Below we give a version of the Introduction to the article. We encourage the reader who finds the topic interesting to read the full article, which goes into considerable technical detail, at the link given above.
Art, Mathematics, Music and the Physical Universe
Our knowledge of the external physical universe obviously comes entirely from our senses, mostly by seeing, hearing and touching.
As a beginning let us consider the sense of vision. When we look at an object such as a plant we observe the green of the leaves and the colour of the flower. But these are solely subjective. The flower does not in itself possess colour but just has a certain characteristic physical and chemical composition that produces the colour we see. The actual observation of colour occurs as a consequence of the light of the Sun being partially absorbed and partially reflected by the material of the flower. The reflected light rays from the flower arrive at our eyes where they fall on the retinas leading to impulses that travel down the optic nerves to the brain where electric currents are produced that give rise to our seeing colour. If the person observing the flower has impaired colour vision, like the chemist and founder of modern atomic theory John Dalton, he or she will not observe the colour that a normal person will see. Thus what we see is necessarily not a complete or accurate observation of the external world.
A truly interesting manifestation of this has been produced by the Magic Eye pictures [A New Way of Looking at the World (1964)], or Single Image Random Dot Stereograms, that give rise to remarkable visions of three-dimensional objects by employing coloured dots, the 'Salitsky Dots', on a flat piece of paper.
As in the case of vision, there are no actual sounds in the physical world. For example when one of the hammers of a piano hits a string it causes it to vibrate, disturbing the air in its neighbourhood. This causes a wave motion to travel through the air to our two ears causing the eardrums to vibrate sending signals down the nerves to the brain where electric currents are produced that lead to the experience of the sensation of sound.
Again our experience of the sounds of thunder and the flashes of lightning during a thunder-storm are just sensual impressions of the complicated physical phenomena taking place in the lower atmosphere of the earth.
As a consequence it seems that the external world is not what we observe with our senses but is purely an inference from our observations. Thus our senses provide us with just an image of the world around us and not the actual universe, an inference not unfamiliar to the Greek philosopher Plato. Nevertheless the images that are produced in our brain are so plausible that we behave as if the images of the objects that we perceive, with their visual attributes of colour, their texture and feel, and the sounds that arise from them, are the actual physical objects themselves.
In the past this led to many misunderstandings about the universe. As an example our hands, feet and the rest of our body feel warmth and cold providing a subjective estimate of temperature although temperature is accurately measured by thermometers using, for example, the expansion and contraction of liquid mercury in a fine tube. At the beginning of the 18th Century it was believed by Georg Stahl that heat was transferred by means of a substance called phlogiston but this was proved to be false by Antoine-Laurent Lavoisier towards the end of that century, heat being actually transferred by the kinetic energy of vibration of matter.
A mathematical theory of heat was developed by Sadi Carnot who introduced the heat engine cycle in 1824 and Rudolf Clausius who in 1850 introduced the second law of thermodynamics and then in 1865 introduced the concept of entropy that may be regarded as a measure of disorder. This led to the mathematical development of the subject of Thermodynamics and eventually to Statistical Mechanics with the work of James Clerk Maxwell and Ludwig Boltzmann in the 1870s who introduced the fundamental formula for the entropy S = k log W where W is the number of distinct accessible states of an enclosed dynamical system in equilibrium and k is Boltzmann's constant. W can also be regarded as the thermodynamic probability of the system.
The second law of thermodynamics states that the entropy S of an enclosed dynamical system has the property of increasing with time, or alternatively that heat cannot be transferred from a cold body to a hot body without other simultaneous changes taking place in the system of bodies or its environment. The idea of the arrow of time arises from the understanding that the entropy of the universe increases with the flow of time.
The Mental World or Consciousness
It is proposed that the entire external physical universe is an inference from our sense data. Furthermore our individual physical body is also just an inference from our senses even though its existence is not regarded as being in question.
Assuming the existence of our body and the physical world in general, since otherwise we revert to the philosophy of solipsism that leads us nowhere, our sense data arising from seeing, hearing, touching and so on is produced by our nervous system and brain. It is conceivable that our sense data are simply a manifestation of the electromagnetic field produced by electric current flows in the brain and that our thought processes are entirely controlled by them including all of our feelings such as pain and happiness, our thoughts, memories, dreams and decisions. Thus when we make a decision to carry out a certain action this is just a function of the physical processes in the brain and the decision is not made by our conscious mind but by the physical brain. In fact it has been shown that the physical brain seems to make a decision about an action before we become conscious of making it which, if really true, is quite a remarkable discovery.
It is worth remarking that the pianist Vladimir Ashkenazi made the revealing comment in a 1970 film that when he is playing the piano at his highest potential he does not consciously control his fingers at all and that it feels as if his body is playing by itself. Indeed the speed of his fingers on the piano keyboard is often so rapid, such as when he is playing a Chopin study, that it is obviously impossible for his performance to be carried out in a purely conscious manner and that it must be controlled by his physical brain.
The brain controls our entire physical activity by means of the nervous system coupled to our body and decides everything that we do with the aid of the conscious mind by which we perceive the world. We describe our view of the physical universe as well as our feelings, innermost thoughts and desires by means of speech, the written word, and by the creative application of art, music and mathematics.
From this point of view consciousness can be regarded as the process by which the physical universe can perceive itself. In particular Homo Sapiens have been able to look into the far reaches of space as well as backwards in time almost to the beginning of the universe, examine microscopic phenomena, describe the world by means of art and photography, music and mathematics, and modify the behaviour of the external world by the application of science and engineering.
