What is Fractal: Definition and 96 Discussions

In mathematics, a fractal is a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at different scales, as illustrated in successive magnifications of the Mandelbrot set. Fractals exhibit similar patterns at increasingly smaller scales, a property called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.
One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.Analytically, most fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still topologically 1-dimensional, its fractal dimension indicates that it also resembles a surface.

Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".The consensus among mathematicians is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures, and sounds and found in nature, technology, art, architecture and law. Fractals are of particular relevance in the field of chaos theory because the graphs of most chaotic processes are fractals. Many real and model networks have been found to have fractal features such as self similarity.

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  1. Mashiro

    I Expressing any given point on plane with one unique number

    Currently, as far as I know, the two main ways to express any given point on a plane is through either cartesian plane or polar coordinates. Both of which requires an ordered pair of two numbers to express a point. However, I wonder if there exists such a system that could express any given...
  2. Fractal matter

    Is fractal universe with determinism possible?

    I'd like to know your opinion. Do you think the universe could posses the following qualities altogether? 1) All quantum fields are effective fields 2) Gravity is an effective field 3) Local realism (possibly superdeterminism) 4) There is a preferred frame of reference It seems to me, that...
  3. K

    I MOND from galaxies that are fractal

    I found these papers, the first is that galaxies are fractal based on observation Fractal Analysis of the UltraVISTA Galaxy Survey Sharon Teles (1), Amanda R. Lopes (2), Marcelo B. Ribeiro (1,3) ((1) Valongo Observatory, Universidade Federal do Rio de Janeiro, Brazil, (2) Department of...
  4. M

    Turbulence of square fractal grid

    Hi all, Whats the significance/application of square fractal grid turbulence studies?
  5. K

    Drawing Fractals from IFS Equations

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  6. S

    Binnig's fractal evolution applied to multiple universes?

    Gerd Binnig, Nobel laureate in physics in 1986, proposed in his article "The fractal structure of evolution" [1] that everything in the universe, including its laws, had changed and became what we have got today through a process which mixes some concepts from darwinian evolution and fractal...
  7. sbrothy

    I Barrow black holes with fractal surface?

    This should maybe go in the Beyond The Standard Model forum but since it's a paper about quantum cosmology I'll put it here. Feel free to move it if it's too speculative but that's exactly my question. That is: if it is... Perusing "The Area of a Rough Black Hole" - -...
  8. M

    A Golden ratio fractal phase conjugation

    So, I recently heard about the Golden ratio fractal phase conjugation and I was wondering if it could possibly explain the solar system formation. Meaning, if you can imagine a torus energy dynamic shape (with the sun in the middle) maybe it could be explained by this theory?
  9. Cheesycheese213

    B What is the Area of a Sierpinski Triangle?

    I was trying to find some sort of pattern in the triangle (below) to graph it or find some equation, and I thought maybe measuring something would be a good idea. I was okay just calculating the area for the first few iterations, but then I got confused on how I was supposed to represent like...
  10. Ryu

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    So I had a topic which I would like to fact check from an informed scientific source. Basically there is an argument about whether or not an object that naturally exists in a fourth dimensional space, would by default have more than countably infinite times the energy of a 3 dimensional Object...
  11. C

    Welcome to the PF Community: A Guide to Exploring the World of Physics

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  12. Fractal dimensions-3blue1brown

    Fractal dimensions-3blue1brown

    How fractals have weird dimension definitions.
  13. nomadreid

    Is fractal nature dependent on the number of steps?

    In looking at the definition of a Hausdorff dimension of a space S = inf{d>0: inf{Σirid: there is a cover of S by balls with non-zero radii} =0} where i ranges over a countable set, it would appear that it would be acceptable to take the index set to be finite, but I am not sure how you would...
  14. S

    I Fractal Dimension & Degree of Freedom

    I have been learning about the idea of "dimension" of a shape in terms of how its "mass" scales down when we cut it into self-similar parts. For example, However, the term "dimension" is closely linked with the idea of degree of freedom. So my question is, is there any sense in which the...
  15. qnach

    I What are the differences between Sierpinski/Vicsek fractals?

    What are the differences between Sierpinski/Vicsek fractals?
  16. nomadreid

    I Physically relevant: fractals, phi?

