Archives for posts with tag: Benoit Mandelbrot

Fractals are complex geometric patterns that repeat themselves at every scale. Trees and ferns are fractal in nature. The branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Man has always found fractals beautiful without realizing why. Why are clouds and mountains beautiful? Trees? Flowers? Snowflakes? I believe it is because of their fractal geometry. If you google “fractals found in nature”, you will see objects that I have photographed throughout my whole life. One natural fractal I had never photographed, however, were “Brownian Curves”, and although most people would say this series are simply photographs of smoke, the mathematics behind them are truly beautiful.

This slideshow requires JavaScript.

After viewing Enrique’ series of smoke images, the complex and fractal inspired works led me to shoot more off my own smoke images with the intention of montaging them into my final video. the next gallery is a display of a small number of my smoke images. In the capture of these images i created a small studio environment, using lighting and background to achieve the best results. The smoke was created using incense  sticks. When photographing a small number of the smoke images, i places coloured acetate in front of the lighting set ups to create colour casts over the smoke.

This slideshow requires JavaScript.

A second series of image by Enrique has caught my attention also. “Connected” is a series of 22 images that portray the intertwined nature of life. Gravity is an invisible glue that binds all the objects in the space-time continuum that we inhabit. If we could see that invisible glue, it might look like these images. Yet, gravity is not the only force that creates bonds between us. “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”

In the world of networking websites, such as Facebook, 6 degrees of separation is no longer a theoretical principle. The man next to you on a bus is not a complete stranger. Similarly, every vertex in a network of bubbles is connected to every other intersection. Some times the path is very apparent, but often it is not. Yet, the connections remain. And like people, every node is unique. But put together, a larger entity appears. this series of images leads my consistently back  to the thoughts of; In a universe where everything is connected, should we be surprised by coincidences?

The subject matter depicted in these images is no coincidence either: soap bubbles. The dual hydrophilic (‘water-loving’) and hydrophobic (‘water-hating’) nature of soap allows it to modify the surface behavior of the water in which it is dissolved, and thus create bubbles. But it also links substances that otherwise would not mix, like oil and water. In every day life we use this property to wash our hands clean. But life itself could not exist without the connections that these molecules make. The walls and membranes of plant and animal cells are covered in soap-like molecules that allow them to network with each other and interact with the world around them, and to create larger, multicellular organisms like the fruits and vegetables that form the backgrounds of every image in this series. In using these concepts and techniques, Enrique subtly, even if unintentionally relates his work to fractal geometry.

This slideshow requires JavaScript.

 

Advertisements

Perhaps best known in the west for In the Hollow of a wave off the coast of Kanagawa (left), also called The great wave, from Thirty-Six Views of Mt. Fuji, in the 1830s and 40s the Japanese artist Hokusai made wonderful woodblock prints that present fractal aspects of nature with a sophistication rarely matched even today..  He did not have any mathematical training; he left no followers because his way of painting or drawing was too special to him.  But it was quite clear by looking at how Hokusai, the eye, which had been trained from the fractals, that Hokusai understood fractal structure.  And again, had this balance of big, small, and intermediate details, and you come close to these marvelous drawings, you find that he understood perfectly fractality.  But he never expressed it.

 

The Great Wave off Kanagawa

The Great Wave off Kanagawa 

 

 The below link is to a brilliant  publication regarding The Great Wave painting.

http://rsnr.royalsocietypublishing.org/content/63/2/119.full.pdf+html

Related articles

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.

 

318348108_fd1383e033

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

366728330_8f15645d0d_m    366728669_28f15ae45d_m

These are black and white Persian Rug examples

366686572_e7e139130b

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.

hindi

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”

366663277_8e7dfd15f4_o

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.

 

 

 

 

 

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

biomimicry-koch-snowflake-537x402

 

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.

 img66

vanaelst_cantorset

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

“Fractals are not just artificial constructs, they shape us and the world we live in.” (Gleick – 1987).

