Fractal fonts

The letter A above consists of many Sierpinski triangles, blank spaces of which are colored black. Using of triangular grid to form letter gives some Gothic effect. Font designer Frodo7 created a font Sierpinski Black using such principle, which consist of 87 characters inluding Latin capital and lower letters and numbers. A sample pattern of the font is represented below. Artist created several variations of this font with different kinds of filling of letters.

Frodo7 also created another font based on fractal Fractal font, where letters look like wide tape, which curl into form of lowercase letters. Letters are filled with fractal pattern of black and white rectangles. Sample letters and fractal pattern, which is used for filling letter, are represented below.

Another fractal font was created by Tony de Marco. It is based on simple raster font, where letters are divided into pixels. Every pixel is separate fractal of regular located circles.

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If you know another fonts based on fractals, please let me know, I will happy to update this post.

New set of fractal trees - 2

Today, I represent a small but nevertheless interesting set of fractal trees of various kinds.

Two mystical fractal trees by Jacob Ankney (Jeddaka).

A new variation of Pythagoras tree with curl as base figure for duplication in fractal, which was created by IDeviant.

And, in conclusion two fractal images from Shutterstock. The first can be bought here, the second is no longer represented there.

Fractal architecture

Not only abstract images can be created with fractals, but also very impressive images of strange towers and temples. Some time before I posted fractal image of a fractal temple. Now, the next some surrealistic images of fractal towers are represented below.

Medieval fractal (by Ramiro Perez)

Sunset Castle (by Ramiro Perez)

Ivory Tower - 2 (by Stefan Vitanov)

Airy (by Stefan Vitanov)

More images of fractal architecture can be found at

http://rfractals.net/?page_id=64 (Ramiro Perez)

http://www.fractalexperience.com/show_gallery.php?gallery=18&range=5&page=1 (Stefan Vitanov)

Mandelbulb

The Mandelbrot set is one of the most known fractal. It can be seen on many sites and images over the Internet. It was represented in many variations. Today, many fractal artists create beautiful images, which are based on it. But all this time it remained only a two dimensional fractal.

Of course, many artists created 3D images with it. Below you can see the Mandelbrot fractal (to the left) and such pseudo-3D. But as you can see, the figure to the right is the same 2D fractal, in which values of the fractal are representes as levels above the base plane.

It remained two-dimensional until Daniel White and Paul Nylander constructed a three dimensional analog of the Mandelbrot set, using an hypercomplex algebra based on spherical coordinated, when in classic Mandelbrot set algebra of complex numbers were used. They called their creation Mandelbulb.

You can read about history of creation of Mandelbulb and many images of it at http://www.skytopia.com/project/fractal/mandelbulb.html. Here, there are very large image of the fractal wholly and images of some it's parts.

The Mandelbulb opended the door to a new class of wonderful fractals.

Artforms of Nature and fractals

The nineteen century German biologist Ernst Haeckel is famous for his fantastically illustrated book Artforms of Nature. The copyright for this book from 1904 has now expired and thanks to Wikimedia Commons it is available for everyone to appreciate.

Haekel's artistic interpretation of the biological forms he studied have a clarity of symmetry and detail that has been a source of inspiration for many artists and engineers over the years. They provide the perfect subject matter for Photoshop plugin Pixel Bender Fractal Explorer. The plugin was created by Tom Beddard.

Below, you can see the original image of Ophiodea (to the left) and fractal based on the original images, which created with help of Pixel Bender Fractal Explorer. The fractal has the same symmetry as the original image.

Another example, the Phaeodaria and two fractals, which are based on it. Fractals have another type of symmetry, but, nevertheless, they are look very natural.

The Great Wave off Kanagawa

Some time ago, talking about fractal waves I referred to the artwork of XIX century The Great Wave off Kanagawa by japanese artist Hokusai. It was very interesting to see in old artwork selfsimilar shapes like in fractals. This artwork is of great importance to Japanese.

So, it was printed on postage stamps and became inspiration for many designer works. Even more it's motif was used in Nintendo game The Demon Blade. It's 2D adventure game with many traditional Japanese scenes. Two of them show the Great Wave in sunny day and in storm.

Another painting based on the famous artwork is The Great Monkey Wave. From afar it looks like an original image but nearby we see that the wave is filled with numerous monkeys.

