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Mathematics in the Arts

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At first glance, mathematics and the arts seem to have nothing in common. Mathematics is above all logical and about calculations and proofs, whereas the arts are more emotional and creative. However, once you scratch the surface of each, you find they are linked in more ways than you would imagine. The aim of this Entry is to delve into the relationship between the two, and prove that they are more linked than they seem.

A lot of people hate maths - they couldn't stand their maths lessons in school, and tend to pull a face at the slightest mention of the subject. However, there aren't many people in the world who hate the arts – everyone has hummed a tune, painted a picture, read a novel, or seen a play. So how can one of the most hated subjects be linked to one of the most loved? Looking closer, we see that many mathematicians are highly musical, so there we have our first, albeit tenuous, link – the people who do maths are also doing music. However, this entry aims to show that there are many more, deeper, links between mathematics and the arts by looking at each branch of the arts in turn, starting with mathematics in music.

Mathematics in Music

The most obvious use of mathematics in music is in the songs of people such as Tom Lehrer, where mathematical concepts and equations are used as the lyrics of the song. These can be written for comedy value, or as teaching aids (for example to help children learn their times-tables). However, the link between mathematics and music goes much deeper than this.

To Greeks in the time of Pythagoras, the link between mathematics and music was never questioned – music was part of the mathematical quadrivium1 (the other elements of which being arithmetic, geometry and astronomy). For the Greeks, music was all about mathematics – creativity wasn't considered, and only the science of harmony and sound were studied. It had always been known that certain combinations sound nicer than others, but the Greeks were the first to study the notions of consonance (the pleasant sounds) and dissonance (sounds that clash). They were also the first to realise that, say, if you pluck a string, and then pluck another string which is twice the length of the first, that the notes would sound the same, albeit one higher than the other. This is what we would call an octave, and we now know the reason behind it.

The pitch (ie, how high or low the note sounds) is determined by the frequency at which the air vibrates to make the sound. This frequency is measured in hertz (Hz), which is the number of times per second the sound wave vibrates. For example, the note we call middle C, which can be found more or less in the middle of a piano keyboard, has a frequency of around 262 Hz, meaning that anything producing sound waves at this frequency will make a sound like that note. Any note producing a frequency with a ratio of 1:2 to another note will sound the same but at a different pitch, producing an octave. In the western scale of music (C, D, E, F, G, A, B), notes with a frequency ratio of 1:2 are given the same name. Consonant (pleasing) chords2 are those where the frequencies of the notes 'match' - ie, the peaks and troughs of the waves coincide more often than not. The ratios 1:2, 2:3, 3:4 and 4:5 all sound pleasing - thus when the notes C & C, C & G, C & F, and C & E are played together, a consonant chord is created - and when, say, C and D are played together, a dissonant chord is heard. Interestingly, these ratios correspond not only to the frequency of the note, but also to the string-length needed to produce the note. So by shortening a string which produced an C to four-fifths of its length, you can play the note E.

The Golden Ratio

Another link between mathematics and music can be found in Fibonacci numbers and the golden ratio. Fibonacci numbers are easy to define, and are the numbers of a sequence discovered by Leonardo da Piza (otherwise known as Fibonacci). Its first two elements are both 1, and then each subsequent element is formed by adding the previous two together. Thus, the first few elements are 1, 1, 2, 3, 5, 8, 13... The so-called golden ratio is, however, slightly more difficult to define, not least because it is often defined in two different ways. The first definition of a golden ratio is φ = (1 + √5)/2 ≈ 1.61803399, which is the number that the ratios between consecutive Fibonacci numbers tend towards. It can be found in many patterns in nature, such as flowers and pine cones, and has been used in many paintings and other works of art. The second definition of a golden ratio is a more geometric interpretation. You divide a line into golden sections if, when dividing a line into two parts, the ratio between the whole line and the largest part is equal to the ratio between the two parts. This interpretation can also be found in nature, and also in geometrical shapes, such as the ratio between the lengths of a side and a diagonal of a normal pentagon. But are either of these golden ratios used in music? Ratios of this kind can, in fact, be found in many pieces of music, such as Handel's Hallelujah Chorus, where certain climaxes of the piece are found in bars relating to the golden ratio. However, in cases such as this, it is unknown whether the golden ratio was used consciously, or if it resulted from an unconscious idea that it 'sounds better' to use golden ratios in music. Some musicians have, nevertheless, set out to use the golden ratio and Fibonacci numbers in their compositions. It isn't only in musical compositions that the golden ratio can be found – it can be seen in instruments themselves. If you look at a piano keyboard, you see that in one octave there are eight white notes and five black ones, making 13 notes in the octave, with the black keys split into groups of two and three. All of those numbers are in the Fibonacci sequence. In addition, some instruments, such as violins, are constructed according to the golden ratio.

