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# Fractals

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## Q: What are fractional dimensions? Can space have a fractional dimension?

Physicist: There are a couple of different contexts in which the word “dimension” comes up.  In the case of fractals the number of dimensions has to do (this is a little hand-wavy) with the way points in the fractal are distributed.  For example, if you have points distributed at random in space you’d say you have a three-dimensional set of points, and if they’re all arranged on a flat sheet you’d say you have two-dimensional set of points.  Way back in the day mathematicians figured that a good way to determine the “dimensionality” of a set is to pick a point in the set, and then put progressively larger and larger spheres of radius R around it.  If the number of points contained in the sphere is proportional to Rd, then the set is d-dimensional.

(Left) as the sphere grows, the number of points from the line that it contains increases like R, so the line is one-dimensional. (Right) as the sphere increases in size the number of points from the rectangle that it contains increases like R2, so the square is two-dimensional.

However, there’s a problem with this technique.  You can have a set that’s really d-dimensional, but on a large scale it appears to be a different dimension.  For example, a piece of paper is basically 2-D, but if you crumple it up into a ball it seems 3-D on a large enough scale.  A hairball or bundle of cables seems 3-D (by the “sphere test”), but they’re really 1-D (Ideally at least.  Every physical object is always 3-D).

A “crumpled up” set seems like it has a higher dimension than it really does.  You can get around this by using smaller and smaller spheres.  Eventually you’ll get the correct dimension.

This whole “look at the number of points inside of tiny spheres and see how that number scales with size” thing works great for every half-way reasonable sets.  However, fractal sets can be “infinitely crumpled”, so no matter how small a sphere you use, you still get a dimension larger than you might expect.

The edge of the Mandelbrot set “should” be one-dimensional since it’s just a line. However, it’s infinitely twisty, and no matter how much you zoom in it stays just as messed up.

When the “sphere trick” is applied to tangled messes it doesn’t necessarily have to give you integer numbers until the spheres are small enough.  With fractals there is no “small enough” (that should totally be a terrible movie tag line), and you find that they have a dimension that’s often a fraction.  The dimension of the Mandelbrot’s boundary (picture above) is 2, which is the highest it can be, but there are more interesting (but less pretty) fractals out there with genuinely fractional dimensions, like the “Koch snowflake” which has a dimension of approximately 1.262.

The Koch snowflake, which is so messed up that it has a dimension greater than 1, but less than 2.

That all said, when somebody (looking at you, all mathematicians) talks about fractional dimensions, they’re really talking about a weird, abstract, and not at all physical notion of dimension.  There’s no such thing as “2.5 dimensional universe”.  When we talk about the “dimension of space” we’re talking about the number of completely different directions that are available, not the whole “sphere thing”.  The dimensions of space are either there or not, so while you could have 4 dimensions, you couldn’t have 3.5.

## Finding Beauty: Fractal Patterns on Earth as Seen from Space

In a world made small and accessible by technology, it is easy to forget the magnitude of nature’s infinite complexity. But sometimes technology reminds us, such as when trawling planet Earth on Google’s Satellite View, zooming across landscapes partitioned by natural and unnatural boundaries.

While searching Google Earth, Paul Bourke, a research associate professor at the University of Western Australia, discovered an amazing sight—the patterns of the Earth seemed to form a delicate geometric pattern when viewed from the sky. Not only delicate, but almost perfect. Bourke was captivated by the geography—lacy tracks of rivers and mountain ranges stretching across the Earth in unison as if digitally cloned.

Fractals are recognized as patterns of self-similarity over varying degrees of scale. There are both mathematical fractals as well as natural fractals—the former are idealized and found across a range of scales, while the latter generally only exist across a smaller scale range.

Bourke explains that fractals are found in all parts of life, from the brain sciences and astrophysics to geographic formations and riverbeds. “Fractal and chaotic processes are the norm, not the exception.”

“I always knew these amazing natural patterns would be there,” he said. “They are literally everywhere—it’s just a matter of finding them.”

And find them he did. Bourke, an authority on fractals and visualizations, showcases more than 40 different fractals he’s uncovered while zooming through the satellite views of 25 countries. Through his website, he encourages users to submit examples they’ve found in their own browsing, and provides KMZ coordinate files for each image, allowing users to visit the exact views of the fractal features. Bourke’s collection realizes the power enabled by the open-ended tools of modern technology and applies them to a practical and popular aesthetic end.

To see more natural fractal patterns, visit Bourke’s website.

