Critical Paper

Essay Question:
What is Beauty? A critical review of views of Beauty in Science and Art.

Introduction

Beauty is truth, truth beauty, - that is all
        Ye know on earth, and all ye need to know. (J Keats, 1819)

In this essay I will compare and contrast the concept of beauty and truth in science and art. Whilst the joining of the concept of beauty to truth in the above quote could at first glance be argued to underpin the idea of scientific beauty more readily than aesthetic beauty, the exploration of truth in art will also be shown to be equally valid.

In the first chapter I will consider the idea of whether science and art are totally separate cultures and if there may exist any shared territory. Chapter 2 considers the philosophical debate surrounding definitions of beauty and offers a definition for beauty for use in this essay. Chapter 3 examines the science of beauty including the work of Mondrian and a true aesthetic.  It considers whether cross-cultural aesthetic judgement can be arrived at based on statistical evidence, as in the work of Komar and Melamid that uses statistical preferences to create art. Chapter 4 explores whether there is beauty in science and mathematics. I illustrate how it is essential to consider mathematics as it is the language of science. Indeed, Gauss, a leading mathematician of the 18th century, is reported to have said “Mathematics is the Queen of the Sciences” (Linehan, 1932, p. 296).
Chapter 5 reviews Chaotic Beauty.  In this chapter I identify examples of shared space between science and art by looking at fractals and how chaos mathematics reflects the form of nature in a more truthful way than had been previously possible. The conclusion identifies similarity and difference in the perception of and need for beauty in both Science and Art.

Chapter 1 – Two Cultures

Until the Renaissance,(15th Century), Science and Art were considered to be joined in the pursuit of ‘Natural Philosophy,’ but since that time there has been an apparent cultural acceptance that they are two separate cultures.  

C P Snow brought fresh attention to the idea of two cultures when referring to how he perceived the separateness of science and art in his 1950 Rede Lecture, “We have lost even the pretence of a common culture” (Stringer, 1983, p 172). This gap is also lamented in 1984 by Samuel Edgerton (p107) when he states that “science and art have appeared more split asunder than symbiotic.”

Art and science have cross-fertilised ideas and technologies since the perceived split of the cultures. Examples of this include the use of Euclidean geometry in the construction of perspective in the Renaissance and the twentieth century birth of Modern Art. Shearer, (1992, p143), claims this “can be directly linked, respectively, to perspective and non-Euclidean geometries, which originated during those times.” It could even be argued that mathematics is the language of science enabling art to succeed, eg perspective as illustrated below.

Masolino’s ‘St. Peter Healing a Cripple and the Raising of Tabitha’ (1425).

The recognition of the constant interplay between science and art is undervalued in the two culture view, “We are concerning ourselves here with interactions between activities that have, traditionally, been treated separately” (Field 1997, p.1). However examples like the artist’s use of camera obscura long before chemical photography show much knowledge was being transferred in both directions (Hockney, 2001, p 5).

The interplay between art and science in the Renaissance and at the end of the 19th century led to leaps forward in both cultures “on the crucial eves of the two astounding Scientific Revolutions,” (Edgerton, 1984, p 108).

With the current cross-fertilisation apparent in the yearly prizes from organisations such as The Wellcome Trust (Biomedical science) and the ubiquity of fractal images, we can see that the interplay between science and art gives us “an age uniquely conditioned to finding direct aesthetic satisfaction in the geometric and organic patterns of nature” (Rieser pp54). The Wellcome Trust Arts Awards are aimed to “encourage high quality interdisciplinary practice and collaborative partnerships in arts and science practice” (Wellcome, 2011).

Chapter 2 – Definitions of Beauty

In this chapter I will attempt to reach a definition of beauty that can be used through the rest of the essay. This will be achieved through a chronological study at what philosophers and scientists have concluded through time.

The concept of beauty has developed and changed markedly since it became a subject for philosophy. When Socrates (469 – 399 BC) was considering beauty in classical Greece, he started from a functional point of view; “Nothing is beautiful for any other purpose than that for which the thing is adapted” (Socrates quoted in Coomaraswamy, 1941, p339). It must be noted that the requirement for ‘purpose’ is a subjective concept in the arts.

