your achievements I am going to the essence. As an example, consider the definition of the fractal properties of a rough surface.
1. I affirm that pure roughness has no spectrum. Since the computation of the Fourier coefficient is a computation of definite integral with help of simple discrete formulae (left or right rectangles in Russian tradition). To confirm the accuracy of his calculation of Fourier coefficient (definite integral) need to apply the rule of double recalculation (Runge rule in Russian tradition). When we are verifying the accuracy of calculation of Fourier coefficient we can find that pure roughness has no reliable coefficient at all (relative errors of calculation are huge). This procedure (application of Runge rule) is very important because it allows control step of discretization in discrete measurements of surface (step of discretization is a main parameter which defines accuracy of calculation).
2. You affirm that pure roughness has spectrum. You calculate it by statistics on many parallel trace of measurement on rough surface. You cannot control step of discretization on each trace of measurements. But you suspect that big statistics on many parallel trace automatically lead to good accuracy of calculation. But this statement is not correct. You can verify it by simple example: calculate only one Fourier coefficient with help of big statistic. Then select only even (or only odd) values in the same big statistic and calculate the same coefficient. Comparison of these two values will give you a huge mistake which defined by step of discretization.
3. In addition in any case you can’t calculate spectrum of discrete signal. You can find only finite set of Fourier coefficients. But in Lorentz theorems are used marks of the asymptotic comparison O() and for definition of dimension of Mandelbrot fractality alpfa (or 2 alpfa) you have to know only infinite number of Fourier coefficients (full spectrum). This fact can be explained as follows: when you use a finite number of Fourier coefficients, it is similar to using a segment of the Fourier series for approximation of the desired function. But this segment of the Fourier series is always infinitely differentiable function and for him alpha is equal to 1. A full Fourier series on demand of Mandelbrot turn to the continuous function with the coefficient alpha
@A. S. Kravchuk: your latest post is excellent. Many thanks for taking the time to give a detailed view of the fractal properties of a rough surface.
Incidentally, the articles from the Nuclear Physics journal (from my University library) provide interesting applications of fractals. Unfortunately, these articles are not open access and cannot be posted on RG. If you are interested, I can send these articles to you attached to a message.
Thank you very much. I'm sure that this is interesting work. Unfortunately I do not know of Nuclear Physics.
here you can find interesting and useful information (Can Science Be Used To Further Our Understanding Of Art?) http://phys.unsw.edu.au/phys_about/PHYSICS!/FRACTAL_EXPRESSIONISM/fractal_taylor.html
Fractal images came from expanding a point by its neighbor points, calling them space or coordinate system, and with all points that are coming from different positions according to that start point, we can emerge a fixed result based on same formula that we are applying calculations based on that input point; Thus fractal images pointing at the pure unity which they have came from, as a spread pattern.
Still i have defined 2 factors which will make a fractal beautiful: 1) It must made up of recognizable parts(not be purely noise and dots) 2)That those parts have a chaotic pattern between them that make them prettier, which can be the smaller copies of those "parts".
https://www.flickr.com/photos/sevenx/sets/72157632140614971/
Fractals are products of complexity of non-linear dynamics. It was not Mandelbrot who first discovered fractal geometry but some other mathematicians like Wacław Sierpiński (Sierpinski's triangle, Sierpinski's gasket) more than 50 years earlier when the word "fractal" did not exist yet. Just like the steam engine was known 2000 years before the Industrial Revolution. The true secret why fractals invoke ascetic pleasure lies in the unique link between the Feigenbaum constant and the Fibonacci numbers (and the Golden Ratio). That Golden Ratio was used by Masters of Works who relied on heuristics, empirical methods, when almost nobody knew any mathematics. Today I know of at least one luxury jewellery maker who transform absolutely unrivalled pieces of fractal imagery projecting 4-D objects into a 3-D space in gold. _
As Andras J. Pellionisz wrote we are able to assess the beauty at a glance, without a ruler and a calculator. Nature made us practitioners, doers, not theorists. We owe nothing to academizing science. You ride a bike, a 10-dimensional machine without a hint of a mathematical model, and that's OK, otherwise only smart mathematicians could be granted bicycle licences issued by equally smart beaurocrats.
I can't agree with Qefsere Doko Gjonbalaj that fractal geometry is an extention of classical geometry. It is entirely different. You cannot extend linearities into non-linearities. Nor it can be used to make precise models of physical structures. Some structures, perhaps most of them will remain unmodelled forever. Fractalities tend to be multidimensional, often very multidimensional. Certainly we can try reductionism losing the edge of seeing the whole of it. And if you add some portion of randomness into the dynamics, and that randomness is what is unknowledgable, we cannot come up to any precise model.
Still I am glad that you asked your question. It invariably leads to the problem of creativity, what it means to be creative, how some people are more creative than others and why man's creativity fails to be age-bound or algorithmic. I can talk about it using my own experience (again heuristics before theory) as a painter and a classical guitarist. Fractals are in music, they are in visual arts, they are in our organs, practically all life is based on fractals derived from non-linearities and chaos. There is no doubt that humans are fractal sensitive in all respects.
