Archives for the month of: February, 2013

Eadweard Muybridge (1830–1904), a brilliant and eccentric photographer, gained worldwide fame photographing animal and human movement imperceptible to the human eye. Hired by railroad baron Leland Stanford in 1872, Muybridge used photography to prove that there was a moment in a horse’s gallop when all four hooves were off the ground at once. He spent much of his later career at the University of Pennsylvania, producing thousands of images that capture progressive movements within fractions of a second. I have been lucky enough to see this work in exhibition. Since viewing the exhibition I have noticed the increasing influence Muybridge has had upon my own work. In this instance I will be mimicking his techniques in almost all of my own photography. I believe a Muybridge’s studies in motion to have directly influenced me in my choice to shoot and exhibit my work in stop motion.

The Horse in Motion by Eadweard Muybridge.

Although Eadweard Muybridge thought of himself primarily as an artist, he encouraged the aura of scientific investigation that surrounded his project at the University of Pennsylvania. Published in 1887 as Animal Locomotion, the 781 finished prints certainly look scientific, and historically, most viewers have accepted them as reliable scientific studies of movement. The recent rediscovery of Muybridge’s working proofs, however, demonstrates that he freely edited his images to achieve these final results.

The Horse in Motion by Eadweard Muybridge. &qu...

The Zoopraxiscope.

The zoopraxiscope is an early device for displaying motion pictures. Created by photographic pioneer Eadweard Muybridge  in 1879, it may be considered the first Movie Projector. The zoopraxiscope projected images from rotating glass disks in rapid succession to give the impression of motion. The stop-motion images were initially painted onto the glass, as silhouettes. Some of the animated images are very complex, featuring multiple combinations of sequences of animal and human movement. This creation of Muybridge truly caught my attention and imagination upon viewing, I now have the desire to incorporate this into my own work, although it seems unlikely it would be possible or suitable to exhibit this project.

Simulation of a spinning zoopraxiscope

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In a 1904 letter to Emile Bernard, Paul Cézanne wrote, “everything in nature is modeled according to the sphere, the cone, and the cylinder. You have to learn to paint with reference to these simple shapes; then you can do anything.”

In contrast, in the Fractal Geometry of Nature Benoit Mandelbrot takes a different point of view: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

To the extent that the arts are informed by nature, we should not be surprised to find fractal aspects in the visual arts. The extent of this presence, much of it largely unconscious, may be a surprise. Fractals may have become a cliché in modern computer graphics, but they have a long and rich history in art

Although not being defined or labelled until 1976 fractals have been observed  throughout history, manifesting themselves visually throughout different cultures through artefacts and artworks resonating strongly throughout many if not all major world cultures and religions as shown here.

 

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This an image from the Medieval Celtic Book of Kells (597 A.D)

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These are black and white Persian Rug examples

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The above image is a birds-eye architectural plan of an ethiopian village. fractals

are evident throughout African art and culture, however this was my favourite.

islam

The image on the left is a traditional Islamic artwork, whereas the image on

the left is a computer generated rendering of the Mandelbrot set.

The similarities are obvious and evident.

eygption

The above depicts and ancient Egyptian symbolism, again a fractal.

 

judaic

This symbol is a well known Jewish icon.

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For me, most interestingly of the religious symbols is this one from

Hinduism

 The Italian renaissance and dutch matters even took influence from unknowingly observing fractals in nature. by using a method of iteration with the desired material the artist were able to create increasingly realistic natural forms.  For example The Dutch painter Jan Van Goyen (1596-1656) was capable of creating realism a picture with “small efforts” from stains of colors. Compare Van Goyen’s painted clouds with the typical fractal computer-generated “cloud”

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Left is a section of a painting of Goyen’ Two men on a Footbridge

over a stream. To the right is a cloud generated by an iteration computer

system.

The next number of  posts will be regarding artists that have not only worked around the concepts of fractals but also artists that have influenced me within my own work.

 

 

 

 

 

Natural objects exhibit scaling symmetry, a key characteristic associated with fractals. They also tend to be “roughly” self- similar, appearing more or less the same at different scales of measurement.         Sometimes this means that they are statistically self-similar; that is to say, simply put they have a distribution of elements that is similar under magnification. Fractals are unpredictable in specific details yet deterministic when viewed as a total pattern – in many ways this reflects what we observe in the small details & total pattern of life.  One has to see rather than just look to notice the amount of attention that is present in the smallest of details. Below are a small number of my favorite natural occurring fractal images.