For example, Plato asserts in ‘The Republic,’ that the ideal Form of the spoon exists only in the mind of God (Plato, 375 BC, p xv). He asserts that the craftsman can only make a reflection of this Form and further he states that artists present a mirror, a mimesis, of this crafted form without ever reaching the complete truth of these ideal Forms. He describes the artist as a deceiver unable to capture the whole of any concept or Form (Plato, 375 BC p361).

A weakness in the Platonic view of art and beauty is that it only encompasses that which is physically made by man and nature. He does not include conceptual work such as poetry, plays or prose. He writes “the graphic arts are full of the same qualities and so are the related crafts, weaving and embroidery, architecture and the manufacture of furniture of all kinds; and the same is true of living things, animals and plants” (Plato, 375 BC, p103).

Aristotle (384 – 322 BC) also believed in ideal Forms but was not so harsh about the nature of imitation or the idea that Forms were created by God and unreachable by man; “Aristotle’s notion of mimesis is similar to the view of Plato, since they both claim that art imitates nature.” Aristotle is more positive about artists than Plato. He also takes the idea further to include plays and concepts (www.iep.utm.edu). Aristotle also asserts that nature has a purpose, that it is moving toward its natural state, “telos” (Cazeaux, 2000, p7).

The classical view of beauty was represented by reworks and repeating the ideas of the classical philosophers by writers such as Thomas Aquinas until the 18th century; “None of the system builders of the 17th century – not Descartes, Locke, Spinoza, Berkeley, Leibniz – has anything much to say about the beautiful” (D Cooper, 1992, p46).

However, Hogarth (1697 – 1764) was hoping to educate as well as establish a consensus of aesthetic values of beauty. His book ‘Analysis of Beauty’ (1752) is an exercise in discovery as well as explanation. He uses images such as the range of corsets below at the bottom left of the opening plate to initiate debate over beauty and illustrate the beautiful. Armstrong (2004, p5) calls this “the earliest aesthetic experiment.”

Hogarth, The Analysis of Beauty, 1752, Plate 1, London, British Museum.

Hogarth asserts that the serpentine lines of corset number 4 above are the most beautiful and hopes that the reader will always have enough knowledge to pick from the three corsets aligned in the middle (corsets 3, 4, 5) or be thus educated by his point of view.  “And it is in this line which, he thinks, reveals the ‘essence’ – the secret – of beauty” (Armstrong, 2004, p5). Hogarth believes that there is an objective beauty that is dependent on shape or pattern.

Hogarth was a keen follower of the Socratic idea of beauty. He espoused the requirement of fitness for purpose.  In the first chapter of ‘Analysis of Beauty’ he contrasts a war-horse with a race-horse, recognising that both have a “characteristic beauty” with relation to their use (Burke, 1942, p 152). Hogarth recognises two forms of beauty; that of
purely formal beauty and the beauty of fitness; the first is governed by the serpentine line, the second arises ‘chiefly from a fitness to some designed purpose or use.’ (Burke, 1942, p 152)

In opposition to a fitness for purpose aspect of beauty, Immanuel Kant (1724 – 1804) asserts that beauty has no objective form, that it is an internal, subjective judgement of taste. “The main question which Kant asks in the third Critique is how a subjective, aesthetic judgement, a judgement of taste, can claim universal assent” (Cazeaux, 2000, p5).

