Tag Archive: fractal

An icy moon orbits a frozen world. You can decide which is which.

mandelbulb 3d fractal landscape example

An Icy Moon in Orbit Around a Frozen Planet

The mountains in the image above are merely a rendered 3D fractal. If you are not familiar with fractals they are never-ending patterns, infinitely complex patterns that are self-similar across different scales. In other words, as you enlarge a fractal you will see the same, or very similar pattern emerge repeatedly ad infinitum as you enlarge the original fractal. This is called expanding symmetry or evolving symmetry. You are already familiar with fractal patterns, the natural world is full of them. Take a tree, as a simple example. The pattern of the trunk and branches and secondary branches is repeated as you move to the scale of branches, secondary branches and twigs. Another example would be the tiny rivulets of rain or melting snow water running onto successively larger flows until they are a large river emptying into a sea. Physical laws dictate that flowing water moves in similar ways at all scales. The same thing is true for coastlines at any scale; similar patterns reappear over and over and over as you look at ever smaller scales. As a result it is practically impossible to actually measure the exact length of a coastline.

The math behind fractals is very complicated, and I don’t understand it anywhere well enough to explain it. Even the experts are not in complete agreement as to how they should be defines. Even Benoit Mandelbrot, who is generally credited with developing the modern idea of self-similarity, fractals, in nature, once characterized them as, “beautiful, damn hard, increasingly useful. That’s fractals.” If you want to impress people at a party with how smart you are repeat Mandelbrot’s 1982 definition: “A fractal is by definition a set for which the Hausdorff-Bescovitch dimension strictly exceeds the topological dimension.”

So, enough with the heavy-duty science. You can find much more about the science of fractals on the Internet. I generally think of fractals as coming in 2D and 3D varieties. I do some work with both types, examples of which have appeared on this blog. 2D fractals are generally backgrounds or some other element; with the 3D fractals often being the mountainous terrain generated by various software packages.

Abstract, 2D fractals can be generated by a computer calculating a simple equation over and over. Here are a few examples, all made with the Apophysis fractal generator:

multicolor apophysis fractal


swirling, multi-colored apophysis fractal

Gravitational Blueshift

orange and green apophysis fractal with a starry background

Quantum Entanglement

multi-colored apophysis fractal

Structural Iridescence

Then there are 3D fractals. I used a free software package called Mandelbulb to create the examples in this post.

This is a very simple one in which the 3D nature is evident:

simple mandelbulb 3d cube fractal

Simple 3D Cube-Style Fractal

Of course, 3D fractals can be much more involved:

blue mandelbulb 3d fractal

A Blue Hatch, A Red Atmosphere

red-toned mandelbulb 3d fractal

Alien War Helmet

And one I like very much – a habitat on a rocky moon or asteroid. The fractal reminded me of the cartoon-like atmosphere of some of the stories by the great Polish science fiction writer Stanislaw Lem, author of Solaris, The Cyberiad, Tales of Pirx the Pilot, and many others. I added a few elements to complete the scene including a spaceship Pirx would have been proud to fly.

modified mandelbulb 3d fractal in memory of stanislaw lem

Stanislaw Lem Station

Science and art! Cool, eh? You can do this too.

A New Toy

Like most artists working with digital media I use several software applications. Sometimes I may use as many as six or seven programs to produce a single finished image; especially if it involves 3D modeling – whatever works to get the job done. One program I have used for quite a few years now is Project Dogwaffle. I began with the freeware version and was very impressed; it was also a lot of fun. There have been many updates over the years and I upgrade every so often. A few days ago I upgraded to the latest version, known as Howler. No single application does everything, but Howler is a very sophisticated, though generally easy to use program. In addition to the standard painting functions that you would expect you can also make animations. Special settings for “painting” foliage are also included and some time ago the ability to generate 3D terrains was incorporated.

I have been tinkering with my new toy and thought I you might interested in the results. There will be more examples and explanations of how I use Dogwaffle in conjunction with other software posted here in the coming weeks. I am always experimenting and trying to push the envelope of what can be done. My friend Philip Staiger who helped develop various versions of Dogwaffle will also probably be posting some of it and additional material at his website, thebest3d.com. He has also made some very good video tutorials. My thanks to him for his recent assistance.

As I said 3D terrain modeling has been a feature for some time and, like the other capabilities, it gets better with every new version. The ability to work both 2D and 3D in the same software makes some things much quicker and easier.

