Analysis

Reflecting upon my practical work there is still a lot I would like to do. However conceptually I do feel that what I have produced fits my brief, I am neither an artist or mathematician yet I have produced a visual that is created by computer drawing of mathematical equations.

The visualisation is sketchbook style and depicts elements that are found in nature. It also has a process that is similar to nature that alters the image by varying degrees at varying intervals. 

Math and Art Practical - Final Code

I reverted back to black and white after working on improving the colour and realism in the code. This produces a an image more reminiscent of a sketchbook and I think although it is quite different from my initial aim it is far better than having an image that is too colourful and lacking harmony.

Code Improvements... and Difficulties

I have worked hard on applying a gradient fill to the mountain shape... what I have been trying to do is graduate from a random colour value to black... similar to Etienne Saint Amant work that is light at the top moving to dark at the bottom.
I have however been having a lot of trouble with this as using a random value in a for() structure keeps producing different colour values in each polygon that makes up the mountain.

Because I have had difficulty with this and found the previous code over-done I am considering reverting back to grey-scale and returning to a sketchbook aesthetic.

First Code

This is my semi - complete and working code. I have removed the delay() function so it will update quickly and at regular intervals for the sake of demonstration.

There are things I like about this visualisation and things that I don't. In general I am happy with the 3 fractal elements that make up the sketch; the tree, the mountain and the plasma(sky). However I am finding that there is rapid loss of any sense of subtlety or style as the random function is generating all sorts of brightly coloured fills for my shapes and the image is fast becoming alike to the images that I have steered clear of from the beginning.

There are a few things that I am considering trying to reduce the high level of colour of that fast accumulates on the screen. Having already set the alpha level to that there is a layered and muted effect to the colour I want to work on reducing the colour level by integrating some greyscale into the image to avoid the overcrowding seen in this video.

Practical Project - Thoughts and Aims

What I am wanting to achieve in my code is essentially to select one of the 3 fractal types and draw it to the screen.

I will make use of the random function to control the selection of the element to be drawn, its location and the interval between each alteration; it will also be useful in the selection of a colour palette.

I hope that the image will look very different each time it it drawn.

Sample Codes

To see the original codes in action and see where I have downloaded from click the following links Plasma Fractal Generator, Tree Fractal Generator, Mountain Fractal Generator.

Having decided to tackle the task of making a continuous sketch in Processing I did some searching to find out what effects some fractal equations produced and what fractal work had been produced already in Processing.

I was happy to find that there are quite a few sample codes available online to get me started with my project, these are the 3 codes that I intend to start combining and experimenting with.

I like these especially because they are relatively simple and they allow multiple drawings to be created from the same code. The aesthetic that they all have is reminiscent of natural landscape although they appear more 'sketchy' as if they are more of a hand-drawn impression of a natural landscape... all of which suit me as a starting point for my concept.

Research and Processing - The beginnings of the practical project

For my practical project I am really interested in creating 'art' from 'math' so I have been doing some research on computer generated art and especially the field of fractals. I do like the idea of using code to create my project as it is creating a visual from mathematical equations and contextualises just how useful math can be in the creation of visuals and thus in creative practice.

Having seen a few references to the fractal art displayed on the website deviant art, I decided to check it out and came away with a much clearer idea of what I like and what is achievable.

I have developed a serious dislike all the fluro coloured creations that seem to overwhelm the field of fractal created art. There are far fewer subtle and nature-like creations but I did find this image (see left) produced by artist Etienne Saint Amant. What really struck me about  his work is that it so closely seems to resemble hand painted art, what I believes takes this work further is the subtley beautiful animation of the same piece available on his website. What I especially like about this piece is the artfully chosen colour palette and the subtle combination of shapes that appear like brush strokes in the image.


This second image is another from the fractal gallery in deviant art. Again what I like about this image is that it manages to keep a real natural look although the image itself is surreal. The colour palette is really simple yet striking and the image is altogether very effective.






I have decided that I am going to tackle my practical project using Processing so that I might create a visual that is constantly drawing and to the screen following a random process that will produce different results each run time.

Practical Project Plan

For my practical project I want to expand on several of my key points addressed in my essay and conducting my own practical project utilising fractal geometry.