Kant believes that nature is beautiful in itself and it is the manufactured things by humans that are beautiful only in a subjective sense. He uses the term ‘purposiveness’ to describe the lack of purpose to nature; nature just is.
Kant states that for beauty to be experienced in anything apart from nature the experience of the ‘sublime’ pleasure moment must be disinterested in the content or context of the object (Cazeaux, 2000, p7). Obviously this would exclude the theorems of maths and ideas of science as the content and context are required for their beauty to be considered.
David Hume's (1711 – 1776) essay, Moral and Political, 1742, echoes the Kantian assertion of subjectivity and individual pleasure; "Beauty in things exists merely in the mind which contemplates them." (http://www.iep.utm.edu/humeessa/) But this would mean, contrary to Socrates and Hogarth, that nothing can hold beauty without a human mind to contemplate it.
G W F Hegel (1770 – 1831) disagrees with the position of subjectivity, instead positing a spirit similar to the Platonic position of ideal forms existing outside human experience. He also believes in a grand plan, a spiritual underpinning and in the belief that the human race is moving toward this ultimate knowledge. “For Hegel, the artwork is the sensuous display of the mind’s relationship with external reality” (Cazeaux, 2000, p10). Hegel asserts that the content of an artwork should reflect its ideal form, “The importance of art for Hegel resides in its being the substantial expression of ideas.” In this Hegel did not foresee any of the abstract nature of Modern Art,
from the impressionists painting light instead of objects, through the performative, non-representational project of abstraction, to conceptual art’s simple sentences on gallery walls. (Cazeaux, 2000, p11)

Friedrich Nietzsche (1844 – 1900) removed the separation of appearance and reality, stating the world is “constructed through representation”, that art, and our perception of beauty within it is mere anthropomorphism. Of nature and beauty there is an unknowable quantity;
We obtain the concept, as we do the form, by overlooking what is individual and actual; whereas nature is acquainted with no forms and no concepts and likewise with no species, but only with an X which remains inaccessible and undefinable for us. (Nietzche, 1873, p56)

Nietzsche believes that there is no ultimate truth, that beauty is constructed in the mind, created by man not God.

During the Modern age, here defined as from the Industrial Revolution to the end of the First World War, Kantian ideas dominate. Georg Simmel (1900, p299) writes of the disinterested reflection, “When I call an object beautiful, its quality and significance become much more independent of the arrangements and the needs of the subject if it is merely useful.”

The rise of mechanical reproduction leads the philosopher Walter Benjamin to place more importance on authenticity and he asserts that art in the modern age is losing its ‘aura’ (Cazeaux, 2000, p300). It could be argued that developments in science lead to changing views of beauty. The Modernist aesthetic can be characterised as;
dependent upon the techniques and technologies... either as something expressive in its own right or as something which can open up a new, non-imitative perspective on the world. (Cazeaux, 2000, p367)

Although the term ‘Postmodernism’ would not be coined by Lyotard until 1979, it exists as a response to Modernism and demonstrates its distrust for universal truths or meta-narratives:
So Lyotard declares the death of meta-narratives… Whether meta-narratives are invoked to support the sciences (revelation of truth), political movements (emancipation of humanity) or artistic movements (achieving deeper vision) they no longer provide the legitimacy they did in the modern era. (Schroeder, 2005, p.329)
The scientific view put forward by Semir Zeki is that the ideal forms of Plato are related to environmental pressures and a previous knowledge of the form; “neurobiologically, there is not one visual aesthetic sense but many, each one tied to the activity of a functionally specialised visual processing system” (Zeki, 1999, p 87).

A discussion of beauty can be looked at in its opposition to ugliness. Plato believed that to create an ugly thing was an immoral activity, “For in all of them we find beauty and ugliness. And ugliness of form and bad rhythm and disharmony are akin to poor quality expression and character, and their opposites are akin to and represent good character and discipline” (Plato, 375 BC, p103). This sense of ‘goodness’ is discussed by Armstrong in his work “The Power of Beauty” (2005, p59-61). He notes that the architectural beauty inherent in an area such as Karl Marx Allee in East Berlin could be tempered by the knowledge of its use to celebrate Stalin. Here Kant’s pleasure of experience removes any objective sense of beauty with the horrors perpetrated by Stalin.
Taste therefore is what is contingent and transitory in a work of art, what belongs to a period or a nation, what is relative. No subject matter, no motif, no idea, no third dimension, no anatomic knowledge, no plasticity, no ability, no beauty, included in a work of art, have artistic value by themselves. (Venturi, 1944, p272)
This point illustrates a cultural language at work.  Mathematics can also be viewed as a language. Coomaraswamy noted in his 1941 review of Hardy’s ‘A Mathematician’s Apology’ that the apology would be of “greatest interest and value to ‘aestheticians’ since it is here for the first time that the beauty of mathematics has been discussed by a mathematician” (p 339).
The idea of simplicity is often included as an aspect of beauty; that idea of ideal form is reflected in Plato and Hogarth for example. “Mathematicians have customarily regarded a proof as beautiful if it conformed to the classical ideals of brevity and simplicity” (McAllister, 2005, p22). Science can be seen to judge things with reference to Occam’s Razor which states that when 2 or more explanations are possible the scientist should err towards the simplest, that the ‘truest’ will be the most elegant.
Munsterberg was sure that science was well on the way to scientifically formalising most aspects of aesthetic judgement through for example the use of brain scans. He was;
inclined to start with the study of the psychological processes in aesthetic enjoyment. Here alone is evidently solid ground. It was the great day of emancipation for aesthetics when at last it became liberated from metaphysical speculation. (Munsterberg, 1909, p 126)