It was found at http://www.thinkforward.org/gallery_mosaics2.htm .

New set of fractal trees

A new set of fractal trees open images by Nirolo. Nice gradients give mystical glow to trees. Probably, these trees should grow in the foresests of the elves. Curved branches look very unusual for fractals.

Fairy Tree

A tree of Fire and Ice

The next tree is like a lightning. Branches have rough outline like real lightning. But it still variation of Pythagoras tree. The image was created by Cory Ench. Author also created several variations of this fractal.

The last image show us apocalyptic sunset with a fractal tree of very regular form. The image was created by George13.

Cubic pyramid

A nice pyramid of cubes was modeled and rendered by David Pearson (fpsurgeon). Each lower level of the pyramid consists of smaller cubes, which are four times more than above. So we see a simple but elegant fractal. This nice rendering was created as a further development of studing by author an open source program Structure Synth, which provides creating 3D structures from a set of user rules. One of preliminary studies you can see below.
Imagine if the top cube of the pyramid is too heavy, and the whole structure will collapse.

Fractals by Manny Lorenzo

Some time ago I've published one fractal by Manny Lorenzo in the post about Pythagoras trees. Today I would like to show some his fractals.

Let start from new kinds of Pythagoras trees. He created a tree with spheres and a very strange kind of Pythagoras tree, which has infinity symbol as a base part of fractal. It looks unlike Pythagoras tree, but it is true.

Also he created a set of wonderful fractals of Nautilus shells. Nautilus shell is one of known examples of fractals in nature. 

Except these ones he created another fractal shells.
Visit Manny's Flickr phostostream to see more beatiful fractals.

Fractal tilings

Fractal shapes can be used as tiles for filling plane. In most cases variations of the Koch snowlake are used. A simple Koch snowflake is represented to the right. To create a set of tiles, which can be used for filling whole plane, we need another variations of the snowlake which exactly match to all convexes and concaves of the first figure.

Two variations of such kind of tilings were represented on Briges 2008 conference, which took place in Leeuwarden (Holland). Both were realized in wood. We see that several kinds of fractal tiles were used in both cases. 

Koch tiling by Edmund Harris

Pentagonal Koch tiling by Chaim Goodman-Strauss

Images were found in Briges 2008 - Leeuwarden pool at Flickr.

More mathematical issues about fractal tiles you can read in the article about Rauzy fractal.

Fractal waves

On the image above we see artwork "The Great Wave off Kanagawa" by Japanese artist Hokusai, which was published in 1832 as the first in Hokusai's series 36 Views of Mount Fuji. It depicts an enormous wave threatening boats near the Japanese prefecture of Kanagawa; Mount Fuji can be seen in the background. The main reason of publishing this artwork here is highly detailed painted wave. As we know, some artworks, which are close to fractal images by detailed elaboration, were created long before the inventing fractals by Benoît Mandelbrot. Sea waves can be represented by many types of fractals, as you can see below.
Fractal artworks with waves are created by modern artists too. For example, Robert Fathauer depicted a wave fractal with Escher-like fish tiling in his artwork "Fractal Fish - Grouped Groupers" (2001).
Also, a wave with horses figures are depicted on the cover of English rock group Keane "Under The Iron Sea" (2006).

Menger sponge

As the Sierpinski carpet is a generalization of the Cantor set from one dimension into two dimension, the Menger sponge is a generalization of the Sierpinski carpet into three dimensions. Sometimes this three-dimensional fractal called Menger-Sierpinski sponge or Sierpinski sponge. It was first described by Austrian mathematician Karl Menger in 1926.

Like the Sierpinski carpet begins from square, Menger sponge begins from cube. Every face of the cube is divided into 9 smaller squares. This operation divide the cube into 27 smaller cubes. Then center cubes from all faces and the inner center cube are removed, leaving 20. This is a level 1 Menger Sponge. The next levels forms by repeating these steps to all 20 cubes rest. Below you can see first four levels.

Below you can see the Menger sponge with cut off corner, which was designed by Seb Przd.

There's also similar three-dimensional fractal based on tetrahedron, which is a generalization of the Sierpinski triangle into three dimensions. Below, you can see two versions of the Sierpinski pyramid and the Menger sponge in a single image.