Here is the first image I worked up in Howler. I probably should not have put my copyright mark on it as I, more or less, slavishly followed the steps outlined in a video tutorial by Dan Ritchie who developed Dogwaffle. I really like how Howler handles fog.

Desert Sunset

Desert Sunset

To generate a 3D terrain you need a black and white image where the lighter the shade the higher the elevation; I call this a bump map. Here is a bump map for mountainous terrain. A simple, black and white, overhead image – think topographic map where the lightest shades are the mountain tops and the dark shades are lowlands. These are very easy to make by rendering plasma clouds.

black and white elevation map

Black And White Elevation Map

Here is a screenshot of the same bump map inside Howler’s 3D designer filter. Notice the terrain is a light neutral color. On the right you can see a panel with some of the many settings you can adjust.

screenshot of project dogwaffle howler 3d designer

Draft Terrain Render In Howler 3D Designer

The next step is to apply an image map, a texture, to the terrain. This is a simple image map made by again rendering plasma clouds, adjusting the color and adding a rock pattern to make it more interesting and realistic. As with rendering plasma clouds this is quick and easy to do. For the sake of brevity I left out a couple of steps required to add the snow.

terrain image map

Terrain Image Map With Snow

The same image map applied to the terrain bump map in 3D designer.

image map applies to the bump/terrain map

Image Map Applied To The Terrain

By adjusting a multitude of settings – elevation amplitude, illumination, point of view, etc., etc., etc.,  anyone can make a nice scenic view.

Here is another doodle I made. Except for the Northern Lights, which I made with other software, everything in this image was done with the Howler version of Dogwaffle on the same manner as outlined above. You can also make some very nice Northern Lights in Dogwaffle.

example 3d terrain render in project dogwaffle howler

Aurora Borealis Over Snow-Covered Mountains

One other thing you might find to be interesting.

Here’s a fractal image I made some time ago.

fractal magen david/star of david

Fractal Star Of David

After running it though the same process as above – I was curious what would happen – this is the resulting image with the illumination set to make the relief more visible.

fractal image rendered as terrain

Fractal Image Rendered As Terrain

I definitely recommend Dogwaffle to anyone, from beginners in digital arts, to professionals with many years of experience.


As I have noted before; sometimes you just get lucky. I came up with a random, sort of, fractal the other day resembling a highly-stylized human wearing fur clothing. It even appears that there is a wind-blown fur ruff around the parka hood. I thought it would make a nice addition to my series of postage stamps for an independent Alaska; all I needed to do was drop into an appropriate polar background an add a harpoon based on an actual Inuit weapon. The result is below. I hope you like it. “Tuvaaq” is the word for hunter in Inuktitut, the Inuit language.

Click on image for a full-size view.

Tuvaaq - Hunter


Here’s the fractal without the fancy background.

Tuvaaq Fractal

Fractal Hunter

Both versions of the image are available on a number of items at my Zazzle store.

Jack Frost

Jack Frost making art on your windows. I added a wintery background to a random fractal which resembles ice patterns on a window pane. The glowing blobs of color, elements of the original fractal image, are just a nice plus.

The winter solstice arrives tomorrow.

Click on image for full-size view.

Jack Frost

Jack Frost At Work

Those patterns on your windows during winter appear as water vapor crystallizes. The patterns are caused by slight changes in the temperature of the air, but other factors, related to the formation of snow and other forms of frozen precipitation, also contribute to variations in design.

They often begin when a film of condensed moisture is frozen by a puff of cold air, and the glass gets a thin sheet of ice. Perhaps another film of moisture forms and trickles down, cutting ditches and tiny streams through the underlying ice, while another puff of cold air freezes this moisture. New frozen vapor molecules may add microscopic ice crystals. The freezing and melting process repeats again and again. Bit by bit the subtle weather conditions etch the window panes with patterns.

Like snowflakes, frost needs a nucleus around which to form. Ice crystals form on dust, small bumps and imperfections in a windowpane, or the texture of plants. Then frost grows as more ice crystals form on the ones that have gotten started. It can make fanciful feathery or spiky patterns.

In contrast, when water vapor condenses into water droplets on a cold windowpane and these droplets then freeze slowly, the ice has a different look, as seen in the close-up photo of frozen condensation droplets. The large bubble-like shapes in the photo are frozen condensation droplets, and the many small lines on their surface are cracks that formed as the ice froze and expanded.

Ice Whales

As I have noted before I play around wa bit ith fractal generating software. Every so often a fractal image emerges which resembles something in the real world. This one resembles a swirling group of whales under polar ice; perhaps beluga or minke whales which frequent the poles.