Munsterberg appears to look to measurement as holding the key to the science of beauty. This certitude had been mirrored by Albert Michelson, American physicist in 1894, when he declared that all of science was discovered and that it was only a job for the values to be refined and made more accurate, “that the future truths of physical science are to be looked for in the sixth place of decimals." This is echoed by Lord Kelvin in 1900 in his speech to an assemblage of physicists at the British Association for the advancement of Science in 1900 in which he stated, "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.” These beliefs were already being challenged and were destroyed with Einstein’s Relativity and Quantum Mechanics,
The transition from classical to modern physics is often described as a revolution that took place between 1895 and 1927. (Brush, 1980, p 398)

Munsterberg echoes the Kantian idea of ‘purposiveness’ of nature when he asserts that nature is beautiful in itself, regardless of purpose or perception in the human mind, “Whether any such element of nature is comforting to ourselves or painful, is agreeably tickling or disagreeable, does not influence the beauty of nature” (Munsterberg,1909 p126).

 

In the twentieth century, a leading aesthetician, Clive Bell takes on the Kantian view but refines it with the idea of ‘Significant Form’, “a particular combination of lines and colours that stir our aesthetic emotions...These emotions are special and lofty” (Freeland, 2001, p 15). Bell was careful to include definitions of beauty that could encompass modern art in its many forms such as surrealism.

In Dali and Halsman’s book “Dali’s Moustache,” a series of questions are posed for Dali to address: “What is ugliness? Disorder. What is beauty? Harmony” (Dali, 1954, pp 57 – 63). In his painting he regarded himself as “swimming between two kinds of water, the cold water of art and the warm water of science.” He did not accept that science could fully explain reality, “it is a greater illusion than the dream world” (Ades, 2000, pp 10-11).  

 The Oxford English Dictionary definition of beauty is: “a combination of qualities, such as shape, colour, or form, that pleases the aesthetic senses, especially the sight.” This echoes how philosophers have forged definitions of beauty as outlined in this chapter. The Oxford English Dictionary offers a second definition: “a beautiful or pleasing thing” and in its need to please, beauty comes closer to the functionality and rightness for purpose posited by Socrates.  This latter definition linking fitness for purpose is an essential part of considering beauty in science. For the purpose of this essay examining views of beauty in science and art, the definition of beauty must include both elements. Beauty is therefore a combination of aesthetic qualities and fitness for purpose.

Chapter 3 – The Science of Beauty
Measurement features once more in the science of beauty.  In an attempt to show statistics can be seen as a reflection of universal aesthetic, Komar & Melamid in 1992 began the project ‘America’s Most Wanted.’ Their project involved simple questions about the most popular ingredients in paintings.

America’s Most Wanted, 1994, Komar & Melamid

Various ingredients were discovered by opinion polling of the most wanted aspects. These included: recognisable historic figures, running water, the colour blue etc as noted by Dutton (2009) in ‘The Art Instinct.’ He goes on to make the point that if it were a question of favourite foods, one would not want a plate with pizzas, burgers and ice cream necessarily on the same plate;
just because people like George Washington, African game, and children in their pictures, it doesn’t follow that they want them all in the same one.  (Dutton, 2009, p 14)

Opinions were garnered from 14 countries and the World Wide Web, showing that blue is the world’s favourite colour closely followed by green, that paintings of the size of the television or dishwasher were most popular and that scenes of nature were paramount in disparate countries such as China, Kenya and the USA. Interestingly, in the opposite quest of the World’s ‘least’ wanted, it is the wall size abstract paintings the world over (Komar & Melamid, 1994).
  
