Archives for posts with tag: Math

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.

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.

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.

 

 

 

 

 


“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.