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17-salt-pile-shoreline_9359_990x742

fractal_5a    fractal_6a

IDL TIFF file

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fractal_13

lightning-gallery-14

This short video is  a  movie inspired by numbers, geometry and nature by Cristóbal Vila. It explains beautifully and simply the relationship between fractals, numbers and nature. the accompanying music is Wim Mertens – Often a bird which compliments the visuals brilliantly.

Koch Snowflake

The Koch Snowflake is a geometric fractal (fractal produced by self-similar geometric manipulations)

This problem was made by taking an equilateral triangle and dividing each of the three sides into three equal segments, then placing another equilateral triangle facing outward with the base forming each middle segment. The middle segment is then removed. This process is continued and can be contented infinitely.

VonKoch

The below diagram shows how the Koch Snowflake is formed, starting from a single equilateral triangle

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Sierpinski Triangle

The Sierpinski Triangle is one of the most well known and familiar fractals. It is also a geometric fractal (like the Koch Snowflake). Also, both the Koch Snowflake and the Sierpinski triangle use an equilateral triangle as the base.

sierpinski.clear

This fractal is made by starting with an equilateral triangle, and connecting the midpoints of each of the sides to form 4 corresponding equilateral triangles. This process is then repeated for each of the triangles, and each of those triangles, and so on, creating the shape shown above. The first 4 iterations of this process are shown below.

sierpinskytrianglewikipedia

 

Sierpinski Curve

The Sierpinski Curve is a base-motif fractal that looks exactly like the Sierpinski Triangle after an infinite number of iterations. The base for the curve is a straight line at each iteration, the lines are replaced by the motif.

s_arrow_curve2

 Because the Sierpinski Curve is a sweep fractal, the motif is flipped for every other segment before the substitution. Although the curve clearly has two endpoints, the sum of the lengths of the segments forming the fractal is infinite. This is because the motif is 1.5 times longer than the base, which makes the fractal infinitely long after infinite number of iterations.

400px-Arrowhead_curve_1_through_6

 

Cantor set

The cantor set was a mathematical concept that was introduced by the German mathematician Georg Cantor in 1883. This the first Mathematical Monster that I personally encountered, the cantor set starts out as a set of infinite points on a line segment, but with the recursive subtraction of the middle third of each line segment, it eventually becomes more sparse. During the 1980s at IBM, such concepts as the cantor set demonstrated real world applicability when a wide range of disruptive frequencies prevented the transmission of computer data over telephone lines. When simulating the frequencies to generate this noise, mathematicians such as Mandelbrot discovered that within these frequencies there were other smaller frequencies of the same form. This self-similarity only meant one thing — fractals were responsible.

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Julia Set

The Julia set is a set of complex numbers. It is far more complicated than the other fractal mentioned above. It is one of those fractals that cannot be described as a base motif fractal. The Julia set it created by plotting the elements of the set on the complex plane. As with many other fractals the Julia set is often depicted with bright colourings. The colouring is not part of the definition of the fractal. Julia sets are named after Gaston Julia. He was a French mathematician who discovered Julia sets and first explored their properties.

Julia_set

The Link Below is to a short 50-minute documentary. In this film, Hunting the hidden dimension, NOVA takes viewers on a fascinating quest with a group of maverick mathematicians determined to decipher the rules that govern fractal geometry.

From the trees to the respiratory system within your body, fractals exist all around us. Clouds, lightning, and coastlines – these are all examples of fractals. Even we are comprised of a multitude of fractals. Fractals are a part of everything. For centuries, people have recognized nature’s tendency to repeat itself throughout its designs and patterning’s. Take Hokusai’s The Great Wave off Kanagawa for example. One of my all time favorite paintings.  Created with the traditional Chinese woodblock style, you can see within the paining the repeating patterns at the tips of the giant wave as it crashes down on the boats. Within the crest of the large crashing wave, we can see smaller cresting waves… and smaller still from those.  Although mathematicians, artists, scientist and great thinkers throughout history have noticed these patterns, they were unaware of the brilliance they were viewing, Fractals

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Hokusai’s Great Wave

Fractals: A Brief History 

 Antennas, fractal compression, understanding the stock market, and designing more efficient computers have all be possible and dramatically improved because of our understanding of fractals. Before French-American mathematician Benoit Mandelbrot launched his branch of fractal mathematics, our understanding of how things function and why things repeat themselves was limited. 

