Beautiful Fractals

A fractal is defined by its properties. Two of the most important properties of all fractals are :-

1) self-similarity

2) fractional dimension

Self-similarity means that one part of the fractal is very similar to other parts of the same fractal. This can be seen in most fractal art . . . for example the fractal image above is a spiral made of smaller similar spirals, and each of those smaller spirals is itself made of similar smaller spirals, and so on, ad infinitum.

Start with a straight line . . . that has one dimension. Then make the line increasingly twisted in more and more complex ways . . . if the line was infinitely twisted it could fill an area and would thus be two-dimensional. Because of the principle of self-similarity (infinite complexity), a fractal line is part-way between one and two dimensions, so it is a fractal line that is on the way towards filling a space, because the wiggles on the line themselves have smaller wiggles, and those wiggles in turn have smaller wiggles and so on.

This might seem like mathematical abstraction but it has very practical results. For example, take the coastline of an island . . . look at it from far away and lay a piece of string along the coastline, and you will arrive at a length for that coastline. Then zoom in and you will see that where the coastline appeared to be a simple shape from far away, the line along the coast has a lot more detailed wiggles the closer you get. You could continue this increasing detail down to grains of sand along the coastline, and if you lay the piece of string around all the details, you get a LONGER measurement than you did from the initial far view!

There is a border between two countries in Europe that the two countries measure more than 10% differently, because they measure it at different scales!

The property of self-similarity is a very useful one, because it means that the data which represents a fractal need only define the basic motiv once, then define how that motiv is changed when it appears at the next scale down (in practice, the basic motiv is defined BY its relationship to its other versions). So an infinitely complex fractal can be represented by a very small amount of data. This is used in practice in JPEG image compression which uses a type of IFS fractal formula to compress images effectively.

It is a common misconception that fractals are "chaos". While this is true in a few examples (such as white-noise, which is a fractal), the majority of fractals lie on the border between chaos and order.

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The plant above shows obvious fractal self-similarity . . . if you zoom in to a small part of the center, it is a similar shape to the whole plant.

The sound of a waterfall is another example . . . slow the sound down and it sounds the same, rather than different, which is self-similarity again.

Most of the natural world is made of fractals . . . planet orbits, galaxy shapes, cloud forms, plants, ocean waves, etc. are all mostly fractal in essense.

This can even be found if one researches how natural structures are made. For example, the difference between a flat oyster-shell shape, and a tightly-coiled spiral shell, is defined by the angle at which one particle of the shell is joined to the next. Genetics stores the information about that angle, and so can make either shell with just a small change of one piece of data. It used to be thought that the amount of data represented by genes is not enough to build an animal or plant, but if one realises that the construction is done using fractals, the genes can easily hold enough data to represent infinitely complex shapes.

Nature is made of fractals . . . and so our human body senses are specifically suited to appreciating fractals! That's why an infinitely complex fractal image is at the same time intuitively "understandable" by the human brain, so we relate to them easily. This also explains how man-made constructs which are based on linear math rather than fractal principles, are so easily percieved to be boring. This even causes psychological problems . . . attention deficit disorder is really due to a lack of fractals in an environment . . . straight lines are not what the brain is designed to appreciate!

Many nature photographs (and paintings etc.) are really fractal art! Spot the self-similarity in the images below . . .



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Trees are fractal forms.

The great trees of the forest spreading their branches to catch the sunlight above. Each branch a variation on the form of the whole tree. The dappled patterns of the light as it filters through the canopy.






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Another obviously fractal tree.

The sun rising through the mist sillhouettes the great tree, beautiful and iconic, spreading its ever-thinning array of organic curves twisting and multiplying through the space it pervades.







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Fractal clouds

The white clouds billow and float in the ambience of the warm dusk, their shapes related yet each profoundly unique and beautiful in their ever-changing evolution, inhabiting a vast space only seen through their etheric transient existance.

Computer software which can generate fractals has been around since the mid 80's (and is relatively easy to write for oneself), because an infinitely complex fractal can easily be generated by a simple algorithm.

See below for free fractal software.

These 3 fractals are typical of early fractal works in the last 80's or early 90's . . .


