What happens when you repeatedly square a complex number and color-code how fast it runs away to infinity - a beautiful fractal structure on the real-imaginary plane. you get a different pattern for different starting points (this one starts at 0.285+0.01i). Each square = 0.01+0.01i
Transfer this onto watercolor and we're ready for open studio weekend.
The geometric patterns on sea
shells are one of the most intricate and colorful found anywhere in nature. The
Textile Cone sea snail for example is called the “Cloth of Gold” cone (above) for good reason. How did it come to be? What natural processes in the snail's development create such beauty?
The answer lies in the behavior of pigmentation cells on the mantle’s edge as the animal grows in size.
In effect, shell patterns are a recording of the neural activity in the snail's mantle
during growth.
Anatomy of the Textile Cone
We will skip the mathematical model describing neural activity in the mantle and go straight to the net observed effect. Pigmentation on the cells in the growing edge switch on (black) or off (white) based on the state of its three
neighbors (cell above left, above and above right)
Pigmentation patterns of the next generation of cells on the growing edge of a snail's mantle
These set of rules, when repeated over time, causes the following pattern to emerge on the shell:
Resultant pattern on the shell
The rule set described above belongs to a class of systems known as Cellular Automata (CA). CAs were first identified and described by John Von Neumann and Stanislaw Ulam in the 1940s and have since been observed in several fields including physics, biology, economics and sociology. Stephen Wolfram developed a classification scheme for CAs in his 2002 book A New Kind of Science.
The example described above is known as "Rule 30" Cellular Automaton. It's easy to see why it's "Rule 30" once we replace "On" and "Off" with 1s and 0s.
Rule 30
00011110 in binary = 30 in decimal. You can play with all possible combinations, from 00000000 to 11111111 (0 to 255 in decimal). Here are some interesting "rules" and patterns:
Cellular Automata Rules and Resulting Patterns
These patterns are classic examples of emergence. Emergence is the phenomenon
by which entirely new properties or behavior arises from simple local
interactions. In other words, the sum is greater than its parts.
Emergent systems are
characterized by irreducibility – the collective behavior cannot be reduced to
a single property or interaction. This is because in emergent
systems, causation is iterative i.e. the effects are also causes. For this reason
emergent behavior cannot be predicted mathematically. You’ll need to actually
run the experiment to learn the outcome.
Nature is full of emergent phenomenon– snow flakes, ant colonies, sand dunes, bird flock, even the weather.
Examples of Emergence in Nature
Are you an emergent phenomenon? the hundred billion cells in your brain somehow interact with each other to produce consciousness - with which you can read and understand this sentence. Amazing, isn't it?
References
1. Algorithmic Beauty of Sea Shells by Hans Meinhardt
2. Geometry and Pigmentation of Sea Shells by S. Coombes
3. A Model for Shell Patterns based on Neural Activity, Brad Ermentrout, The Veliger, April 1986
I've been experimenting with fractal geometries in origami recently. Turns out paper is the perfect medium to explore fractal concepts like recursion and emergence. Below is a fractal with 4 levels made with Arches watercolor paper (140 lb cold pressed - pick your size). The color and texture seem to work well with the form, not stealing the focus away from the concept itself.
"I could be bounded in a nutshell and count myself a king of infinite space"
-William Shakespeare, Hamlet, 1603
Pick up a strip of paper, twist it one full turn, then tape the ends together. You've got a Möbius strip.
This seemigly simple object has a peculiar topological property - it has only one edge and only one side!Try coloring the edges in two different colors - it cant be done. A two-dimensional creature on the surface of the Möbius strip (say, for argument's sake, an ant with zero height) will perceive the surface as one infinitely long strip.
A Möbius Strip
The Möbius Strip has fascinated artists and mathematicians alike for a long time. What follows is a visual survey of the artwork this humble strip has inspired around the world.
Möbius Staircase - Nicky Stephens
Max Bill - Kontinuität outside Deutsche Bank's Headquarters in Frankfurt
At the Fermilab, Batavia, IL
Topological III - Robert Wilson, Harvard University, Cambridge, MA
Moebius Strip I (1961) - Maurits Cornelis Escher, wood engraving and woodcut in red, green, gold and black, printed from 4 blocks
Moebius Strip II (1963) - Maurits Cornelis Escher, woodcut in red, black and grey-green, printed from 3 blocks
LEGOMöbius Strip, Andrew Lipson
Möbius Gear, Concept by Tom Longtin
Aaron Hover's 3D printed model of Tom Longtin's concept
At this point I'm tempted to talk about Trefoil knots, Klein Bottles and higher dimensional non-orientable surfaces, but I will save them for another post!