I am especially aware of the fact that I am neither an artist or a mathematician and my practical project will be conducted on the premise that I am neither but that fractals are a useful tool that I and others can use in creative practice.  I intend to use mathematical equations to create a work that is aesthetically appealing and applicable to creative practice as a field. 

I want to further explore the use of fractal geometry in creative practice and further the three main points that I addressed in my essay; the shared aim of math and art to explain and explore nature, the validity of process in art and how this can in cases mimic nature very closely and lastly the counter-productive insistence on trying to define inter-disciplinary work as 'math' or 'art'.

To do this I will produce a 'non art' and 'non-math' piece of work that instead aims at being an investigation at how fractal geometry can be useful in creative practice. 

Essay

In what ways can creative practice integrate Math and Art?

Lidy van Deursen

Math and Art 175006

AUT Auckland University of Technology

          The integration of mathematics and art is a highly debated topic; in this essay I will advocate the practice of fractal geometry as a viable and practical example of the synthesis of these two subjects and suggest how they might be effectively applied to creative practice. I will address in this essay the connections between the aesthetics of nature and fractal images, identify the importance of process in creation and advocate fractal geometry as a successful integration between the two topics with reference to my accompanying practical project. For the sake of clarity I will define the word ‘art’ as it will be used in the context of this essay. Any product when under consideration of artistic merit must obey three rules before it is to be considered art. It must be creative, be a product of the human mind and it must be authored. Academics seem to use terminology that is mutually exclusive and seem to believe that anything that borrows from another field cannot belong in their own. Computer generated images are highly useful and increasingly common in creative practice; fractal geometry from which many of these are produced, integrates mathematics and art in a way that is useful in creative practice as well as in each respective field.

          Fractal geometry, referred to as the “geometry of nature” by its creator (Mandelbrot, 1989, p.21); is a geometric understanding of the world. Fractal geometry is more advanced than traditional Euclidean geometry that by comparison is inadequate at describing everyday life as it is only concerned with shapes that are too perfect to be found in the real world (Velichová, 2010). Fractals are, put simply, a geometric form with self-similar, irregular details (Lu, 1993). Fractal geometry is useful because it manages to mimic many of nature’s subtle patterns very accurately; a geometry that reflects the complexity of the natural world is a useful tool in both mathematics and art as it helps both practices with their aim to understand the world around us. Fractals produce an organic and natural aesthetic in an image, computer artists have adopted them to produce images of artificial landscapes with a high level of realism, this makes them highly relevant to creative practice as modern day media is experiencing a shift towards the digital and there are many applications that can benefit from their use. In the 1970’s two areas of study grew simultaneously with the aim of identifying ‘nature’s rules’. Chaos theory is the understanding of natural systems while fractals are used to display and describe the patterns that are left behind by these processes (Taylor, 2002). The images produced by fractals are sometimes called the “new geometry”(Taylor, 2002, p. 3) because they are so different from the traditional geometry that the public is familiar with. These fractals are a unique branch of mathematics because they reveal geometric patterns in the most intricate and seemingly disorderly of systems. Fractal geometry has been useful to the science and mathematical worlds because it helps them to explain these natural world processes but they also incredibly aesthetically appealing because they model the natural world so well. ‘Computer artists’ have used fractals to create realistic and visually interesting artificial landscapes. The multi-disciplinary role of fractals has been best descried by Giliola Guirgola as both “art objects” and “mathematic[al] elements”, this labels fractals as a tool for both art and the understanding of the natural world (Guirgola, 2010, p. 100). Because fractal geometry is effectively a visualisation of nature I believe it is applicable to any practice that examines nature as part of its process. I am in agreement with the early mathematician Pythagoras who “saw maths sitting at the centre of art, life and nature” (as cited in Sims, 2010), and believe that creative practice is informed by all of these fields either directly or indirectly.