## Nanosecond Trading Could Make Markets Go Haywire

From this analysis emerged records of 18,520 sub-950-millisecond crashes and spikes — far more than they, and perhaps almost anyone, expected. Equally as striking as these events’ frequency was their arrangement: While market behavior tends to rise and fall in patterns that repeat themselves, fractal-style, in periods of days, weeks, months and years, “that only holds down to the time scale at which human stop being able to respond,” said Johnson. “The fractal gets broken.”

## Fractal art - Wikipedia

A piece generated in Apophysis.

Showing breakage in space in non integer Multibrot set

A Fibonacci word fractal by Samuel Monnier

Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still images, animations, and media. Fractal art developed from the mid 1980s onwards.[1] It is a genre of computer art and digital art which are part of new media art. The Julia set and Mandlebrot sets can be considered as icons of fractal art.[2]

Fractal art is not drawn or painted by hand. It is usually created indirectly with the assistance of fractal-generating software, iterating through three phases: setting parameters of appropriate fractal software; executing the possibly lengthy calculation; and evaluating the product. In some cases, other graphics programs are used to further modify the images produced. This is called post-processing. Non-fractal imagery may also be integrated into the artwork.[3]

Fractal art could not have developed without computers because of the calculative capabilities they provide.[4] Fractals are generated by applying iterative methods to solving non-linear equations or polynomial equations. Fractals are any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size.[5]

## The Fractal Art Manifesto

As stated by Kerry Mitchell in The Fractal Art Manifesto,[6] "Fractal Art is a subclass of two-dimensional visual art, and is in many respects similar to photography—another art form that was greeted by skepticism upon its arrival. Fractal images typically are manifested as prints, bringing fractal artists into the company of painters, photographers, and printmakers. Fractals exist natively as electronic images. This is a format that traditional visual artists are quickly embracing, bringing them into Fractal Art's digital realm. Generating fractals can be an artistic endeavor, a mathematical pursuit, or just a soothing diversion. However, Fractal Art is clearly distinguished from other digital activities by what it is, and by what it is not." According to Mitchell, fractal art is not computerized art, lacking in rules, unpredictable, nor something that any person with access to a computer can do well. Instead, fractal art is expressive, creative, and requires input, effort, and intelligence. Most importantly, "fractal art is simply that which is created by Fractal Artists: ART."

## Types

A 3D fractal generated using Visions of Chaos

There are many different kinds of fractal images and can be subdivided into several groups.

Fractal expressionism is a term used to differentiate traditional visual art that incorporates fractal elements such as self-similarity for example. Perhaps the best example of fractal expressionism is found in Jackson Pollack's dripped patterns. They have been analysed and found to contain a fractal dimension which has been attributed to his technique.[8]

## Techniques

Fractal image generated by Electric Sheep

Fractals of all kinds have been used as the basis for digital art and animation. High resolution color graphics became increasingly available at scientific research labs in the mid 1980s. Scientific forms of art, including fractal art, have developed separately from mainstream culture.[9] Starting with 2-dimensional details of fractals, such as the Mandelbrot Set, fractals have found artistic application in fields as varied as texture generation, plant growth simulation and landscape generation.

Fractals are sometimes combined with human-assisted evolutionary algorithms, either by iteratively choosing good-looking specimens in a set of random variations of a fractal artwork and producing new variations, to avoid dealing cumbersome or unpredictable parameters, or collectively, like in the Electric Sheep project, where people use fractal flames rendered with distributed computing as their screensaver and "rate" the flame they are viewing, influencing the server, which reduces the traits of the undesirables, and increases those of the desirables to produce a computer-generated, community-created piece of art.

Many fractal images are admired because of their perceived harmony. This is typically achieved by the patterns which emerge from the balance of order and chaos. Similar qualities have been described in Chinese painting and miniature trees and rockeries.[10]

Some of the most popular fractal rendering programs used to make fractal art include Ultra Fractal, Apophysis, Bryce and Sterling. Fractint was the first widely used fractal generating program.

## Landscapes

A 3D landscape generated with Terragen
Main article: Fractal landscape

The first fractal image that was intended to be a work of art was probably the famous one on the cover of Scientific American, August 1985. This image showed a landscape formed from the potential function on the domain outside the (usual) Mandelbrot set. However, as the potential function grows fast near the boundary of the Mandelbrot set, it was necessary for the creator to let the landscape grow downwards, so that it looked as if the Mandelbrot set was a plateau atop a mountain with steep sides. The same technique was used a year after in some images in The Beauty of Fractals by Heinz-Otto Peitgen and Michael M. Richter.

In this book you can find a formula to estimate the distance from a point outside the Mandelbrot set to the boundary of the Mandelbrot set (and a similar formula for the Julia sets), and one can wonder why the creator did not use this function instead of the potential function, because it grows in a more natural way (see the formula in the articles Mandelbrot set and Julia set).