I rendered it with a transparent background and superimposed it on a dark background. I tweaked the colors and added a border. I think it came out really well. I hope you like it.

Click on the image for a full-size view.

A fractal representation of whales under polar ice.


I have not been posting regularly as of late, but I have several wildlife projects in the works as well as some more landscape images from my recent ferry trip. Stand by.

Update – July 6, 2012.

This image received a “Today’s Best Award” from Zazzle.com.

Zazzle Award


I have a software application which generates fractal images. Every so often I set everything to “random” just to see what will happen. Lo and behold, I got an image looking much like the Northern Lights. After adding it to a night scene it looks even better.

Click on image for full-size view.

Fractal rendering of the "Northern Lights"

The aurora borealis shimmers in the northern sky

The aurora borealis (or Northern Lights), named after the Roman goddess of dawn, Aurora, and the Greek name for the north wind, Boreas. Many cultural groups have legends about the lights. In medieval times, the occurrences of auroral displays were seen as harbingers of war or famine. The Maori of New Zealand shared a belief with many northern people of Europe and North America that the lights were reflections from torches or campfires.

The Menominee Indians of Wisconsin believed that the lights indicated the location of manabai’wok (giants) who were the spirits of great hunters and fishermen. The Inuit of Alaska believed that the lights were the spirits of the animals they hunted: the seals, salmon, deer and beluga whales. Other aboriginal peoples believed that the lights were the spirits of their people.

The connection between the Northern Lights and sunspot activity was suspected as far back as 1880. Thanks to research conducted since the 1950’s, we now know that electrons and protons from the sun are blown towards the earth on the ‘solar wind’. (Note: 1957-58 was International Geophysical Year and the atmosphere was studied extensively with balloons, radar, rockets and satellites. Rocket research is still conducted by scientists at Poker Flats, a facility under the direction of the University of Alaska at Fairbanks.

The temperature of the sun’s atmosphere is millions of degrees. At this temperature, collisions between gas molecules are frequent and explosive. Free electrons and protons are thrown from the sun’s atmosphere by the rotation of the sun and escape through holes in the magnetic field. carried eartward in the solar wind, the charged particles are largely deflected by the earth’s magnetic field. However, the earth’s magnetic field is weaker at either pole and therefore some particles enter the earth’s atmosphere and collide with gas particles in the higher layers (thermosphere). These collisions emit light that we perceive as the dancing lights of the north (and the south).

Most aurorae occur in a band known as the auroral zone,which is typically 3° to 6° in latitudinal extent and at all local times or longitudes. The auroral zone is typically 10° to 20° from the magnetic pole defined by the axis of the Earth’s magnetic dipole. During a geomagnetic storm, the auroral zone will expand to lower latitudes. The diffuse aurora is a featureless glow in the sky which may not be visible to the naked eye even on a dark night and defines the extent of the auroral zone. The discrete aurora are sharply defined features within the diffuse aurora which vary in brightness from just barely visible to the naked eye to bright enough to read a newspaper at night. Discrete aurorae are usually observed only in the night sky because they are as bright as the sunlit sky. Aurorae occasionally occur poleward of the auroral zone as diffuse patches or arcs (polar cap arcs, which are generally invisible to the naked eye.

Because the phenomena occurs near the magnetic poles, northern lights have been seen as far south as New Orleans in the western hemisphere, while similar locations in the east never experience the mysterious lights. However the best places to watch the lights (in North America) are in the northwestern parts of Canada, particularly the Yukon, Nunavut, Northwest Territories and Alaska. Auroral displays can also be seen over the southern tip of Greenland and Iceland, the northern coast of Norway and over the coastal waters north of Siberia. Southern auroras are not often seen as they are concentrated in a ring around Antarctica and the southern Indian Ocean.

Auroral displays appear in many colors. Variations in color are due to the type of gas particles that are colliding. The most common auroral color, a pale yellowish-green, is produced by oxygen molecules located about 60 miles above the earth. Rare, all-red auroras are produced by high-altitude oxygen, at heights of up to 200 miles. Nitrogen produces blue or purplish-red aurora. The lights appear in many forms from patches or scattered clouds of light to streamers, arcs, rippling curtains or shooting rays that light up the sky with an eerie glow. Curtain-like structures show field lines in the Earth’s magnetic field .

The auroras that resulted from the “great geomagnetic storm” on both 28 August and 2 September 1859 are thought the most spectacular in recent recorded history. It was reported by the New York Times that in Boston on Friday 2 September 1859 the aurora was “so brilliant that at about one o’clock ordinary print could be read by the light”.