          The processes that constantly change and shape the environment around us are essentially equivalent to the conditions that produce fractals. Process in art can be as significant as the actual outcome of the work; some meticulous ‘artistic’ processes have in cases been shown to produce work that contains fractal structures. An example of this is the “drip and splash” paintings of artist Jackson Pollock. Although his work was produced before the discovery of fractals it has been found to be “composed of distinct fractal patterns” (Abbbott, 2006). Pollock was an artist that had a strong affinity with his process, involving evolution of his work over prolonged periods of time. There is a distinct ‘cumulative’ style to Pollock’s paintings as each piece was worked for months with a vast variation in the intensity of each repainting and duration between bouts (Taylor, 2002). Pollock’s “continuous dynamic” process is very similar to that of nature and has been described as action art by critic Harold Rossenburg. Rossenburg also claims that Pollock’s painting style “doesn’t reproduce nature; [but] is nature”. This claim almost labels the work as more than ‘art’ and is supported by Pollock’s hatred of the signing process which he saw as an “artificial act that recognises the canvas as an artwork rather than a piece of nature” (as cited in Taylor, 2002). Nature is sporadic in its shaping of and changing the natural world and is in essence the ultimate creator. Process in creative practice can in most cases be as relevant as the actual product, fractal images are in essence visualisations of possible natural outcomes whether or not they appear in the natural world and as such are visually appealing to humans.

          Fractal geometry in creative practice has mostly been used in the production of fractal patterns that have found applications in visualisations and computer art. It has in essence become a reluctant part of the ‘art world’ that is both broad and snobbish with definition. Because fractals are both computer generated and often presented to the world in the form of ‘mathart’ they have been surrounded by a lot of scepticism. Claiming that fractal images are simultaneously mathematics and art has been problematic so I believe a new definition is necessary. The issue doesn’t lie in the aesthetic merit of the fractal images, in fact fractals are so life-like that there are fractal image compression techniques that encode a fractal to approximate the original image into a smaller file (Lu, 1993). I am exploring in a related practical project the idea of fractals in fact being a non-mathematical and a non-artistic tool to be used wherever they are useful. I am neither a mathematician nor an artist but I believe they are the best tool to create certain types of images. My belief is that fractals have a high aesthetic appeal which makes them comparable to art; at the same time they have the ability to be manipulated easily through mathematics within a computer so that they are an integration of both mathematics and art without claiming to be one or the other inclusively or exclusively.

          The success of integrating mathematics and art into creative practice lies in the ability to take from either field as is useful, and to be open to borrowing from any practice as seems appropriate or beneficial. Fractals are highly useful to creative practice as a visualisation tool; they make optimum use of the patterns of modern mathematics and enable aesthetic adjustability, characteristic of art; they are also easily created in a computing environment. My practical project has been developed in tandem with this essay and explores the way that mathematics and art can be brought together to make something new that is an effective visual display but asserts the fact that it makes no claims to be either math or art, but instead is informed by both these practices. Fractal geometry is a cross disciplinary and advantageous practice, it can be easily integrated into creative practice and I predict that its uses will only increase as we move further and further into the age of computers as our primary tool.


References:


Abbott, A. (2006). "Fractals and art: In the hands of a master." Nature 439(7077): 648-650.

Giurgola, G. (2010). Creative Mathematics and Rational Art in Virtual World. Aplimat 2010. S. U. o. T. i. Bratislava. Slovak University of Technology in Bratislava, Aplimat: 91 - 102.

Lu, G. (1993). "Fractal image compression." Signal Processing: Image Communication 5(4): 327-343.

Mandelbrot, B. B. (1989). Fractals and an Art for the Sake of Science. Leonardo. Supplemental Issue, MIT Press. 2, Computer Art in Context: : 21 - 24.

Sims, J. (2004). "Notes on a mathartist." The International Review of African American Art 19(3): 52-55.

Taylor, R. (2002). Fractal Expressionism - Where Art Meets Science, Santa Fe Institute.

Velichova, D. (2010). Chaos in Math and Art. Aplimat 2010 S. U. o. T. i. Bratislava. Slovak University of Technology in Bratislava, Aplimat: 677 - 686.

Abstract

In what ways can creative practice integrate Math and Art?