The three pictures show landscapes formed from the distance function for a family of iterations of the form z2 + az4 + c. If, in a light from the sun. Then we imagine the rays are parallel (and given by two angles), and we let the colour of a point on the surface be determined by the angle between this direction and the slope of the surface at the point. The intensity (on the earth) is independent of the distance, but the light grows whiter because of the atmosphere, and sometimes the ground looks as if it is enveloped in a veil of mist (second picture). We can also let the light be "artificial", as if it issues from a lantern held by the observer. In this case the colour must grow darker with the distance (third picture).

## Artists

The British artist William Latham, has used fractal geometry and other computer graphics techniques in his works.[11] Greg Sams has used fractal designs in postcards, t-shirts and textiles. American Vicky Brago-Mitchell has created fractal art which has appeared in exhibitions and on magazine covers. Scott Draves is credited with inventing flame fractals. Some artists, such as Reginald Atkins, create fractal art for relaxation.[3] Carlos Ginzburg has explored fractal art and developed a concept called "homo fractalus" which is based around the idea that the human is the ultimate fractal.[12] Merrin Parkers from New Zealand specialises in fractal art.[13]

## Exhibits

There has been fractal art exhibits at major international art galleries.[14] One of the first exhibitions of fractal art was called Map Art. It was a travelling exhibition of works which originated from researchers at the University of Bremen.[15] Mathematicians Heinz-Otto Peitgen and Michael M. Richter discovered the public not only found the images aesthetically pleasing but that they also wanted to understand the scientific background to the images.[16]

In 1989, fractals were part of the subject matter for an art show called Strange Attractors: Signs of Chaos at the New Museum of Contemporary Art.[9] The show consisted of photographs, installations and sculptures designed to provide greater scientific discourse to the field which had already captured the public's attention through colourful and intricate computer imagery.

## Fractal - Wikipedia

A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension[1] and may fall between the integers.[2]. Fractals are typically self-similar patterns, where self-similar means they are "the same from near as from far"[3] Fractals may be exactly the same at every scale, or as illustrated in Figure 1, they may be nearly the same at different scales.[4][5][2][6] The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.[2]:166; 18[4][7]

As mathematical equations, fractals are usually nowhere differentiable, which means that they cannot be measured in traditional ways.[6][2][8] An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface. [2]:15[1]:48

The mathematical roots of the idea of fractals have been traced through a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century, 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 21st century.[9][10] The term "fractal" was first used by mathematician Benoît 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.[2]:405[7]

There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. The general consensus 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 in great depth[2][4][5]. Fractals are not limited to geometric patterns, but can also describe processes in time.[6][11][3] Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds[12] and found in nature[13][14][15][16][17], technology[18][19][20][21], and art[22][23][24].

## Introduction

The word "fractal" often has different connotations for laypeople than mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception. The mathematical concept is difficult to formally define even for mathematicians, but key features can be understood with little mathematical background.

The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the infinite regress in parallel mirrors or the homunculus, the little man inside the head of the little man inside the head...). The difference for fractals is that the pattern reproduced must be detailed.[2]:166; 18[4][7]

This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractional or fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived. A regular line, for instance, is conventionally understood to be 1-dimensional; if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the curve in Figure 2. It is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. The fractal curve divided into parts 1/3 the length of the original line becomes 4 pieces rearranged to repeat the original detail, and this unusual relationship is the basis of its fractal dimension.

This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable". In a concrete sense, this means fractals cannot be measured in traditional ways.[6][2][8] To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring a wavy fractal curve such as the one in Figure 2, one would never find a small enough straight segment to conform to the curve, because the wavy pattern would always re-appear, albeit at a smaller size, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. This is perhaps counter-intuitive, but it is how fractals behave.[2]

## History

Figure 2. Koch snowflake, a fractal that begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral "bump"

The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the way.[9][10] According to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).[25] In his writings, Leibnitz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them[2]:405. Indeed, according to various historical accounts, after that point few mathematicians tackled the issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters".[10][9][8] Thus, it was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable.[9]:7[10] Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass[10], published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals.[9]:11-24 Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called "self-inverse" fractals.[2]:166