Screenshot’s from http://awp.diaart.org/km/painting.html Komar 1994

Piet Mondrian also searched in his work for a universal aesthetic. His choice of using just line and blocks of colour was similar to Malevich (Zeki, 1999, p109). Mondrian’s reduction of colour and form is seen as an attempt to reduce to the simplest form.
Artists at the beginning of the 20th century, such as Mondrian, actually echoed the pursuit of human’s domination of nature with geometry in their art. (Shearer, 1992, p148)

That Mondrian is the most ‘true’ aesthetic compared to computer generated ‘pseudo-Mondrians’ has often been tested. In McManus’s (1993) study at University College London, when asked to select the actual Mondrian painting from real and imitations, it was found that students picked the real Mondrian image significantly more often and the majority believed it to have the greatest beauty (McManus, 1993, pp 86 – 88). McManus concludes in his study that as the subjects had significantly greater than chance alone of selecting the original Mondrian.  It suggested his study “may encapsulate some universal principal of compositional order which can be detected.”

The 1920’s saw the rise of quantum mechanics as a truer description of the world and nature despite its bizarre acceptance that particles can be seen to act as waves or particles depending on the observation made. Further quantum mechanics shaped the understanding that Kelvin’s view of merely refining results would never give ultimate scientific truth, statistics became the measurement tool (Stewart, 1989, p30).

 

Chapter 4 – The Beauty of Science

The Pythagorean School, started by Pythagoras (circa 580 – 500 BC) had the philosophy that “all is number” and has been credited with the first proof of a theorem. Pythagoras’s religion was numbers. “He worshipped whole numbers and simple fractions. And he took his religion seriously” (Hoffman, 1998, p45).
Since the time of Pythagoras the expectation that mathematical and scientific theorems can exhibit beauty has been accepted. Kemp (2000, p 104) reminds us “that pure geometric forms exhibit aesthetic perfection has an ancient ancestry.”

Although Plato was not the first to identify the 5 regular solids that now bear his name, Emmer (1982, pp277) states that Plato was the first that can be found to describe them fully in his dialogue Timaeus. The 5 Platonic solids: tetrahedron, cube, icosahedron, octahedron and dodecahedron are all the regular solids in the universe in 3 dimensional space, through all time. The mathematical proof of this confirms the truth of this theory and from that truth can be found a mathematical beauty;
The theorem stating that in three dimensions there are only five regular solids (the Platonic solids) is generally considered to be beautiful. (G-C Rota, 2005, pp4)

The Golden Section or ratio lauded by Da Vinci and others is a proportion of line that is, purported to be most pleasing to the eye. It is first seen in the Pythagorean Pentagon (Boyer, 1968, p 50) and, although it would not be discovered for 1500 years, it is a product of the Fibbonaci sequence. Thus the ‘truest’ art rule is based on the maths of nature. “The Golden Rectangle is said to be one of the most visually satisfying of all geometric forms” (Bergamini, 1963, p 94).

Fibonacci Sequence - http://www.toddholoubek.com/classes/livingart/?page_id=29
The Fibonacci sequence was created in response to a mathematical problem about how many rabbits you would get over an amount of time. The sequence continues by adding the 2 previous results together: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... and can be seen in the spirals in sunflower heads and throughout nature. The ratio that 2 consecutive members of the sequence gives is 1.618... which is the same as the proportions in Pythagorean Pentagon and Da Vinci’s Golden Section (Stewart, 1989, pp 257 – 261). Our appreciation of beauty in nature can be seen in this mathematical form. Furthermore, it could be argued, our appreciation of other forms as beautiful (conforming to these proportions) is also mathematical.


