What is Fractals: Definition and 62 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. P

    How Does a Developer Dive into Fractal Geometry?

    Hello everyone! My name is Psyklon and I work as a full-stack developer at my job. In my free time, I mostly generate things on my computer like fractals, cellular automata, and various other generative arts. Since I first read about fractals I was fasciated by them, so I started to dive...
  2. nomadreid

    I The limit of the perimeter of a Koch snowflake as s(0) goes to zero

    On one side, if I have any finite value of s = the side of the original triangle of the Koch snowflake iteration, then the perimeter is infinite, so intuitively On the other hand, if I looked at the end result first and considered how it got there, then intuitively (Obviously at n=infinity and...
  3. fresh_42

    Challenge Math Challenge - June 2021

    Summary: Lie algebras, Hölder continuity, gases, permutation groups, coding theory, fractals, harmonic numbers, stochastic, number theory. 1. Let ##\mathcal{D}_N:=\left\{x^n \dfrac{d}{dx},|\,\mathbb{Z}\ni n\geq N\right\}## be a set of linear operators on smooth real functions. For which values...
  4. K

    Drawing Fractals from IFS Equations

    Attempt The interval sketch is obviously a line from 0, 1. F1 would cut the interval line by a third similar to the Cantor set. F2 would cut the interval line by a third and then there is a transformation that moves to the point (1/3, 1/3). F3 would cut the interval line by a third and then...
  5. qnach

    I What are the differences between Sierpinski/Vicsek fractals?

    What are the differences between Sierpinski/Vicsek fractals?
  6. 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...
  7. mister mishka

    Art My Video Art (fractals, video feedback loops, etc)

    Hey, I just thought I would share some of my video art. They are meant to be visuals going alongside Techno / House music in the nightclubs of Berlin. However I got a lot of ideas from reading up on physics, like diffraction patterns, fractals, polarization of light / materials, lenses, video...
  8. W

    Do not try this at home kids......

    Found this interesting video - thought there would be interest here... VIDEO LINK
  9. Michele Zappano

    Fractal Symmetry & Noether's Theorem: Chaos & Conservation

    I was reading a Steven Strogatz book and he said that the self similarity of fractals is a symmetry. Has any conservation law been linked to this type of symmetry using Noether's Theorem?
  10. P

    B Make & Share Your Fractals - Get Ideas & Inspiration Here

    Fractals! Fractals are great . . . . I like making them and looking at them and using them. They are really art in every way. I was just wondering if anyone else here made fractals, too. Do you have any of your favorite examples of fractals? I’m going to be doing something for a little project...
  11. itssilva

    Continuum hypothesis and fractals

    Wikipedia: "The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers" ; i.e., let cardinality of integers = ℵ0 and cardinality of reals = ℵℝ; then there is no ℵ such that ℵ0 < ℵ < ℵℝ . But what about...
  12. J

    What is the role of fractals in procedural generation?

    I've been reading a lot about computer games like Dwarf Fortress (of which I am also a huge fan) and No Man's Sky which generate the bulk of their content automatically by generating it procedurally. It often seems that when the methods of procedural generation are discussed, fractals end up...
  13. R

    In search of Julia Sets and ChaosPro Training

    Hello everyone. I am looking for some free software to produce a Julia Set that would allow me to enter the equation. I would prefer it to be downloadable software, but if a web applications all there is that will do. It's preferable that a color scheme can be chosen, or at least used. Also, I...
  14. nomadreid

    Are fractals a dead-end area of maths?

    This is not a high school question, but it seems to be too broad to fit in any of the other categories. Fractals are cute (nice pictures that can also be used to give better graphics, and also "shocking" that one can define a non-integer dimension), can be used to estimate lengths or volumes of...
  15. A

    How are fractals modeled by math?

    http://upload.wikimedia.org/wikipedia/commons/2/21/Mandel_zoom_00_mandelbrot_set.jpg So I saw this interesting picture of a fractal. Is this simply art or is it modeled by a specific function?
  16. J

    Why are fractals and chaos theory synonymous?

    I'm doing a presentation in a few weeks on fractals and chaos theory. To me, their link is more intuitive than mathematically/physically sound, and I'm really struggling to put the link into words. I've tried googling it, but no where seems to give a satisfactory explanation of the link...
  17. B

    Fractals- complex but having minimal entropy?