    There are two subjects which pop up a lot as having physical examples (or, more precisely, where their approximations have), but many (not all) of them seem rather indirect or forced. For example: [1] phi (the Golden ratio) or 1/phi: (a) trivia: sunflowers and pineapples giving the first few...
  17. Ventrella

    A Binary fractal tree with equidistant leaves on a circle

    Does there exist a binary fractal tree… (reference: http://ecademy.agnesscott.edu/~lriddle/ifs/pythagorean/symbinarytree.htm ) …whose leaves (endpoints) lie on a circle and are equidistant? Consider a binary fractal tree with branches decreasing in length by a scaling factor r (0 < r < 1) for...
  18. Ventrella

    A Examples of fractal structure in prime partition numbers?

    Regarding the recent discovery by Ken Ono and colleagues of the fractal structure of partition numbers for primes: a great lever of intuition would be to see a diagram, or any presentation of the numbers that reveals this fractal structure. Perhaps the fractal structure is somehow hidden in a...
  19. T

    B Clarification about Fractal Dimensions

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  20. berkeman

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    So the quality of my over-the-air digital TV reception has been getting worse over the past few months. Probably the digital TV antenna that I'm using is getting old and starting to degrade. Last night I resorted to trying different reflector combinations to try to boost the signal for the...
  21. C

    I Why does physical space have to be three-dimensional?

    There's a question that's been in my mind for quite a while but I cannot figure out what the answer is. I't probably an ill posed question but I will ask it anyway: 1.- Do we know what the dark-matter statistical distribution in our Universe is (at large scales)? 2.- In case we do, could...
  22. T

    I Dimension using box counting technique

    On an exam we just took, we were asked to find the dimension of a set using the box counting technique. So choose an epsilon, and cover your object in boxes of side length epsilon, and count the minimum number of boxes required to cover the object. Then use a smaller epsilon and and count the...
  23. Euler2718

    MATLAB Problem with Generating Barnsley Fern Fractal in MATLAB

    The code is as followed: function fern() AI = [0 0 ; 0 0.16]; AII = [ 0.85 0.04 ; -0.04 -0.85 ] ; AIII = [ 0.2 -0.26 ; 0.23 0.22 ] ; AIV = [-0.15 0.28 ; 0.26 0.24 ];BI = [ 0 ; 0]; BII = [ 0 ; 1.6]; BIII = [ 0 ; 1.6]; BIV = [0 ; 0.44]; N = 10000; I = 50; H = zeros(N,2); for n=1...
  24. Blidaru Bogdan

    A transfer matrix method for the analysis of fractal quantum

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  25. ssamsymn

    Find a Fractal Object with Known Boundary Term

    For my work, I need to check my calculations with an example of a fractal object. I searched on the internet, there are some examples of fractals with their hausdorff dimensions, but no boundary terms related. Also found some 1-d examples, but I need d>3 dimensional objects since my calculations...
  26. O

    MHB Is a fractal always a predictable pattern?

    It never goes crazy and starts doing it's own unpredictable number pattern or sequence, right? Thanks in advance.
  27. D

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  28. A

    MHB Is Fractal Binary a New Math

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  29. Greg Bernhardt

    Art Amazing Fractal Art - Favorite Designs

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  30. H

    Exploring the Relationship Between Hausdorff Dimension Definitions

    Hi, I am trying to understand why do the two versions of Hausdorff (fractal) dimension are actually the same.I refer to the definition by coverings and the definition by ratio of two logarythms. http://en.wikipedia.org/wiki/Hausdorff_measure...
  31. H

    What is the concept of fractal dimension and how is it computed?

    Hi, Can someone give me a link to a clear and relatively basic and short matirial introducing the notion of fractal dimension (Hausdorff dimension)? Thank's in advance.
  32. J

    Sierpiń́ski's fractal and calculating the total blank space

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  33. P

    What the best beginner textbook for chaos, fractal, & random analysis?

    Hi all, I'm trying to self-learn about chaos, fractal, or anything that correspondence to random analysis (maybe with some material from statistical physics). Anyone know what the best textbook for these fields?
  34. E

    Golden Ratio Dragon Fractal (figured it out)

    Back in november I asked the forum about this fractal: http://en.wikipedia.org/wiki/File:Phi_glito.png At the time I couldn't figure out how to make it. Since then I've figured it out. I used MS Excel. I'm not completely satisfied though. There are some gaps between the major...
  35. G

    Capacitance of a Fractal Surface

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  36. M

    Magnetic Pendulum Fractal Basin Boundaries in Mathematica

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  37. N

    What is the representative fractal equation for a power law distribution?