The aurora is thought to have been produced by one of the most intense coronal mass ejections in history, very near the maximum intensity that the Sun is thought to be capable of producing. It is also notable for the fact that it is the first time where the phenomena of auroral activity and electricity were unambiguously linked. This insight was made possible not only due to scientific magnetometer measurements of the era, but also as a result of a significant portion of the 125,000 miles (201,000 km) of telegraph lines then in service being significantly disrupted for many hours throughout the storm. Some telegraph lines, however, seem to have been of the appropriate length and orientation to produce a sufficient geomagnetically induced current from the electromagnetic field to allow for continued communication with the telegraph operator power supplies switched off.

Both Jupiter and Saturn have magnetic fields much stronger than Earth’s (Jupiter’s equatorial field strength is 4.3 gauss, compared to 0.3 gauss for Earth), and both have large radiation belts. Auroras have been observed on both, most clearly with the Hubble Space Telescope. Uranus and Neptune have also been observed to have auroras.

Foxy Loxy

A lucky fractal image which just happens to resemble the head of a red fox. I added a brownish background and just a few brush strokes to finish it off.

Image available on a variety of gifts at my Zazzle store.

Click on image for full-size view.

A fractal image which resembles a red fox.

Red Fox

The red fox (Vulpes vulpes) is both the largest of the true foxes and the most widely distributed carnivore in the world. They are found throughout the entire northern hemisphere from the Arctic Circle to North Africa, Central America and Asia. The species is Eurasian in origin and colonized North America shortly after the Wisconsian glaciation about 10,000 years ago. Its range has increased alongside human expansion, having been introduced to Australia, where it is considered harmful to native mammal and bird populations.

Adult red fox have a year-round red coat that is typically much more striking during the winter months; a washed out orange to cherry red. The red portions cover the head, shoulders and back, and the rump may be either red or a light gray. Jet black marks the legs and ears and the chest and throat are typically a light gray to white.

Red foxes have elongated bodies and relatively short limbs. Males are generally larger than females. Adults may average from 8-12 pounds, and vary in total length from 48 – 57 inches. The tail, which is longer than half the body length (70% of head and body length), is long, fluffy and reaches the ground when in a standing position. Red foxes are very agile, being capable of jumping over 2 metre high fences and swim well. Vixens have three pairs of teats, though vixens with 7, 9 or 10 pairs are not uncommon.

Red foxes have binocular vision, but their sight reacts mainly to movement. Their auditory perception is acute, being able to hear black grouses changing roosts at 600 paces, the flight of crows at 1/4-1/2 km and the squeaking of mice at about 100 metres. They are capable of locating sounds to within one degree at 700–3,000 Hz, though less accurately at higher frequencies. Their sense of smell is good, but weaker than that of specialized dogs.

Apart from its large size, the red fox is distinguished from other fox species by its ability to adapt quickly to new environments and, unlike most of its cousins, is not listed as Endangered anywhere. Despite its name, the species often produces individuals with abnormal colorings, including albinos and melanists. Forty-five subspecies are currently recognized, which are divided into two categories: the large northern foxes, and the small, primitive southern foxes of Asia and the Middle East.

Red foxes are social animals and either establish stable home ranges within particular areas or are itinerant with no fixed abode. Red foxes family units share a joint territory. In favourable habitats and/or areas with low hunting pressure, subordinate foxes may be present in a range. Subordinates are mostly young from the previous year, who aid in rearing the breeding vixen’s kits. Non-breeding vixens will guard, play, groom, provision and retrieve kits, an example of kin selection. Red foxes may leave their families once they reach adulthood if the chances of winning a territory of their own are high. If not, they will stay with their parents, at the cost of postponing their own reproduction.

After a gestation period of about 52 days, females give birth to litters that vary in number from 1 to 12, with 3 – 6 being common. Young are born blind and helpless, and are weaned by week 12 when they learn to hunt for themselves. Both males and females play a major role in food acquisition for growing pups.

Dens are usually found in abandoned woodchuck or other small mammal burrows, widened to suit the needs of a family of foxes. Their basic structure consists of the main channel, with a chamber or a widening of the main channel, lined with grasses and other forbs to make a dry refuge and birthplace for their pups. Dens vary in location, and may be found among the root systems of large trees along the banks of streams or gullies, in or beneath hollow logs or hedgerows, or anywhere a woodchuck might decide to dig their burrow. It is common to find a den with multiple entrances. As temperatures in the den increase with the onset of summer, red fox will move the pups into a new den site every few weeks to minimize exposure to parasites such as fleas.