Lidy van Deursen

Math and Art 175006

AUT Auckland University of Technology

          This research project will be focused on exploring the ways that math and art can be integrated in creative practice with a particular focus on the field of computer-generated art and its role in the creation of fractals. Hailed as “the new geometric language” by Mandelbrot, mathematician and ‘father’ of fractal geometry; fractals have given rise to a new form of art.
          “Euclidean geometry concerned mainly with perfect abstracts non-existent in nature, ha(s) no way of describing items in our everyday lives” (Velichová, 2010). I will be expanding on ideas presented by Mandelbrot, Velichová and Taylor amongst other academics to explore the ways that fractal geometry can be used to explore connections between math and art and extend geometric connections beyond that of traditional ‘Euclidean’ geometry.
          The research and practical will seek to explore the synthesis of math and art with the aim of proving them to be of equal use in the field of fractal geometry; the realism produced and the potential to mimic nature will be forefront in my investigations and analysis of this topic. Opinions of academics in related fields as well as the differing definitions of true ‘math’ and ‘art’ will be given due consideration throughout the examination.

Literature Review

In what ways can creative practice integrate Math and Art?

Lidy van Deursen

Math and Art 175006

AUT Auckland University of Technology

          This literature review aims to compare and contrast the current knowledge on the similarities and differences in the fields of mathematics and the arts. These two fields might initially seem very different but in recent years there has been a lot of debate over whether they are in fact connected; this literature review aims to explore these alleged connections. History suggests that in the past math and art have at times been considered as parts of the same practice and it is interesting to see in modern times their connections being reexamined in the search for greater understanding of both fields as well as the possibility of integrated practice.

           Historically it would seem mathematics and art have previously been considered as parts of the same, whole although they have “been on opposite poles when describing the real world” (Velichová, 2010, p. 677). In ancient Greece mathematics fell under the category of the fine arts and both artists and scientists were grouped together and considered bizarre people interested in the useless, strange and outrageous. The mentality of the time was that “artists belonged together with scientists” (Velichová, 2010). In support of this it is suggested in the paper ‘Fine arts and Geometry’ that man is endowed with a certain ‘geometric instinct’ that enabled him to satisfy his innate need for construction (Ladopoulos, 1970). From this it would be understood that geometry is something that comes naturally to man. The author also claims that the birth of fine art should be attributed to the “internal need” of man to represent and express (Ladopoulos, 1970). This claimed geometric aptitude can be applied to both construction of the artistic and the practical, placing early geometry between what might be considered the early fields of art and science.
             Consistency of proportion has been noted since hundreds of thousands of years before Euclid introduced his famous geometric theories. The human hand has been proven to correspond with the proportions of the golden ratio (the proposed most attractive proportions in nature and design) (Samoila, 2010). Scientists find the study of the human hand accountable for artifacts being produced in this special ratio. This occurred from about the time when primates evolved into humans, long before any education would inform them of the significance of this ratio. “Using the hand for hundred thousand years implied also carefully observing the hand and its inner proportions” (Samoila, 2010, p. 169). The Fibonacci sequence is closely linked to the golden ratio and they have been claimed to “attribute to the contribute to the harmony in nature” (Carafoli, 2009, p. 246).
           Pythagoras is cited by professor and proclaimed ‘mathartist’ Sims as “the first thinker to advocate proofs” and “teach the connections between mathematics, nature and the arts.” (Sims, 2004, p. 52) Pythagoras theorem is a key visual mathematics-teaching tool for Sims.

           Giliola Giurola believes that “the relationship between mathematics and art has always been clear” and that mathematics has many applications when applied to reality. (Guirola, 2010, p. 92) Chakravartty agrees with this adding “in both art and the sciences, successful representation is a matter of fitting or approximating things in the world” (Chakravartty, 2010, p. 40). When asked to think about the combined study of mathematics and the arts the first connections one might guess at is geometry in art and the use of visual diagrams in mathematics to help with the visualisation and understanding of concepts. In ‘Geometric Symphonies’ Paul Cézanne is quoted saying “in nature everything is modeled according to three fundamental modality: the sphere, the cone and the cylinder. You need to learn to paint these very simple figures, then you can make everything you want.” (as cited in Comitá, 2010) From this artists’ statement it is clear that he considers the world to be made up of assignable geometric entities that can be replicated for use in art. Sims is also in agreement, calling the conviction that mathematics and art are opposites a “disabling counter-productive myth” (Sims, 2004, p.53) He instead encourages understanding that both fields are fundamental expressions of our consciousness and are “connected by the voice of nature” (Sims, 2004, p. 53)
          For the combination of mathematics and art, digital technology has now advanced to the point that it is the “the primary choice” of mathematical tool used in the production of art (Schattschneider, 2003) In the same article compasses, rulers, grids and computers are cited as physical tools for the creation of art that rely on the power of mathematical relationships and processes to give them their creative power. (Schattschneider, 2003) Visual art is also beneficial in highlighting mathematical concepts by giving pictorial support with the recognition of mathematical principles.