    Fractals are just many iterations of a very basic formula, so they can be described with little information, and yet they are extremely complex given enough iterations. Can they be described as low entropy despite their complexity?
  18. G

    MHB Fractals in relatively prime integers

    Greetings, humans! (Tongueout) I'm from Ukraine. My English is very bad. So I will use a Google Translate. In 2002, I came up with an interesting piece. I was only 14 years old. I was thinking about fractals and chaos theory, and did not want to learn. Did not want to learn, and were forced to...
  19. D

    Start learning more about fractals

    Hi, I'm curious to start learning more about fractals and am wondering what some of the classic/decent texts in the field are. Any suggestions would be much appreciated. Thanks!
  20. P

    How can I make my own fractals?

    Hello everyone. You probably all know about the Mandelbrot set. I am not here to ask questions about the set itself, but how I can graph fractals similar to it. Is there a program where I can input my own equations? Or would I have to be a java expert to do so? The fractals that I want to make...
  21. M

    Undergraduate courses for the study of fractals

    If one wanted to study fractals and their properties, which upper level undergraduate mathematics courses should one take? Or which sequence of generally offered courses should one take? Just to name a few courses that I guess would be related and are offered at the University I study at are...
  22. D

    Where Can I Find a Good Fractals Programming Tutorial?

    I want good tutorial on programming fractals using computer languages like C++,Java,Python,C#,Perl,Delphi,OpenGL. I want to program various fractals using any above languages for that I need some tutorial. Please suggest me some of the resources which are useful.
  23. P

    Teaching Fractals to Ninth Graders

    (sorry if this is in the wrong section) I teach a ninth grade class at a public high school and had a student ask me a question about fractals. We were reviewing complex numbers in Algebra 2 when she asked me how to write the equation of a fractal. How would I explain this or go about teaching...
  24. Matt Benesi

    Continuity of real portion of cosine variant of 3d fractals

    By continuity I mean an unbroken fractal. With certain variants, one ends up with sharp gaps in the fractal. mag=({x^2+y^2+z^2})^{n/2} yzmag=\sqrt{y^2+z^2} \theta= n *atan2 \;\;(x + i\;\;yzmag ) \phi = n* atan2\;\; (y + iz) new_x= \cos{(theta)}\;*\;mag new_y=...
  25. B

    Anyone familiar with fractals?

    Part of our core mathematics course is a completing a 12 page essay on a chosen topic and I decided to write about Fractals. After a bit of research I started to question whether there was enough material at undergraduate level to be able to complete a comprehensive essay on the subject...
  26. Newtons Apple

    What is the significance of the Mandelbrot set and fractals?

    Can someone explain the Mandelbrot set to me? I know the equation is zn+1 = zn2 + c But what does this mean? Whats its basically saying here? All I know about it really is the existence of fractals, but why are fractals so mathematically important?
  27. Z

    What are self-similar pictures of bifurcations and fractals ?

    What are "self-similar pictures of bifurcations and fractals"? The context is: 1.5 Deterministic Chaos Chaos brings to mind many things: butterflies in South America changing world weather; self-similar pictures of bifurcations and fractals; and an inability to describe irregular...
  28. S

    Problem involving simple fractals (Koch snowflake problem)

    Homework Statement Construction begins with an equilateral triangle with sides of length one unit. In the first iteration triangles with length one third are added to each side. Next, triangles of length 1/9 are added to all sides, etc., etc. Is it possible for a bounded region to have...
  29. P

    Dimensional regularization and Fractals - Is it a crackpot idea?

    I came across a curious site on this topic: http://e-infinity-energy.blogspot.com/2011/01/t-hooft-veltman-dimensional.html On one hand, the blog history is filled with non-mainstream ideas. (They invented a new subfield called E-infinity.) On the other hand, the people there seem to be tenured...
  30. H

    Do Fractals have anything to do with Brownian motion?

    I watched a show on Fractals and it sort of remind me of Brownian motion. So my question is has anyone ever used fractals to explain Brownian motion?
  31. T

    Fractals and chaos - what do they have in common?

    I really like fractals and what I find even more amazing, is that something so beautiful can be so useful. I've heard that they can be used for example in chaos. I've gone through about 10 websites and all were about fractals and chaos. But I couldn't find a single sentence that would tell me...
  32. M

    Resistance of Fractals: Sierpinski Triangle

    Let assume an exact mathematical fractal on a surface, for example Sierpinski-triangle, made of material with homogeneous conductivity. What do you think, it has zero, finite, or infinite resistance between two points (for example two corner of the triangle)?
  33. mnb96

    Lorentz attractors and fractals

    Hello, as far as I know a "fractal", by definition should manifest self-similarity or at least statistical self-similarity. This usually takes the form of scale invariance. Can anyone point out where is the self-similarity in the plots of Lorentz attractors? Thanks.
  34. J

    Fractals and Area summation in Mathematica

    Homework Statement Begin with an equilateral triangle T of side length 1 At the middle of each side of T place an equilateral triangle whose side lengths are 1/3 Repeat this process ad infinitum By summing an appropriate series, show that the area A of the fractal obtained above is finite...
  35. B

    Link between fractals and chaos

    I have studied fractals and think of them as things that exhibit self similarity at different characteristic scales. For example a grain of sand looks a bit like a pebble, which looks a bit like an outcrop, which looks a bit like a mountain. If you count the number of grains of sand there will...
  36. D

    What mathematical subjects should I study after calculus to understand fractals?