Whilst-contemplating algebra one day Mandelbrot claims he began seeing vivid, geometric images in his mind. He soon realized that he had actually transformed algebraic formulas and equations into the pictures he was seeing. However, he kept this discovery to himself, for he figured it would not help him in the long run.

 Classical mathematics had its roots in the regular geometric structures of Euclid and the continuously evolving dynamics of Newton. Modern mathematics began with Cantor. Georg Cantor, a Russian mathematician. In 1883, he created the first mathematical monster: He formed a straight line and broke it into thirds, erasing the middle portion. He repeated this process time and time again, in an attempt to fathom the mathematical mystery. He soon discovered that the length between the lines grew infinitely small and appeared to approach zero. However, the length never actually shrank to zero, and the pattern continued to repeat itself. Mandelbrot realized that these were types of fractals. He also saw different uses for them other than just “Mathematical Monsters.”

Julia Set – 

A short while later, Mandelbrot stumbled upon another monster, the Julia set. Julia was named after the French mathematician Gaston Julia who studied it during the First World War. Gaston Julia tried numerous times to iterate an equation in a feedback loop; however, each time he plugged a number into the formula, he would get a new number. Unable to create an image, he gave up; it was too much for one man to accomplish. Luckily for Mandelbrot, he had computers to aid him in his problem solving and was able to graph Julia using an IBM computer. It was a breathtaking image.

Mandelbrot Set – 

It was time for Mandelbrot to create his own equation. In 1980, he took all of the Julia sets and used them to create one iconic image. He named this image “The Mandelbrot Set.” Little did he know that it would become the most famous of all fractals. The Mandelbrot set and fractals were embraced by artists but not by mathematicians. His peers criticized and ridiculed him, claiming he was no good at math and that his fractals were merely nothing. His friends did not even speak to him. They called his fractals “artifacts from the computer” and called them utterly useless. Ignoring the responses from his colleagues, Mandelbrot created his soon-to-be-famous book, The Fractal Geometry of Nature. In it, he demonstrated how to measure things in nature and provided more ideas on the applications to his newly discovered branch of mathematics. Soon afterwards, fractals were everywhere. 

 Now, as Mandelbrot points out… “Nature has played a joke on the mathematicians,” believing the 19th-century mathematicians may not have been lacking in imagination, but Nature was not either. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turn out to be inherent in familiar objects all around us.

Fractals Today 

In the early 1990s, fractal antennas were introduced. One of the very first of these was in the shape of the infamous “Koch snowflake.” It worked extremely well and had the advantage of being far smaller than a normal antenna with a greater surface area, leaving the antenna able to receive a greater number of frequencies.

 Mobile phone companies were also having similar issues at this time: every single portion of the phone ran at a different frequency. However the introduction of the fractal antenna enabled companies to all these frequencies to run on one single phone, thus kick starting and enabling the mobile phone revolution that continues today.

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Fractal Antenna

Fractals are also used heavily in movies. For example: It is possible to create natural-like scenes such as mountains and even entire planets using only iteration. Fractals premiere in the film industry came in Star Trek II: The Wrath of Khan, where a planet was made entirely from a fractal. Even the last Star Wars movie had fractals. In one of the scenes, the protagonists and antagonists fight in a chamber filled with lava. That lava was created purely using fractals. The artists added swirls to the 3-D model and shrunk them. They repeated this layering each upon another until the entire lava background was made of fractal swirls.

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 Star Wars – Fractal Lava

 

Fractals are not only used in technology but are being found applications throughout our modern world, recently being used to understand the human body. For example: our heartbeats do not actually act like metronomes; instead, they fluctuate to several extremes, forming familiar patterns exactly like fractals do. These fractals also show up in blood vessels, the nervous system, the respiratory system and brain patterns, even the movements of the eye. All these discoveries are helping doctors in diagnosis and treatments.  

As you can see, throughout several different fields, fractals have initiated revolutions of all kinds. We are fractals, and fractals are our future as well.

The link below is to an article: How Mandelbrot’s fractals changed the world written by Jack Challoner for the BBC.

 http://www.bbc.co.uk/news/magazine-11564766