          Modern day computers have allowed the study of geometry to extend beyond its previously hand drawn past. The Euclidean Geometry which has existed as the basis of taught geometry through the ages “has no way of describing objects in our everyday lives” (Velichová, 2010, p. 681) Euclidean Geometry, being only concerned with perfect abstracts non-existent in nature became inadequate in describing the real world. Bernoit Mandelbrot writes about geometry and its application to the world: “Fractal Geometry is not just a chapter in mathematics, but one that helps Everyman to see the world differently.” (as cited in Velichová, 2010 ) Chaos theory and fractal geometry has allowed computer art to become more realistic and in the opinion of Guirola “fractals are not only mathematic elements” but are also “art objects thanks to their variety and their pleasant graphic aspect.” (Guirola, 2010, p.100) In contrast to this James Elkins presents a strong argument about the way that fractals are used, he believes that by the time fractals are seen by their audience they are being experienced at a “double remove”. Elkins cites the fact that fractal art is presented independent of its mathematical context and with an overlay of unrelated colours that are not determined by the equation itself. (Elkins, 2009, p. 45) Although Elkins claims to see why fractals are considered an important discovery in geometry and that they are so popular in art (because they mimic nature so well) he maintains a level of sobriety when it comes to them being simply imported into art. Fractals are also considered by Elkins to be of limited use to drawing, as their construction is limited to the computer.

          Some common arguments cited for the synthesis of art and mathematics include beauty, proportions (including the golden ratio and Pythagoras theorem) and the search for truth and the understanding of nature being a common goal of the two practices.
           Firstly Beauty is an interesting argument as a shared goal between the two practices, Elkins believes that beauty is no longer relevant to modern art, he writes that it is rare to hear an artist describe his work as beautiful and that words of a similar nature would sound “insufficient, in an art critic’s mouth”(Elkins, 2009, p. 37). The word beauty is too unfocused and well intentioned for Elkins to consider it a “workable bridge” between the two practices. Carafoli in his article talks about both mathematics and art having two goals that must be fulfilled, both aim at beauty but they must also search for truth, and in science’s case the truth must also be testable (Carafoli, 2009). Carafoli does however bring up a pivotal question on how one might decide upon beauty? He suggests that for the most part our conception of beauty is superficial, and wonders if there are a tangible set of values that might be more precise in assigning beauty fairly. (Carafoli, 2009). One suggested method is that of George David Berchoff who presented his equation in 1928 that claimed to be able to judge the aesthetic ‘value’ of an object by dividing the amount of order of an object by the level of complexity it possesses (Routio, 2007).
          Proportion is another subject that is commonly broached in the art and mathematics debate. Pythagoras theorem, the golden ratio and the closely related Fibonacci sequence have all been suggested to be superior ratios and potentially the most appealing ratios to the human eyes. A study conducted by Georg. Th. Fechner studied the aesthetic preferences of people that were untrained in the aesthetics field and interestingly concluded that none of the proposed superior proportions (notably including the golden ratio and Pythagoras) were consider more beautiful than the others (as cited in Routio, 2007). This is contradictory to the opinions voiced in the other articles that I have read. Another study by Dio, Macaluso & Rizolatti concluded that when brain activity was monitored and participants were presented with a series of sculptures, “the registered activity in the brain clearly favoured the sculptures in which the proportions were those of the golden ratio.” (as cited in Carafoli, 2009). The documentary ‘The Human Face’ would agree with the later. Experiments were shown where a mask was made to conform to the golden ratio and was then fitted to photographs of a selection of people who were ranked by attractiveness, the fit of the mask on the face was proportional to the perceived attractiveness of the individual with the most highly ranked fitting the mask the best (George & Rossiter, 2001).
          Something that has been pondered by most authors in this study has been when art and math combine are they art or are they math, are they both or are they neither. “Some kinds of art can be called ‘art and mathematics’ and they have an obvious but superficial meaning” (Giurola, 2010, p. 98) Giurola goes on to point out that there are two ways to use art and math together, one is to use mathematics as a tool of art and the other is to paint a mathematical topic which he believes is the more effective path. Elkins doesn’t believe in effective combination of art and science, he points out that what little science there is in art it is altered to serve artistic purpose and therefore ruins the scientific content, he goes as far as to say that “science per se will not appear in art, because without art it would only be science.” (Elkins, 2009, p. 39) Another article confirms that math can only exist in a visual artwork if “appeals to mathematicians” and “encodes a mathematical structure”(Velichová, 2010).