Fractals for me, explain all natural phenomenon. While fractal geometry was conceived at the beginning of the 20th century, it was not until the advancement of the super computer that we have been able to see the complete implication and brilliance of fractals. I don’t claim to be an authority or expert on the subject, but I’ll try and explain here what little I know.

The notion behind fractals is moderately humble and apparent when described simply. But the arithmetic used to cultivate those concepts is not as easy. A fractal is a geometric figure with two distinct properties. Primarily, it is Irregular, fractured or fragmented. Moreover, it is Self-similar; that is, the figure appears similar no matter how great the scale of magnification. These objects display self-similar structure over an extended, but finite, scale range.

Alex Grey

Alex Grey

Benoit Mandelbrot, now known as the ‘father or fractals’, devised the term fractal to define such figures, stemming from the Latin word “fractus” meaning broken, fragmented, or irregular. He also outlined astonishing parallels in appearance concerning some fractal sets and numerous natural geometric patterns. Consequently, the term “natural fractal” refers to natural phenomena that are similar to fractal sets, such as the path followed by a dust particle as it bounces about in the air.

“Fractal Geometry plays two roles. It is the geometry of deterministic chaos and it can also describe the geometry of mountains, clouds and galaxies.” – (Benoit Mandelbrot – 1984)

The majority of objects in nature do not adhere to simple traditional geometric forms. Clouds, trees, and mountains typically do not resemble circles, triangles, or squares.  Within the natural world there are no straight lines or smooth edges. A sunflowers pattern for growth, the faultless symmetry of a microbe, the striped coat of a zebra, the barreling of ocean waves, or the harmonized turns and swoops of a flock of starlings twirling amongst trees prior to landing on a telephone wire. How can all those individuals part of the flock evade collisions or confrontations with their neighbors? How do they orchestrate these elegant, precise and instantaneous movements in such a sizeable group?These are a small number of thousands of additional examples are the kaleidoscope of patterns and forms that nature gifts us over a lifecycle.

mountain fractal   4497160728_bf070998f2   wave fractal   FractalClouds

  bacteria fractal   sun_small    abstract-zebra-stripes-colour-black-size-8774-8004_medium    starlings_flock450

Take a tree, for example. Preference a specific branch and examine it thoroughly. Then select a collection of leaves on that branch. In Chaology (the study of chaos) all three of the matters described – the tree, the branch, and the leaves – are identical. For many, the term chaos insinuates randomness, unpredictability and possibly even untidiness. Chaos is actually extremely structured and adheres to certain patterns and algorithms. The complications occur in finding these elusive and sophisticated patterns. One purpose of examining chaos through fractals is to grasp the patterns in the dynamical organization found in nature that superficially appear unpredictable and incomprehensible.  To many Chaologists, the examination of chaos and fractals is beyond just an innovative and fresh field of science that fuses mathematics, theoretical physics, art, and computer science – it is a revolution. It is the breakthrough of a new geometry, one that helps us in defining and understanding the infinite universe we live in; one that is in constant motion, not as static depictions in textbooks. Today, fractal geometry has increasing implementations and applications, from predicting stock market prices to making new discoveries in theoretical physics.

Fractal Trees

Timm Dapper

Mathematicians have attempted to describe fractal shapes for over one hundred years, but with the processing power and imaging abilities of modern computers, fractals have enjoyed a new popularity because they can be digitally rendered and explored in all of their fascinating beauty. However beautiful these renderings are, for me they don’t compare to fractals that form in nature. Are visual intakes are saturated with the computer generated images representing fractals, so much so that for many they define fractal geometry. I feel that a reason for creating my video may be an attempt to disperse the brilliance of fractal geometry.

                                                                               Tiera4414aa    pastel-fractals-background
Computer generated Fractals

For a clear introduction to fractals (including an interesting fractal-generating application for Macintosh), go to:

astronomy.swin.edu.au/pbourke/fractals/ fracintro.