    Hello, I had a question about data which is represented by a fractal distribution. I know that the linear regression lies in the plot of log(N) vs. log(x) for which the ratio represents the fractal dimension as the limit of x going to infinity. However, how would one get the representative...
  38. nomadreid

    Riemann's zeta function fractal because of Voronin?

    Riemann's zeta function "fractal" because of Voronin? I am not sure which rubric this belongs to, but since the zeta function is involved, I am putting it here. I noticed a comment (but was in too much of a hurry to remember the source) that, because of the "universality" of the Riemann zeta...
  39. C

    Linearising Eqn for Fractal dimension

    Homework Statement I am doing an experiment to determine the fractal dimension of hand compressed aluminium spheres. I cut a square of foil of some length ##L## and known thickness, ##t##. I do this a few times, varying ##L##. The radius of the hand compressed spheres, $$r =...
  40. E

    Dragon Curve Fractal Using Golden Ratio

    I've been fooling around in MS Excel trying to reconstruct this fractal: I haven't had any issues here making it. I totally understand the algorithm for generating the left turn/right turn ordering. What I really want to know is how this version is generated: Original image...
  41. N

    Recursion to print a fractal pattern

    C++ Recursion to print a "fractal pattern" I guess I really don't understand how to create a initial case for my recursion Can some advice to where I would need to change the code ___________The problem: Create a recursive function to draw this pattern, given a maximum number of stars (which...
  42. dexterdev

    Is universe a 3 dimensional fractal .?

    Is universe a 3 dimensional fractal...? Hi all, Yesterday I was reading on fractals (2 dimensional). It was interesting to know leaves etc are examples of fractals ( for example : fern). So I thought if there is a possibility for 3 dimensional fractal. And as in any case which is distorted...
  43. M

    Fractal Question :Rodin Scupture to Low density Mesh to 2D to Iterative Function

    Hello there, Subdivision modeling is a tool which all 3D Modeling Softwares uses. Its about creating a low density mesh and computer creates higher density mesh until your box turn into a sphere.I found something about finding iterative fractal function of a 3D object for example Rodin...
  44. S

    Mandelbrot's Fractal: Why the Simple Set Creates Mesmerizing Shapes

    Do you know why a simple set used for construction of the Mandelbrot's fractal results in a jaw dropping shape?
  45. D

    How to find fractal dimension of Gosper Island

    Hi, I'm not sure if this is the right place for this...if it isn't if I could be redirected/if a moderator could move my post to the right place I would greatly appreciate it. In any case, I am trying to understand fractal dimensions. I read through wikipedia's description and I believe I...
  46. J

    Is a Fractal Dimension of 0.65 Possible in Nonlinear Oscillator Systems?

    First post! I'm investigating chaos in non linear coupled spring oscillators. After generating a poincare' map of said system i wanted to see if the map was fractal. i proceeded to use a box counting method in order to calculate a fractal dimension. I generated a plot of log(number of...
  47. C

    Fractal Geometry and the Foundations of Maths

    Hey Everyone, I just wanted to ask for a bit of help on this research assignment I have to do. I have to show how Fractal Geometry contributes to the theory that Mathematics was invented. I have been looking into fractal dimensions and the fact that the dimensions we have labelled (1,2 and 3)...
  48. V

    Finding Fractal Behavior - Methods & Statistical Analysis

    Hi, I have been looking for fractal behavior in a data set. I've used the box method to determine fractal dimension by looking at the inverse of box size and the number of boxes needed to enclose the object. These two variables seem to be fairly accurately predicted by a power law...
  49. C

    How do I compute the fractal dimension of the Koch curve using Matlab?

    Homework Statement I have a problem trying to code these two programs, one is related to the obtain the fractal dimension of the Koch curve by using the method of the box counting dimension and also using the Grassberger-Procaccia algorithm Homework Equations For the box counting...
  50. B

    Fractal Geometry: Uses, Math & Fascinating Patterns

    The very first time I ever heard about fractals was in my junior year in high school in my Algebra II class when we were studying complex numbers. I was fascinated by these wonderous objects and I've had many questions about them ever since. Though two of my main questions have always been...