Golden ratio Parthenon: http://www.nilesjohnson.net/calculus-topics-2.html

During the Renaissance, Euclidean geometry was reflected in the rise of perspective. The discovery and use of perspective in the Renaissance radically altered the art that was being produced. As J. V. Field notes in her book ‘The Invention of Infinity – Mathematics and Art in the Renaissance,’ the artist’s use of perspective also directly affected learned mathematics (Field, 1997, p 7).

Paul Dirac quoted in Rieser’s Art & Science (1972, p18) “that beauty was a safe guide in the search for truth and, further, that it has always been found that highly successful rules are also beautiful.”  
Newton’s assertion in the Principia that if he had seen further it was due to standing on the shoulders of giants. His beautiful rules build on historical scientific achievements.  This exemplifies the additive nature of the scientific method;
There are 2 kinds of truth: truths of reasoning and truths of fact. Truths of reasoning are necessary and their opposite is impossible; those of fact are contingent and their opposite is possible. (Leibniz, 1714)

Leibniz shows the pragmatism of science, how results can overturn previous truths.  Newton’s universal law of gravitation (1687), heralded as beautiful by Matthew Collins (2010), was replaced by Einstein’s ‘General Theory of Relativity’ (1915). However Newtonian gravitation still works on a human scale which is why it is used in creating the course for spaceships to the moon and other bodies in the solar system, but Einstein’s theory is required to explain quantum events and massive objects such as black holes. Newton’s truth has been refined by Einstein but it still works. Newton is accurate to the 6th decimal place but science is yet to discover whether Einstein differs at all from experiment.

The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. (G H Hardy 1941, pp 24) 

This quest for beauty in mathematics is in both the theorems and the proof. Rota (2005, p9) claims that it is rare to have beauty in both, that the most beautiful theorem will not necessarily have an elegant proof. He also makes the point that when mathematicians are seeking solutions that beauty is not foremost in their minds; “Mathematicians work to solve problems and invent theories that will shed new light, not to produce beautiful theorems or pretty proofs” (Rota, 2005, p 9). McAllister argues the contrary that,
Some mathematicians claim also that beauty acts as a guide in making mathematical discoveries and that beauty is an objective factor in establishing the validity and importance of a mathematical result. (McAllister, 2005, p15)
The nature of the proof itself will have an aesthetic quality. This beautiful truth can be seen in contrast to the solution for the 4 colour problem. The problem consists of a simple question: what is the least amount of colours that are required on a map of various territories represented by a colour such that no two colours are touching? The solution, as the title of the problem suggests, is four, but there is not an elegant proof for this, the solution is come by a computer crunching through thousands of possible shapes and set ups. Mathematicians refer to this as an ugly solution; no new knowledge is gained by such an exhaustive search only possible through the mindless repetitions of a computer (McAllister, 2005, p 18). Paul Erdos, the 20th century’s most prolific mathematician, (Hoffman, 1998, p5) referred to the 4 colour solution as lacking insight;
I’m not an expert on the four-colour problem but I assume the proof is true. However, it’s not beautiful. I’d prefer to see a proof that gives insight into why four colours are sufficient. (Hoffman, 1998, p44)

Science’s quest for beauty in its theories continued through the 20th century. This is exemplified in the work of American particle accelerator in the 1980s and 90s. The ‘particle zoo’ that was discovered in Fermilab’s tevitron on smashing protons together required a “beautifully elegant” theory to explain them. To this end mathematicians were asked by Professor Jacobo Konisberg of Fermilab to address this problem and the ‘Standard Model of Elementary Particles’ was created by Professor Frank Wilczek in MIT. By using only 6 quarks, 6 leptons and 4 forces the zoo could be explained. The top quark was predicted in the standard model but due to its infrequency in nature was not discovered until 1995 when millions and millions of collisions had been analysed. (BBC Horizon, 2011)

Standard Model of Elementary Particles, Fermilab.