All artistic structure is essentially 'polyphonic'. It evolves not in a single line of thought, but in several superimposed strands at once. Creativity requires a diffuse, scattered kind of attention that contradicts our normal logical habits of thinking. It requires defining what it is and what it is not. Fractals express what something is exactly. But if we cannot express what something is exactly, we can say something about what it is not. Hence light against darkness, colour against no colour, the indirect rather than direct expression. Fractals are apophatic, mentioning without mentioning, they focus also on what cannot be said directly.
Statues are carved by subtraction. If you delve into the depth of fractals you will find that their pictorial representation is in fact something which has no area at all. You are looking at almost an emptiness highlighted by smudges of genius and those lines and shapes of genius are infinite in length (see the Peano curve) and in the variability of patterns. The statue of David by Michelangelo largely considered the masterpiece of all masterpieces was carved by simply removing everything that was not David (by the words of the sculptor). Therefore, fractals ARE masterpieces simply because they are what they are by simply removing everything that is not fractal.
However, all true art exhibits the soul, the inner part of the creator and if we consider what is ultimately most important, the creation or the creator, I personally have no doubt that no matter how beautiful art can be, its imagery is less important than that of the artist's soul, the artist's mind, bringing us to a perplexing paradox that we may refuse to hang a beautiful work of art at home in spite of its wide acceptance of craft and mastery.
Now coming back to fractal imagery which has gained its wide acceptance of beauty, we need to think for a while about the creator who is not an academic in myopic glasses with a PhD or more, nor a naive painter playing with complex numbers and a galore of colours but an entirely different entity whose name is God. Thus, indirectly I have proven that God is admired also by those who do not believe in Him. You may call it Koszarny's paradox.
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How do fractals contribute to beauty? Very interesting question!
My answer to the question is, fractals contribute to beauty through a whole in which there are far more small things than large ones.
The beauty I refer to is a new kind of beauty, discovered and defined by Christopher Alexander (1993, 2002-2005), or so called objective beauty. I have recently developed a mathematical model of wholeness, with which the beauty can be measured or computed (Jiang 2015).
Alexander C. (1993), A Foreshadowing of 21st Century Art: The color and geometry of very early Turkish carpets, Oxford University Press: New York.
Alexander C. (2002-2005), The Nature of Order: An essay on the art of building and the nature of the universe,Center for Environmental Structure: Berkeley, CA.
https://www.researchgate.net/publication/272159333_Wholeness_as_a_Hierarchical_Graph_to_Capture_the_Nature_of_Space
https://www.researchgate.net/publication/299337109_A_Mathematical_Model_of_Beauty_for_Sustainable_Urban_Design
In addition, this question is closely related to the following one.
https://www.researchgate.net/post/Do_we_need_a_new_definition_of_fractals_for_big_data_Or_must_fractals_be_based_on_power_laws
Article Wholeness as a Hierarchical Graph to Capture the Nature of Space
Presentation A Mathematical Model of Beauty for Sustainable Urban Design
@Bin Jiang :... fractals contribute to beauty through a whole in which there are far more small things than large ones.
I agree with this incisive observation. Evidence of the truth of this observation can been seen in collections (clusters) of small things that are more likely to capture the imagination than big things, which are harder to grasp.
@Paul Ben Ishai
"...the artist will point out that the nature of beauty cannot and should not be encased in an equation, no matter how elegant."
You refer to traditional beauty, or subjective beauty - beauty in the eyes of beholder
However, objective beauty can be measured and computed based on the mathematical model of wholeness that I recently developed (Jiang 2015).
https://www.researchgate.net/publication/272159333_Wholeness_as_a_Hierarchical_Graph_to_Capture_the_Nature_of_Space
https://www.researchgate.net/publication/236887293_A_New_Kind_of_Beauty_Out_of_the_Underlying_Scaling_of_Geographic_Space
Article Wholeness as a Hierarchical Graph to Capture the Nature of Space
Article A New Kind of Beauty Out of the Underlying Scaling of Geographic Space
@Cj Nev
"Personally, I do not think the Mandelbrot set contributes to beauty."
To answer whether the Mandelbrot set contributes to beauty, we must agree upon what beauty is. If beauty refers to something in the eyes of the beholder, the beauty is subjective. If beauty exists in the deep structure and emerges from the scaling pattern of far more small things than large ones, the beauty is objective. Having said so, the Mandelbrot set does contribute to the objective beauty.
https://www.researchgate.net/publication/270634544_Headtail_Breaks_for_Visualization_of_City_Structure_and_Dynamics
https://www.researchgate.net/publication/272159333_Wholeness_as_a_Hierarchical_Graph_to_Capture_the_Nature_of_Space
Article Head/tail Breaks for Visualization of City Structure and Dynamics
Article Wholeness as a Hierarchical Graph to Capture the Nature of Space
@James F Peters
"Incidentally, the articles from the Nuclear Physics journal (from my University library) provide interesting applications of fractals. Unfortunately, these articles are not open access and cannot be posted on RG. If you are interested, I can send these articles to you attached to a message."