In conclusion there is a long history of mathematics and art proving to be a useful combination and the modern re examination of this combination opens the doors to more collaboration in future ventures. As discussed earlier in the review there are two proposed ways to use mathematics and art together. The first is to use the useful parts of one topic to benefit the end purpose of a particular project or subject, this is commonly seen as every piece of art that uses geometry could be said to be borrowing from the field of math. The Second way that Giurola suggests is that the two might be integrated is to make an artistic work about a mathematical concept. This is again a problematic suggestion as this might then defeat the purpose of art.
          The current research on mathematics and art is largely united in exploring the potential the two topics have together but seeming divided in their opinions to if a work can be both simultaneously mathematics and artistic. Luckily though we have been presented with the question about in what ways can creative practice integrate math and art and the evidence seems overwhelmingly in favour of there being many instances of overlaps and cases where one practice has solved the other’s problem.

References:


Carafoli, E. (2009). Scientific and Artistic Creativity: In Search of Unifying Analogies. The Two Cultures: Shared Problems. Milano, Springer Milan: 239-264.

Chakravartty, A. (2010). "Truth and Representation in Science: Two Inspirations from Art." Beyond Mimesis and Convention 262.

Comitá, A. (2010). Geometric Symphonies. Aplimat 2010. S. U. o. T. i. Bratislava. Slovak University of Technology in Bratislava, Aplimat: 29 - 36.

Elkins, J. (2009). Aesthetics and the Two Cultures: Why Art and Science Should Be Allowed to Go Their Separate Ways. Rediscovering aesthetics: transdisciplinary voices from art history, philosophy, and art practice. Stanford, California, Stanford University Press.

Giurgola, G. (2010). Creative Mathematics and Rational Art in Virtual World. Aplimat 2010. S. U. o. T. i. Bratislava. Slovak University of Technology in Bratislava, Aplimat: 91 - 102

George, S. and N. Rossiter (2001). Survival of the Prettiest: The Human Face. Retrieved August 18, 2010, from http://www.youtube.com/watch?v=1AZe9g2Huz0

Ladopoulos, P. D. (1970). "Fine Arts and Geometry." The Journal of Aesthetics and Art Criticism 28(4): 535-540.

Routio, P. (2007) "Beauty of a Product: Arteology, the science of products and professionals." Retrieved August 20, 2010, from http://www2.uiah.fi/projects/metodi/155.htm

Samoila, G. S. (2010). The Harmonic Geomerty of Art. Aplimay 2010. S. U. o. T. i. Bratislava. Slovak University of Technology in Bratislava, Aplimat: 165 - 174.

Schattschneider, D. (2003) "Mathematics and Art -- So Many Connections." Math Awareness Month - April 2003 Mathematics and Art. Retrieved August 20, 2010, from http://www.mathaware.org/mam/03/essay3.html

Sims, J. (2004). "Notes on a mathartist." The International Review of African American Art 19(3): 52-55.

Velichova, D. (2010). Chaos in Math and Art. Aplimat 2010 S. U. o. T. i. Bratislava. Slovak University of Technology in Bratislava, Aplimat: 677 - 686.

Math and Art Blog

This is the first post in my new Math and Art blog. The purpose of this blog will be to document my research and the progress of my own essay and supporting practical project exploring the how math and art can be used together in creative practice.