Red fox are highly mobile and can cover long distances on a daily basis. Travel of greater than 6 miles is not unheard of. Range expansion occurs during the winter months, presumably due to a decreased availability of prey, and contract during the rearing season. Displayed feces and scent posts marked with urine are evidence that red fox are wary of other foxes, and as a result territories seldom overlap.

Primarily nocturnal, red fox may occasionally be seen during the day. The activity of females during daylight hours increases with the feeding demands of growing pups; otherwise daytime is spent resting in regular spots, oftentimes above ground.

Red foxes are omnivores withhas a variable diet, likely coinciding with local prey populations and seasonal availability of small mammals and birds. Small mammals such as mice, squirrels, woodchucks, and rabbits comprise the majority of their mammalian diet, while birds such as grouse, nesting waterfowl, and other ground-nesting birds and their eggs are the most important avian food items in their diet. Other opportunistic food items such as nestling songbirds, various amphibians and reptiles, invertebrates such as earthworms, and carrion are all consumed as the opportunity presents itself. Additionally, red fox have a sweet tooth as they have been noted to consume ripening grapes and apples in the early autumn.

Red foxes prefer to hunt in the early morning hours before sunrise and late evening. When hunting mouse-like prey, they first pinpoint their prey’s location by sound, then leap, sailing high above their quarry, steering in mid-air with their tails, before landing on target up to five metres away. They typically only feed on carrion in the late evening hours and at night. They are extremely possessive of their food, and will defend their catches from even dominant animals. Red foxes may occasionally commit acts of surplus killing; during one breeding season, four foxes were recorded to have killed circa 200 black-headed gulls each, with peaks during dark, windy hours when flying conditions were unfavorable. Losses to poultry and penned game birds can be substantial because of this. Red foxes seem to dislike the taste of moles, but will nonetheless catch them alive and present them to their kids as playthings.

Use of food caches is common for this species. Foraging behaviors most commonly seen include erratic movements in open grassland, head and ears erect searching for the slightest rustle of grass or a glimpse of fur. Once a prey item is located, a fox will freeze, presumably to zero in on the location, followed by a quick aerial pounce and capture of the prey.

The species has a long history of association with humans, having been extensively hunted as a pest and fur bearer for centuries, as well as being prominently represented in human folklore and mythology. Because of its widespread distribution and large population, the red fox is one of the most important fur bearing animals harvested for the fur trade.

Because of their abundance, red foxes are among the most important furbearing animals harvested by the fur trade. Their pelts are used for trimmings, scarfs, muffs, jackets and coats. They are principally used as trimming for both cloth coats and fur garments, including evening wraps. The pelts of silver-morph foxes are popular as capes, while cross foxes are mostly used for scarfs and very rarely trimming. The number of sold fox scarfs exceeds the total number of scarfs made from other furbearers. However, this amount is overshadowed by the total number of fox pelts used for trimming purposes. The silver morphs are the most valued by furriers, followed by the cross and red morphs respectively. In the early 20th century, over 1,000 American fox skins were imported to Britain annually, while 500,000 were exported annually from Germany and Russia. The total worldwide trade of wild red foxes in 1985–86 was 1,543,995 pelts. Foxes amounted to 45% of US wild-caught pelts worth $50 million.

North American red foxes, particularly those of northern Alaska, are the most valued for their fur, as they have guard hairs of a very silky texture which, after dressing, allow the wearer unrestricted mobility. Red foxes living in southern Alaska’s coastal areas and the Aleutian Islands are an exception, as they have extremely coarse pelts which rarely exceed one third of the price of their northern Alaskan cousins. Most European pelts have very coarse textured fur compared to North American varieties. The only exceptions are the Nordic and Far Eastern Russian pelts, but they are still inferior to North American pelts in terms of silkiness.

Fractal Experiment

Here is another fractal image.  I was able to tweak the fractal so that it looks like conifer fronds bound together with straw or some other natural cordage.  Rendered with Apophysis; colors tweaked a bit with Irfanview.  It came out rather well, I think.

Naturalistic fractal image

Fractal Snowbird

For a change of pace I have been experimenting with fractal graphics. Fractals can be quite elaborate, but I generally prefer more simple images. The work below was produced with Apophysis, a freeware fractal generator.

Click on image for full-size view.

Fractal Snowbird