Physicist Max Tegmar states on ‘Horizon – What is reality?’ (2011) that every time that there is a messy, complicated solution within science “eventually someone has come along and unified it all into a beautifully elegant solution.”
The need for elegance and simplicity in mathematics was summarised by Poincare (1854-1912) who
lauded ‘intuition’ as the key to his ‘ability to perceive the whole argument at a glance’ and to determine where the ‘beauty’ of truth within nature’s mathematics [existed]. (Kemp, 2000, p 73)
Mathematics was dominated by Euclidean geometry until the 19th century. “Geometric axioms about the physical world unquestioned since Euclid in 300BC were suddenly invalid” (Shearer, 1992, p147). There was a shift however through the work of Riemann (1826 – 1866) who published his paper in 1857 on multi-dimensional geometry. Euclidean geometry was then seen as insufficient to describe reality.  Euclidean geometry’s ‘fall’ can be exemplified by considering the sum of angles in a triangle on a plane surface and how the angles of a triangle do not add up to 180 degrees when examined on certain curved surfaces, on a globe for example.  After the publication of Riemann’s geometry our view of the space that we live in was radically altered.  This is noted by Henri Poincare, “our choice among all possible conventions is guided by experimental facts” (Shearer, 1992, p143).
The artist Marcel Duchamp (1887 – 1968) was a friend of mathematician Henri Poincare. He used mathematical ideas in his art.  Adcock, 1984, p257, declares, “Poincare demonstrated that one’s choice of geometry was conventional. Duchamp demonstrated that one’s choice of art was conventional.” Duchamp presented objects or ‘ready mades’ to the art world.  For example he presented a urinal entitled Fountain (1917).These items were chosen for their lack of beauty, “they were visually neutral – he neither liked them or disliked them” (Adcock, 1984, p257). Duchamp presented the ready mades to the art world “as a challenge and that now they admired them for their aesthetic beauty” (ibid). Within his work ‘Bride Stripped Bare by Her Bachelors, Even’ he reflects the new geometry of n-dimensional space (Holton,2001,p131).

The Bride Stripped Bare by her Bachelors, Even [The Large Glass] (1915-1923)
© Tate © Succession Marcel Duchamp/ADAGP, Paris and DACS, London 2006

Chapter 5 – Chaotic Beauty
Previous chapters have considered the science of beauty and the beauty of science. The focus has been on simple elegant rules or truths such as Mondrian’s art or Einstein’s E=MC2. So one could ask: Can there exist beauty in chaos, which might appear messy or unpredictable? Nature can be described as chaotic yet is still undoubtedly beautiful.

Chaos theory shows that from simple mathematical laws being iterated, complex and chaotic results can occur;
The ability of even simple equations to generate motion so complex, so sensitive to measurement, that it appears random. Appropriately it’s called chaos. (Stewart, 1989, p 12)

Poincare discovered chaos in his solution to the three body problem which looks how the Sun, Earth and Moon would act through time under Newtonian laws. After repeating calculations hundreds of times he found chaos.  Had he been able to plot millions of points he would have discovered fractals.  Mandelbrot (1993, p11) says that they “could not possibly have arisen before the hardware was ready and the software was being developed.”

Benoit Mandelbrot was the first to use the term “fractal” to describe the imagery associated with chaotic systems. He adapted the Latin adjective ‘fractus’ meaning to break or create irregular fragments. In his book ‘The Fractal Geometry of Nature’ he bemoans geometry’s efforts to represent nature as failed, “Clouds are not spheres, mountains are not cones, coastlines are not circles.” Yet by using the mathematics of chaos, nature can be better reflected, “the advent of fractal geometry, a model which brings geometry closer to the spirit of nature than ever before” (Mandelbrot, 1977, p 1 – 5).
Fractal Fern - The Rockefeller Institute 2010

Fractals reflect one of the most important aspects of nature. They are self similar on various scales. This can be seen most easily in the fractal fern above and also in the shapes of clouds. Mandelbrot describes his book as “both casebook and manifesto. Furthermore, it reveals a totally new world of plastic beauty” (Mandelbrot, 1977, p 3).

