Archives for posts with tag: Fractal

A Brief Concept.

The concept behind this work it to create a 3-4 minutes stop motion video relating to my interest in the natural fractals throughout world, its process and our relationship to them. To depict my concept I will have a wide range of images, from the dispersion of natural dyes in water to fractals from the natural world. Every image will blend and flow with the last.  Also each image will be one of a series of its own, so when hopefully leaving a seamless blend between all images. As the work is going to be a video I wish to play music to accompany and accentuate the work, this should mean that after editing the final video should be around 3.30 / 4 minutes long.

Displaying Work

I would like to see how the video would look projected on a large scale, as this would be a new method of presenting for me. However as there is accompanying music and this is a group exhibition projecting may not be plausible due to needing a set of speakers for the music. I could always present similarly to before on a screen of some kind and a set of headphones, however finding a more suitable screen than previous exhibitions may also not be plausible.

There is one other method of presenting I am currently researching, but would immediately be a drain on resources. Using the phenomenon of video feedback If you link a camera to a TV and then direct the camera at the TV, you get an infinite regression of images. However, you can use the same feedback phenomenon with multiple displays to make a natural forming fractal. By displaying multiple smaller copies of what the camera sees, photographing that cluster of copies, and then repeating the process, you essentially create the self-similar structure seen in fractals. By moving and rotating the camera and projectors, you can create a very wide variety of fractal images. 
This method of presenting would be perfect in theory, however the use of multiple projectors may not be possible, along with not being able to try it out before hand firstly to see if I understand and can get it working. Secondly, to see the aesthetics when projected in this way and if it would enhance and contribute to the work.

Here is a video of the final method of presenting working.

http://www.youtube.com/watch?v=Jj9pbs-jjis

UPDATE: Unable to obtain the relevant equipment i will displaying a HD version of the video upon a Mac screen with headphones. This will allow the viewer to be engulfed fully by the visuals and accompanying soundtrack completely.

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There are many example of fractals throughout the musical world.  But First, The Fibonacci sequence, a fractal set in its own right cant is seen throughout the structure of the natural world. These few examples show the diversity and beauty of this one simple equation. The Fibonacci sequence can also been seen in the composition of one of the worlds most famous painting, the Mona Lisa…. Which was painted by da Vinci. The Fibonacci Sequence is an equation in which the spiral shape found in nature can be seen. It shown Below.

 

spiral

Musical scales are based on Fibonacci numbers

 

The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as

 

There are 13 notes in the span of any note through its octave.

A scale is composed of 8 notes, of which the

5th and 3rd notes create the basic foundation of all chords, and are based on whole tone, which is 2 steps from the root tone, that is the 1st note of the scale.

Note also how the piano keyboard scale of C to C above of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2. While some might “note” that there are only 12 “notes” in the scale, if you don’t have a root and octave, a start and an end, you have no means of calculating the gradations in between, so this 13th note as the octave is essential to computing the frequencies of the other notes.  The word “octave” comes from the Latin word for 8, referring to the eight whole tones of the complete musical scale, which in the key of C are C-D-E-F-G-A-B-C.

 

Musical compositions often reflect Fibonacci numbers and phi

Fibonacci and phi relationships are often found in the timing of musical compositions.  As an example, the climax of songs is often found at roughly the phi point (61.8%) of the song, as opposed to the middle or end of the song.  In a 32 bar song, this would occur in the 20th bar. Many composer such as Beethoven, Mozart and Bella Bartok have been linked to fractals within there compositions.

 

Musical instrument design is often based on phi, the golden ratio

Fibonacci and phi are used in the design of violins and even in the design of high quality speaker wire.

 

violin 3D3H 360

This Website goes into the great details of fractals in music, however its fairly complicated and hard for me with my limited musical experience. However I have included it as its very imformative!

http://solomonsmusic.net/fracmus.htm

 

Cymatics

Essentially not music based but this video of cymatics for me proves the link between sound and patterns perfectly. This video amazed me!

 

My Choice of Music

After Much deliberation and hours upon hours of listening to music ive research into music realting to fractals I have decided to have Tool’s Lateralus as my sound track

Firstly here are a few of the Tracks that were shortlisted for one reason or another.

 

What Phi (the golden ratio) Sounds Like – Micheal Blake

 

Bela Bartok – Music For Strings

 

Bence Peter – Fibonacci Piano piece

However this is my music of choice – With a very helpful video which explains all the reasons why for me!