I am also interested in reading this paper. Could you kindly share it with me? Thanks.
@James F Peters
"Evidence of the truth of this observation can been seen in collections (clusters) of small things that are more likely to capture the imagination than big things, which are harder to grasp."
Thanks for your nice comment!
Not only see big things from small things, but also see small things from big things.
I believe the universe is fractal, what we geographers see can be applied to what biographers see, or to what nano-scientists see, or to what astrophysicists see....
Dear Bin Jiang,
To answer whether the Mandelbrot set contributes to beauty, we must agree upon what beauty is. If beauty is referred to something in the eyes of the beholder, the beauty is subjective. If beauty exist in the deep structure and is emerged from the scaling pattern of far more small things than large ones, the beauty is objective. Having said so, the Mandelbrot set does contribute to the objective beauty.
That's exactly what I agree with you. One can feel it without mathematical rigours that we loose the edge of aesthetics at a scale too large, or a distance too far. I would say that aesthetics is scale-deceptive and we may make a wrong judgement by being not enough inside the deep structure of the whole entity. The nearest good example is an oil painting - the original work versus its cheap reproduction. Or in a more abstract mode, take someone's soul. You cannot make any overall judgements about that person unless you see those small details, how those tiny bits contribute to generating a picture of the true oneself. In art, and we are still talking about beauty, the truth and beauty are often interchangable. The truth is like a mathematical proof. Like solitons, the beauty (or truth) is information in its purest form, you hear a rider riding a horse, yet there is no rider and not even a horse. That is why we often say, I do not know why I like it. It is no wonder since we see no rider, nor a horse. Art is fractal. The painter has so many degrees of freedom, still he needs to control every agent employed so that it does not stand out weirdly from the whole, and the better the artist, the subtler the problems which need to be solved in a masterly fashion. The painter cannot extend the use of one single agent infinitely, its use is subordinated to the whole, it ends somewhere, it is eventually hidden or completely discarded. Look at fractals, how they comply with such rules. Who is taking care of the whole of it? You need someone who says I will do these things with these tools and materials and I take full responsibility for the deed. If I fail, then I will try again, and again until I get what I want. When I am done, I will tell you. Fractals do not come with one click. Those efforts to render them intrinsically beautiful require n-clicks where n goes almost to infinity. If you know the secret formula how to produce fractals, but you are a student, not a master, you cannot make a short cut, you need to fill the phase space with enough details like painting with numbers. A good master knows the shortcuts. He does not paint by numbers. That's the difference between a master and a copyist or a reproductionist. Behind every fractal stands an idea. And the grandest of all is the idea of life. Art and life share the common fractal thread.
Fractal (Mandelbrot in the definition) - mathematical shameless lie. Fractality parameter can not be set experimentally. Those people who have established the value of the fractal - liars. They did not research, and fitting results. American pastoral science must die. Princeton University - stash ignoramuses, loafers and Mathematical idiots.
@Paul Koszarny
I agree with you that fractal geometry is not just an extension of Euclidean geometry. Both are very different, yet related. Euclidean geometry concerns individual sizes, while fractal geometry focuses on all, or whether there are far more small things than large ones. Euclidean geometry is an individual view, while fractal geometry is a holistic view. I can use the following pictures to show the difference: with the left picture we measure the height of trees, whereas with the right picture, we measure all lengths of branches, realizing that there are far more small branches than large ones.
@A.S. Kravchuk
An example is the treatment of statistics. In this case, the forecast made by one researcher in accordance with approved model completely coincides with the forecast of another person.
In the application of fractal theory, I know that forecasts for the same objects by different researchers are entirely different. They coincide only if they are adjusted.
Therefore fractals have no connection with Physics in principle.
I am really abhorred by the above. What forecasts?
What approved models?
What are you talking about? Try making a forecast of any segment of the financial market with statistical tools. Condition 1: Use only an approved model. Condition 2: Invest all your money in the direction of your forecast. Condition 3: Invite your neighbours and friends to do the same. Optional1: Make a hedge fund.
The remaining conjectures sound like soundbites to me. There may be some hidden message there but search me, I do not see it in my mind's eyes.
I still rememberthat we arenot deliberating the definition of the bicycle while enjoying a ride.
Based on maths you described maths but did not explain why science and art do not tally.
Science is not art? I am afraid you know very little of art itself. Really very little. I wonder how would you explain the wandering eyes in oil paintings or a moving foot. Try an approved statistical model to create a calendar from 365 randomly scattered pages and compare it how your result coincides with the outcome of any other person.
The kind of symmetry that we observe in fractals are their self-similarity between scales.Some would say that beauty entails symmetry, and symmetry has aesthetic value to it.
Mr Ruis
Thank you for sending me your fractal work.
Best regards
Qefsere Doko Gjonbalaj
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