Mandelbrot gave a new mathematical way to accurately represent nature; “We turn from mathematicians concerned with Art for Art’s sake to men that celebrate Nature by trying to imitate it” (Mandelbrot, 1977, p407). Though he explains fractals are represented in his outputs only as points of black and white, the range of colours evident in the common images of the Mandelbrot set and other fractals are a human construct from the aesthetic judgement of the programmer that created them (Mandelbrot, 1977, p 7).

Fractional dimensionality has been recognised in the work of American artist Jackson Pollock. “Recent research shows that Pollock’s drip paintings exhibit elements of pattern and symmetry congruent with fractal images” (Cazeaux, 2000, p7). Pollock himself could not have been fully aware of it in 1950 when he painted One (No. 31).

Jackson Pollock; One: Number 31, 1950 © 2010 Pollock-Krasner Foundation

Richard Taylor at the University of New South Wales has analysed Pollock’s patterns and shows that the paintings “are fractal” (Taylor, 1999). He proves the fractal nature of Pollock by comparing its dimensionality with known fractal objects such as the length of a coastline. Taylor is certain of Pollock’s importance and ability to reflect nature;
Pollock’s contribution to the evolution of art is secure. He described Nature directly. Rather than mimicking Nature, he adopted its language – fractals – to build his own patterns. (Taylor, 1999)

When interviewed about whether his paintings were trying to be ‘abstract not figurative,’ Pollock replied, “I don’t paint Nature, I am Nature.” He was concerned with the rhythms of nature; “I work from the inside out, like Nature” (Collings, 1999, p44).

The idea that Mondrian was successful in simplifying and capturing all that is important in nature and a universal aesthetic has been denied with the new geometry of chaos, 
In view of what we now know from the new fractal model, what was discerned by Mondrian as nature’s most fundamental principle – the right angle in its vertical position – now reads as Euclidean and outdated, having very little to do with nature’s universal rules. (Shearer, 1992, p149)

The so-called unreasonable effectiveness of mathematics in the natural sciences is celebrated; “We have tried to make mathematics a consistent, beautiful thing, and by so doing we have had an amazing number of successful applications to the real world” (Hamming, 1980, p87).

Conclusion

The clichéd phrase that beauty is in the eye of the beholder, like most clichés has a grain of truth in it, but as I have shown in this essay there is a universality to truth and beauty that can be applied to both science and art.
Modern art could be accused of turning its back on beauty but maintaining a quest for truth, eg Tracey Emin’s Bed (1988). Within art, the ‘loss’ of beauty since the beginning of the 20th century has been replaced with a version of truth. Duchamp’s Fountain and the Cubist movement had different aims in their art.

In his review in the Art Bulletin of Hardy’s Apology, Coomaraswamy (1941, p339) praises the attempt to define mathematical beauty but appears aggrieved that the art world does not hold such a well defined idea about beauty.
Taste is obviously a consideration in aesthetic judgements of art. Yet the same concerns abound in mathematics too. “It may be that different mathematicians at the same time hold to different aesthetic judgements” McAllister (2005, p16).  He also emphasises that tastes evolve through time.
Although the concept of beauty has been shown to be malleable through time and across disciplines, certain concepts continue to dominate: simplicity, elegance and a fitness for purpose. There is also the need for the subjective experience within the viewer to be recognised as where the beauty really exists.

Scientific beauty has a need for truth, elegance and simplicity, but science’s own pragmatism ensures that even ugly solutions are used until their beautiful counterparts are found. Quantum mechanics is the most accurate and true description of nature yet achieved. Professor Anton Zeilinger says that it is “mathematically beautiful, it describes everything, it just doesn’t make sense” (BBC Horizon 2011).

The beauty of Newton’s Universal law of gravitation was refined and replaced by Einstein’s Relativity but the inverse square law applied by Newton is still being used, its truth and beauty undiminished.

As the geometry of Euclid fell, a new art followed into n-dimensions, Cubism, the Vortists etc. echoing the revolution at the Renaissance with the advent of perspective. Science and Art were moving ahead together.

I have considered views of beauty in art and science. I have examined the existence of two cultures and a possible shared space.  Though the aims of the two cultures are markedly different they both proffer beauty. I have shown that the common aspects of beauty in art and science include aspects of truth, form and fitness for purpose.  

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