Spirafract2 by James_Falconer
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One of my own Fractint fractals.

The infinite spiral, complex and beautiful, like an alien chain stretching forever into the cosmos.





Five-star-fractal by James_Falconer
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Another Fractint image of mine.

Fascinating organic branches, reminiscent of a "brittle star" starfish, twisting towards the trident end-formations, glowing with warmth and the energy of growth, sending out feelers into its surroundings.





Fractal beauty by James_Falconer
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A spiral fractal I made using Fractint.

Twisting formations spiralling inwards to the black hole, bright with the vibrating life-force of infinitely detailed growth.

There are many different types of fractals. The first well-known fractal ever discovered is still one of the most popular types. This is the Mandelbrot set, the algorithm of which was presented to the public in 1985 by Scientific American. The two images above, "Fractal beauty" and "five-star fractal" are both derived from the Mandelbrot set (or its Julia inverse).

The Mandelbrot set is computed by seeing how quickly the iteration of a particular algorithm leads to infinity (or stability) . . . the x and y axes of the image are parameters which are put into the equation, and the colors are representations of what happens to the algorithm when it is iterated with those particular parameters.

The inverse of the Mandelbrot set is the Julia sets, which are also very well known in fractal imagery. For more on this, please see Mandelbrot Set at Wikipedia

However, there are many other types of fractals, such as the "3D Strange attractor" (sometimes called "flame fractal") which maps the orbits of points around certain focal points.



Fractal 4, by Sixtonmelon (Giclee print from ImageKind)
This fractal is a typical 3D Strange Attractor.

Orbiting sparks bright in the cold of space.




A lot of abstract art, in particular "organic abstract" art (see this great Lens by my friend Chris Melchior), is fractal in nature.

It is also interesting to see how the use of fractals in art is part of the long-term progression in the Western Arts . . . for example this can be seen in music with the ambient revolution which is a quantum change from the last thousand years of Western Art Tradition music (which for a thousand years up to the ambient revolution was about NOTES rather than SOUND, ie. symbols rather than the reality they represent), and in the progression to abstract art which began with Turners works which were really abstracts (with a small recognisable element).

For anyone who's interested in more details in the ambient revolution, here's an excellent article on it Ambient music

Fractals are the main contept that the natural world actually uses, and after a thousand years of progression through increasingly sophisticated use of SYMBOLS, the Western Art Tradition is now somewhat embracing the conceptual reality of the physical-universe itself rather than the mere symbolisation of it. It's a fasctinating time we live in :)

Yes, YOU can make your own fractals, and sell them, without spending a penny!

Currently there is lots of fractal sofware to choose from. One of the best known, although rather old now, is Fractint, a free program which anyone can use to create their own fractal images. I've just got Chaospro which seems to do similar things to Fractint and more, with an easier to use interface, and it's FREE!

Fractint can even be used to generate amazing video sequences of evolving fractal shapes . . . all it takes is an understanding of some basic high-school math, and some ability with Excel (or similar) and some patience! The results can be amazingly beautiful.

Once you've created some really nice fractals, you can sell them, without any costs at all . . . all the fractals on this page are for sale, on sites such as Zazzle and CafePress which are print-on-demand websites for cards, T-shirts, even shoes and skateboards! Also, Imagekind, another type of Print-on-demand site, offering high quality giclee prints . . . there are free membership options there.

Once you've got products up on such sites, why not make a Squidoo Lens about them, and write articles and/or make videos to get traffi to your lens?

Update . . . I've found yet another great program for creating your own fractals (plus it does LOTS more), and it's FREE . . . it's called GIMP (which stands for GNU Image Manipulation Program), and you can download it at gimp.org. It can create some amazing fractals, including some of the more usual types, a wide variety of "flame" fractals, and IFS. Best of all, it's also a top-of-the-range image editing program, like Photoshop but better overall and FREE! Check it out :))

Bright Ring by James_Falconer
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One of my favourites from my own fractal designs. I wrote the software myself for this one, based on the "3D Strange Attractor" algorithm.

Intuititively simple yet with infinitely complex detail, the smooth flow of the ring juxtaposed with the chaotic vibrations of the joint.