    I am interested in studying fractals, but I'm not exactly sure what mathematical subjects I need to be familiar with in order to do that. I am currently learning calculus. What do you suggest I study after calculus?
  37. Z

    Are prime fractals, or have a fractal geometry ?

    are prime fractals, or have a fractal geometry ?? my idea is, if we consider the geometry of primes could we conclude they form a fractal ? , for example if we represent all the primes using a computer, it will give us a fractal pattern. according to a paper...
  38. L

    Textbooks on Fractals and Chaos

    Hi! I'm perusing fractals out of personal interest. Although I come from physics, I have a background in math-major undergrad mathematics (complex analysis, a bit of modern algebra). I find Mandelbrot's Fractal Geometry of Nature relatively light reading, but I am interested in a more standard...
  39. A

    Fractals and Chaos: What's the Connection?

    Hi, I've read a little bit about fractals, being self repeating shapes. Is there a connections between fractals and chaotic systems? Thanks
  40. N

    What are the best books for understanding chaos theory and fractals?

    I am looking for two different pairs of books. This first pair would be an introduction into the chaos theory and fractals(seperate books). The second pair, I am looking for complete mathematical theory on each, preferably that is pretty self contained and in depth. My background in...
  41. C

    Can fractals help solve the mystery of the universe's unified theory?

    I recently had a idea after watching nova's special on mendel fractals. I support a string theory as well as M theory. Just like most of us nerds we all want the unified equation that einstein sought after till his last days. Recently fractals were used to describe heart beat's. What was...
  42. A

    Exploring the Intersection of Fractals and 3D Game of Life: A Scientific Inquiry

    HI all. Just looking for some ideas or pointers. I wish to pass on an idea to a friend. I want to look at Platonic shapes in 3D evolving through various iterations. I've played with pretty fractals and I've played with the 2D and 3D Game of Life games. both nice diversions :) on a rainy...
  43. N

    What Are the Best Books to Learn About Fractals?

    Hi all. Was hoping to get some good recommendations for texts on fractals, maybe starting with introductory texts moving on up to the more mathematically intense works. I'm also following this thread: Topology-prerquisites, as I'll probably need to brush up a little. Just for the...
  44. W

    What does it take to study fractals?

    So, what should I have under my belt to study it, in particular both point-set and geometric. I already have CalcI-III Linear Algebra, Differential Equations. I am guessing I have a long ways to go. Please feel free to recommend some literature. Also what does it take to study fractals?
  45. T

    Looking for fractals texts suitable for guided self-study.

    I have finally convinced one of my professors to sponsor an honors research/reading course over the summer, but I need to find a suitable book. I own quite a number of older layperson's books on fractals including the standards by Mandelbrot and Barnsley and even Peitgen/Jürgens/Saupe's Chaos...
  46. X

    Are fractals related to string theory?

    I have just seen some pretty pictures of flame fractals with reference to string theory. I was just wondering if or where they fit in. thanks :smile:
  47. C

    Creating fractals from nothing?

    For anyone here who is familiar with fractals and self similarity: Is it possible to find local self similar patterns from a random arrangement of objects which can apply to the global arrangement of these objects? For example, we have people walking in a city. The positions are somewhat...
  48. F

    Fractals: Infinite Perimeters & Plank Scale

    Are the perimiters of fractals infinite? Or does Plank scale prohibit that? Thanks, FRQ
  49. H

    Exploring the Mandlebrot and Julia Sets: The Beauty of Fractal Patterns

    I am sure the same goes for you lot, I am fascinated by the complex patterns of fractals and recently found out it is generated by extremely simple algorithms (which takes weeks to run). What do I actually need run some algorithms that generates fractals?
  50. L

    How Do Fractals Relate to Successive Terms in Mathematical Sequences?

    I have a portfolio due tomorrow and I don't understand the wording of the q applying the relationship between successive terms of Nn, Ln, Pn, an, and An, use a spreadsheet or similar software to find the values of each parameter up to the value n where An+1 = An to within one millionth of a...