One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand drawn images of a similar function, which is now called the Koch curve (see Figure 2)[9]:25[10]. Another milestone came a decade later in 1915, when Wacław Sierpiński constructed his famous triangle then, one year later, his carpet. By 1918, two french mathematicians, Pierre Fatou and Gaston Julia, though working independently, arrived essentially simultaneously at results describing what are now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors (i.e., points that attract or repel other points), which have become very important in the study of fractals (see Figure 3 and Figure 4).[10][9][6] Very shortly after that work was submitted, by March of 1918, Felix Hausdorff expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have noninteger dimensions.[10] The idea of self-similar curves was taken further by Paul Pierre Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the Lévy C curve.[notes 1]

Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings hardly resembling the image in Figure 3).[2]:179[8][10] That changed, however, in the 1960s, when Benoît Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,[26] which built on earlier work by Lewis Fry Richardson. In 1975[7] Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical Mandelbrot set pictured in Figure 1 captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".[27] Currently, fractal studies are essentially exclusively computer-based.[25][9][8]

Figure 4. A strange attractor that exhibits multifractal scaling

## Characteristics

One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"[2]; this is generally helpful but limited. Authorities disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of self-similarity and an unusual relationship with the space a fractal is embedded in.[2][6][28] [4][3] One point agreed on is that fractal patterns are characterized by fractal dimensions, but whereas these numbers quantify complexity (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns.[29] In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension[7]. It has been noted that this dimensional requirement is not met by fractal space-filling curves such as the Hilbert curve.[notes 2]

According to Falconer, rather than being strictly defined, fractals should, in addition to being differentiable and able to have a fractal dimension, be generally characterized by a gestalt of the following features[4]:

• Self-similarity, which may be manifested as:
• Exact self-similarity: identical at all scales; e.g. Koch snowflake
• Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the Mandelbrot set's satellites are approximations of the entire set, but not exact copies, as shown in Figure 1
• Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals; the well-known example of the coastline of Britain, for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines, for example, the Koch snowflake[6]
• Qualitative self-similarity: as in a time series[11]
• Multifractal scaling: characterized by more than one fractal dimension or scaling rule
• Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have emergent properties[30] (related to the next criterion in this list).
• Irregularity locally and globally that is not easily described in traditional Euclidean geometric language. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls"[1].

As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimension, and is fully defined without a need for recursion.[6][2]

## Common techniques for generating fractals

Figure 5. Self-similar branching pattern modeled in silico using L-systems principles[17]

• Strange attractors – use iterations of a map or solutions of a system of initial-value differential equations that exhibit chaos (e.g., see multifractal image)

## Simulated fractals

Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space. Models may simulate theoretical fractals or natural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal analysis. Some specific applications of fractals to technology are listed elsewhere. Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals.

Modeled fractals may be sounds[12], digital images, electrochemical patterns, circadian rhythms[33], etc. Fractal patterns have been reconstructed in physical 3-dimensional space[20]:10 and virtually, often called "in silico" modeling[32]. Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above.[20][6][11] As one illustration, trees, ferns, cells of the nervous system[17], blood and lung vasculature,[32] and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques[17]. The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects. A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithm.

## Natural phenomena with fractal features

Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees.[34]

Examples of phenomena known or anticipated to have fractal features are listed below:

• coastlines
• snow flakes
• various vegetables (cauliflower and broccoli)
• animal coloration patterns.
• heart rates[13]
• heartbeat[14]

## In creative works

A fractal that models the surface of a mountain (animation)
Further information: Fractal art

Fractal patterns have been found in the paintings of American artist Jackson Pollock. While Pollock's paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work.[24]

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns.[39] It involves pressing paint between two surfaces and pulling them apart.

Cyberneticist Ron Eglash has suggested that fractal-like structures are prevalent in African art and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.[23][40]

In a 1996 interview with Michael Silverblatt, David Foster Wallace admitted that the structure of the first draft of Infinite Jest he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (aka Sierpinski gasket) but that the edited novel is "more like a lopsided Sierpinsky Gasket".[22]

## Applications in technology

Main article: Fractal analysis

### Fractal-generating programs

There are many fractal generating programs available, both free and commercial. Some of the fractal generating programs include:

Most of the above programs make two-dimensional fractals, with a few creating three-dimensional fractal objects, such as a Quaternion. A specific type of three-dimensional fractal, called mandelbulbs, was introduced in 2009.

## List of fractals by Hausdorff dimension - Wikipedia

According to Falconer, one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension.[1] Presented here is a list of fractals ordered by increasing Hausdorff dimension, with the purpose of visualizing what it means for a fractal to have a low or a high dimension.

## Realistic White Clouds on Blue Sky in AS3 Flash

In this tutorial we show how to use two important methods of the BitmapData class,
perlinNoise and ColorMatrixFilter. By combining the two, we obtain a realistic
dynamic cloud effect. The color schemes of the sky and the clouds can easily be changed.