Chinese dragon by James_Falconer
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One of my own image-processed fractals.

The evocative curves of a Chinese dragon, familiar yet other-worldly in its strange alien grace glowing hot in the center while reaching out to the cold dark spaces.






Fractal 6, by sixtonmelon, (giclee print from ImageKind)

Particle storm orbiting gravitational focusses, long loops and intensely chaotic vibrations, cometry paths reaching serenely into the depths of distant space from the bright chaos of the center.





Fractal 725 by KiDAcE
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Bright bubbles, tiny spheres, their force-fields spacing themselves into fascinating patterns in the glowing spaces.





Troglodyte by Kiyoshi
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Crab-like, insect-like, rotating spiral of multiple jointed limbs, beautiful and strange.






Eye of Horus, by LeahMcNeir (giclee print from ImageKind)

Webs of light relating floating patterns in a chaotic space.






With a wink to Rembrandt by AnneVis (giclee print from ImageKind)

Light glinting off translucent rotating forms orbiting a central focus in shimmering movements.






Time Saddle by lucesalegres (giclee print from ImageKind)

Lines of light form curves in space, glowing forms coalese from the relationships between intense directionality.

SpaceFlower by James_Falconer. Create post cards at Zazzle

A spectacular dazzling flower of light evolves from a basket-work of illuminated tendrills.





Fractal 642 by KiDAcEPoster printing by Zazzle

Patterns of patterns in the shimmering brightness of conceptual dimensions.






Tangle of Silk Scarves by AnnG (giclee print from ImageKind)

Stripes and twists, convoluted glowing planes touch and spin.





Convolutions by mistywisp
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Cellular nautilus forms hold smaller nautilus spirals, glowing with bright life and ever-changing evolutions.






Red Bush by therightpic (giclee print from ImageKind)

A red frost-bush against the blues of deep-space night. Webs of organic growths glow in the deep.





Unborn by SophiaDeer
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Fragile, delicate, the growing organic spiral of new life, the glow of life in the dark deep.






Blazing Sky by jennabee25 (Giclee print by ImageKind)

Organic tendrils of deep color reach into the vast fields of misty light.





Genesis by creativ82
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A glowing flower of light reaches upwards, bursting with vitality and bright shimmering potential





Against a blue sky by mistywisp
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The profound beauty of an organic form of delicate light develops in gentle spaces.





Fascinator by mistyw

Dry leaf-like forms grown from spiralling organic formations, their edges catching the light amongst the deep darknesses.

Fractal Art by Vicky. A really great fractal art Lens . . . lots of large fractal art images, including some of my favourites, plus fractal animations, and links to other fractal stuff. Great lens!

Digitalgreenlife fractal art Zazzle Gallery Showcase. Some beautiful fractals, including reflection "landscape" type fractals.

Great Calendar's Fractals Plus. Some nice fractal calenders (I like the "blue fractals" calender with reflected fractal), and other calenders too.

Fractal Art. LOTS of thumbnails of fractal images in Flickr, plus fractal products from other places.

PSYCHEDELIC PSYTRANCE FRACTAL T-SHIRTS & GEAR. Some great T-shirts, powerful symetrical designs, some fractal.

Fun with Fractal Designs. Some nice fractal products, links to fractal software, galleries and tutorials. List of Zazzle fractal galleries (add your own!).

How to make Fractals. Some nice fractal wallpapers, some links to free fractal software.

Wallpaper Fractals - Fractal Art Some links to fractal wallpaper (without images) and fractal books at Amazon.com

Fascinating fractal. Some large fractal images.

Exploring Fractals. Some info on fractals, and fractal software and galleries.

Fractal Graphics Art Posters. Some nice fractal images, posters, books, links, etc.

Digital Abstraction Zazzle Store by spidergod. Lots of fractal products on Zazzle and Ebay, also fractal books on Amazon, and fractal YouTube vids.

My Top 5 Fractal Artists on Zazzle. Lots of fractal art on Zazzle, plus fractal videos, and comments on fractals and Zazzle.

Apophysis : a fractal flame program. Free "fractal flame" software.

R.W. Designs: Fractal Art on Zazzle. Some fractal products, plus links on fractals.