“Composition in Red, Blue, Yellow”, Piet Mondarin 1937-42
“Number 1A”, Jackson Pollock, 1948
Both these paintings are at the Museum of Modern Art in New York. Mondarin’s painting (first one above) is simple, clean and structured. Pollock’s on the other hand, is tangled, messy and complex. Which one do you like better?
Mondarin spent weeks contemplating the precise arrangement of his lines. He was considered an intellectual of his time, who wrote thoughtful essays about his paintings. Pollock, on the other hand, simply rolled out his canvas on the floor of his studio and ran around dripping household paint on it from a stick, in what seemed to be a completely arbitrary process to observers (a video documenting his process below). Pollock was mostly drunk and didn’t bother justifying his work. Once he was quoted as saying “my concerns are with the rhythms of nature”.
Pollock's Process
In 1999, a U. Oregon physicist named Richard Taylor discovered that Pollock’s paintings are indeed fractal shapes[1]. It turns out Pollock was creating fractal patterns with his seemingly arbitrary painting style.
Fractals, of course, are the self-similar patterns found everywhere in nature. These patterns repeat themselves at different levels of magnification. Think cauliflower, tree branches, snails, lungs, snowflakes.
Romanesco Broccoli
Mandelbrot[2] showed in 1975 that fractals can be expressed as self-similar mathematical functions. For e.g. the Julia set
$$z = z^2 + c$$
produces the following plot on the real-imaginary plane when c takes a complex value $$c = -0.8 + 0.156i$$
Example Julia Set
Fractals occupy the bizzare world of fractional dimensional values, or the “roughness” of a fractal. In our familiar world of Euclidian geometry, points have a dimensional value D = 0; lines have a dimensional value D = 1; surfaces have a dimensional value D = 2 and solids have a dimensional value D = 3. Fractals have dimensional values that fall in between these whole numbers, i.e. fractions, sometimes even irrational numbers (!).
For example, the cantor set (below) has a fractal dimension D = 0.63, which means that its behavior is somehwere between a point and a line. The Koch curve at D = 1.26 behaves more like a line, whereas a Sierpinski carpet at D = 1.89 behaves more like a surface. The Menger Sponge, at D = 2.73, is 'almost' a solid.
Cantor Set (D = 0.63)
Koch Curve (D=1.26)
Sierpinski Carpet (D = 1.89)
Menger Sponge (D = 2.73)
The most common way of measuring the fractal dimension (also called fractal density) of an image is the “Box Counting” method. Roughly, the value of D can be obtained by dividing up the image into smaller and smaller squares, comparing N(L)the number of occupied squares to Lthe size of the square. For fractal behavior, N(L)should scale according to the relationship
$$N(L) \backsim L ^ {-D}$$
Natural scenery, too, come in varying values of D. Typical fractal dimension values found in natural scenery summarized in the table below. Now, Taylor surveyed 120 people to see what patterns they preferred, 113 of them chose fractals over non-fractals. Even in fractals, it turns out that people prefer images with a dimensional value of around 1.3.
Natural Pattern
Fractal Dimension D
Galaxies (modeled)
1.05 – 1.25
Cracks in ductile materials
1.25
Geothermal rock patterns
1.25-1.55
Woody plants and trees
1.28-1.90
Waves
1.3
Clouds
1.3 – 1.33
Snow flakes
1.7
Retinal blood vessels
1.7
Electrical Discharges
1.75
Common D values in Natural Scenery
Fractal Density vs. Aesthetic Preference
It's remarkable that Pollock was creating fractal patterns in his paintings long before fractal geometry was officially “discovered” by Mandelbrot in 1975. Taylor also analyzed several Pollock paintings for D values and discovered that as years went by, the complexity, i.e. fractal density of his paintings went up as well. Here is a plot of D over time for Pollock’s paintings[5]:
Fractal Density in Pollock's paintings plotted against the year in which they were painted
Is this the secret to the aesthetics of Pollock’s paintings?
The other interesting question is, are we conditioned evolutionarily to appreciate images of a particular fractal density, because that’s what we see all around us?
Also, how do we expand this intuition to three dimensional objects? Do humans have an aesthetic preference for fractals in three dimensions as well i.e. solids?
Only one way to find out. In the next few months, I'm hoping to repeat Taylor's experiment for for fractals less than 2 dimensions. At the same time, I will attempt to design an experiment to test aesthetic preferences for solids as well. More to come!