Our conscious mind is controlled by a censor in the brain that decides what we are allowed to know or to remember from the past that which is retained in the unconscious mind and this, it must be accepted, can subvert our motives as well as our understanding of the world.
The human brain has developed mathematics enabling it to provide a purely logical and symbolic description of the external universe whereas it employs art and music as well as speech to describe the world and our feelings in an intuitive and frequently beautiful way but in a considerably less logical manner although nevertheless often based on geometry in the case of pictorial art, sculpture and architecture, and symbolic notation in the case of music.
In the case of architecture the design of some monasteries is partially based on the square-root of two which is an irrational number, that is it cannot be expressed as the ratio of two integers, a result first proved by the Greek mathematician Eudoxus (409-356 BC) and shown in Book V of the Elements of Euclid. For example, the length of the nave of the Gothic church of Tintern Abbey on the river Wye, was constructed in the 13th Century to have the same length as the diagonal of the cloister, a design which relates to the mathematical result using the theorem of Pythagoras that the diagonal of a unit square has the length √2. Thus the length of the nave is √2 times the length of the side of the cloister. Moreover the width of the church is the same as the length of the cloister edge and so the nave length is also √2 times the church width. The presbytery and chapel at the east end of the church has the shape of a square whose dimensions are the same as the cloister.
It is significant that all art forms are not simply the imaginative work of our minds but involve the manipulation of the physical world by means of our bodies controlled by our brains and using the senses to observe our creative work. Thus architecture involves the use of engineering skills and the manufacture of machines from various materials, as well as mathematics, to construct buildings; sculpture involves the use of our hands controlled by our brains to shape physical objects; painting and drawing involve the use of our hands to manipulate paint with a brush or a pencil on a suitable surface such as canvas and paper guided by our visual sense; literature involves writing or typing on paper employing our hands; and playing music involves using our hands on a keyboard instrument, stroking a string with a bow on an instrument such as a violin or cello, or our mouths to blow air through a tube such as an oboe guided by our sense of sound. In every case there exists cooperation between our brains together with our conscious minds, our senses and our physical bodies.
Also it is important to realize that our appreciation of the mathematical structure of the physical world is generally involved in much art work and the intriguing relationship between mathematics, art and music is the subject of the following section.
Art, Mathematics and Music
The drawing and painting of people and animals, an activity of artists since primitive times, is clearly associated with symmetry, and thus geometry, since the human form and the forms of living creatures in general are obviously highly symmetrical in shape. Indeed the beauty of the human form, and also of an animal such as the horse, is firmly based on their symmetrical proportions. Thus artists such as Leonardo da Vinci and Albrecht Dürer were deeply concerned with the proportions of the human form from a truly geometrical point of view.
Further, sculpture and architecture have a strongly geometrical basis as is evident from the wonderful work of the Italian Renaissance men Brunelleschi, Alberti and Michelangelo.
The geometry of perspective used by the Renaissance artists and subsequently by many artists in later times has been subjected to detailed analysis by several authors, for example in the book by Martin Kemp [The Science of Art (1990)] devoted to optical themes in western art.
The repeating patterns and designs used in Islamic art are strongly based on the geometry of the fundamental regular polygons such as the triangle, square, pentagon, hexagon, octagon and so on, up to polygons with 24 sides, and also on other beautiful geometrical shapes.
The subject of astronomy has always been dependent to a great extent on geometry. Johannes Kepler, the astronomer who discovered that the planets move round the sun in ellipses, was greatly interested in the subject of geometry. He examined how a plane could be completely filled with equal regular polygons such as the equilateral triangle, the square and the hexagon with six equal sides, and also studied the subject of polyhedra or convex solid bodies with flat faces.
In 1985 an International Congress on M C Escher was held in Rome. The graphic work of Escher has been of profound interest to many mathematicians for a considerable time and has led to many kinds of mathematical challenges, geometrical investigations and research [M C Escher: Art and Science (1986)]. His work was much influenced by the repeating patterns used by the Islamic artists of Granada in Spain.
Escher was able to use the symmetrical shapes of living creatures such as birds, fishes and insects of various kinds as well as flowers to create his wonderful periodic patterns. In this context it is well worth recollecting that scientifically minded people believe that the shapes of all living creatures and vegetation, including plants and trees, have come about by the action of Darwinian evolution arising from their successful adaptation to the ambient environment in which they flourished over long periods of time often leading to their beautiful symmetry and consequently their use as models for human art. Thus we could say that art is founded, at least in part, on the past history of life on Earth.
Music and mathematics also have a close affinity as is readily apparent from the two books of 24 Preludes and Fugues composed by Johann Sebastian Bach using the equal-temperament scale, also known as the well-tempered Clavier. One remembers that Pythagoras did fundamental work on the foundations of the musical scale and on the vibrations of strings, and that the French mathematician Mersenne was also a musician and was the originator of the equal-temperament scale.
Artists have frequently shown a great affinity with music and their pictures often include singers, musicians as well as musical instruments. The modern artist Wassily Kandinsky originally wished to be a musician and another artist of the same generation Paul Klee was a talented violinist throughout his life.
For all these diverse reasons it is highly interesting as well as instructive and revealing to investigate the relationship between art, music and mathematics in its various aspects.