 

Tool – Lateralus

 

http://www.fibonicci.com/fibonacci-tools-lateralus/

Although this track is perfect it is however not to everyone’s tastes… after this troubling me for some time I’ve found a number of cover version which adhere to all of Fibonacci’s inspiration within the musical composition, whilst having a more popular appealing sound.

 

Rockabye Lullabies – Renditions of tool

Covered using only xylophones!

 

String Quartet Tribute to Tool

A Brilliant and moving cover by a string quartet!

 

UPDATE: I have chose the string quartet version, for me it has a much more atmospheric nature along with having more depth and layers. I will now edit the track from 9.32 in length to a more reasonable length using Audacity, ready to edit the image to.

Many artists have also observed fractals in the natural world transferring them into their own works. Yet another artist renowned across the world who incorporated fractals into his work was Salvador Dali. Primarily and predominantly within his painting the ‘visage of war’ (1940), this painting created during Dali’s small stay in California regards concepts of the Spanish Civil War . A the primary interest in this case comes from within the eye sockets and mouth of the face in the painting. There are similar smaller faces within the eye socket and mouth and again within the eye sockets and months of these faces. This suggests a pattern of self-similarity and infinite repetition again and the key characteristics of fractal geometry. Through this infinite pattern Dali seems to be mimicking the infinite pain and anguish of all. Descharnes echoes this in stating “Eyes filled with infinite death”(Descharnes, r, Dali, Harry Abrams, NY 1985) when referring to the unrelenting repetitive nature of the self similarity within Dali’s painting. AS you can see from the second set of images shown below, Dali’s painting mimics a circular form of the cantor set.

pic26

cantor_set_salvador_dali

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.

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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.

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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.

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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

 

Leonardo da Vinci was a man of multiple dimensions of talents. Not only was he a painter but he also had impeccable skills as a sculptor, an architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer. His unquenchable curiosity was equaled only by his powers of invention. He is widely considered to be one of the greatest painters of all time and perhaps the most diversely talented person ever to have lived, to which I would personally have to agree.

Main areas of his studies that have taken influence within this works are:

Studies in Flight

Leonardo made “the flight of birds” the basis for his mechanical approach and he studied the function of the wing, the air resistance, the winds and the currents. Leonardo also studied air-resistance, currents, winds and the laws of equilibrium. This intricate and detailed study into the flight of birds has been a reoccurring influence visibly noticeably within the majority of my previous work.

    English: Design for a flying machine Codex Atl...

Studies in the Rules of Proportion

The building blocks of basic geometry underlay the beauty of natural form. Leonardo provided ingenious illustrations for the treatise that his mathematician friend, Luca Pacioli, wrote on the five regular or ‘Platonic’ solids and their variants. He also strove on his own behalf to solve a series of classic problems in flat and three-dimensional geometry. Divine geometry in nature was most apparent in the action of light. Here, all the powers of nature act mathematically and obey the rules of proportion.

Proportion was also expressed in number, most notably in the harmonies of music. Proportional formulas allowed Leonardo to work complex variations on weights suspended from balances and to show why the quest for a perpetual motion machine was doomed to fail.

Anatomical study of the arm, (c. 1510)

 

Studies in Motion

Leonardo’s vision of the natural world was extraordinarily dynamic. Force was the key to the vision. The application of force was necessary for anything to move. Motion gave life to all things but also exercised a huge destructive potential.

The human body was at the center of his vision. Bodily movements expressed the ‘motions of the mind’. These motions were essential for the painting of convincing narratives. Leonardo’s ‘cinematographic’ images of little figures in action portray the continuity of motion in space in a way that no one had captured previously. As an engineer Leonardo’s supreme ambition was to amplify human motion so that man-powered flight might become possible. The key, as always, lay in nature, above all in the study of flying creatures and their anatomy.

 

Rearing horse

 

Studies in the Body and Earth.

The theory of the microcosm and the macrocosm was ancient. It stated that the human body contained in miniature all the operations of the world and universe as a whole. Leonardo wrote of analogies between rocks and bones, soil and flesh, rivers and blood vessels. He spoke of the ‘body of the world’, ‘veins of water’ and the ‘tree’ of blood vessels.

This analogy served as a tool of explanation. He explained that the old are enfeebled because of the tortuous and silted up nature of their blood vessels. He investigated the nature of water in motion and its behavior in the ‘body’ of the earth.

Heart and its Blood Vessels

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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.

 

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.