One musician who was interested in the link between mathematics and music was Wolfgang Amadeus Mozart. It has been said that he was intrigued by the world of mathematics, and his musical aptitude hardly needs to be discussed here. Mozart often divided his compositions into two sections - the Exposition, where the main theme is introduced, and the Development and Recapitulation, where he develops and revisits the theme. Modern analysis of his music has shown that the golden ratio can be found in the division between the two sections in almost all of his works, where the Exposition is the shorter section. Yet we reach a problem when we analyse all of Mozart's pieces that were in this form, as - while many of them come as close to the golden ratio as it is possible to be in whole numbers - some of them deviate from this hugely. This means we can never be sure whether Mozart - who enjoyed and respected mathematics - used the golden ratio consciously, or if he just did what 'sounded right' and it just happened to fit into the pattern we are looking for.

Mathematics in the Visual Arts

Having shown a link between mathematics and music - even if it is a subconscious one - let us now move on to looking at mathematics and the visual arts; that is, painting and sculpture. The golden ratio, φ, or phi, can also be found in works of art. In 1876, Gustav Fechner conducted a survey which involved showing people a selection of rectangles and asking them which they preferred - which was most pleasing to the eye. The result was that 75% of people preferred those rectangles with ratios close to the golden ratio; that is to say, rectangles for which the height divided by the width (or vice versa) results in a number close to the golden ratio of φ.

But has this golden ratio actually been used by artists? Many people have claimed that rectangles fitting this ratio can be drawn around certain key people and objects in the works of Leonardo Da Vinci, but few can be precise about the size and positioning of these rectangles. It is certainly possible that Da Vinci used golden rectangles in his art, as he was a close friend of Luca Pacioli, who published a three-volume treatise on the golden ratio in 1509 - yet there is no evidence either for or against his conscious use of the 'golden rectangle.' If he did indeed use rectangles such as these, and if its use is accidental, this backs up the fact that golden rectangles are 'nicer' and seem more natural. It isn't certain by any means whether Leonardo did actually use them, or whether those who are looking for golden rectangles find them because they want to find them, rather than because they are actually there.

There is evidence that some artists have used the golden ratio intentionally, such as Salvador Dali in his work Sacrament of the Last Supper. One instance of this ratio can be found in the pentagons behind the figure of Jesus, which, as already stated, contain golden ratios. It is not known why Dali chose to include golden ratios in his work, however.

Fractals

Golden ratios aren't the only link between mathematics and art. Another important link is the fractal which is, according to the dictionary, a term coined by Mandelbrot in 1975. It refers to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration. In simpler terms, it is something that looks the same no matter how close you get to it, such as a fern leaf, or a piece of broccoli. For both of these, you can take a small section, and it looks very similar to the whole thing, albeit at a different scale. That, in essence, is a fractal. These can be art in themselves, and can be very beautiful to look at. Fractals are generally made by computer programmes that take a shape and carry out repeated operations on this shape, so the shape becomes more complicated with each iteration.

Yet referring to fractals as art is like colouring a graph in pretty colours and calling it art. Are fractals actually used in art, or are they representations of mathematical ideas that just happen to look nice? In fact, fractals play a large part in art - especially in computer-generated works. If an artist wants to create a realistic landscape on a computer, they might draw fields, trees, a nice blue sky, and maybe even a few sheep. However, it will never quite look real – the human eye can tell that there is something 'wrong' with the picture. Due to the fact that things like trees are fractals in real life, they really need to be fractals in your art. Computer art that includes fractals is often mistaken for photographs, as our eyes aren't used to seeing objects, like trees, as fractals in anything other than photographs and in real life. Another property of fractals that make them useful in computer art is that you can inject an element of randomness into them, meaning that all the trees are different - which is very important as it would be pretty obvious if all your trees were exactly the same.

Perspective

The third aspect of mathematics with art is perspective. Without perspective there would be no sense of depth in any work of art. It is possible to create works of art in perspective without knowing anything about the mathematics of it at all – if an artist sits in front of whatever they are trying to depict and draws exactly what he sees, his art will be in perspective. The problem is that it is not always possible to sit in front of what you want to draw for the length of time it will take to complete the work, and sometimes the artist wants to move objects around in the composition which can't physically be moved - for example, that oak tree might look better to the right of the castle rather than where it is. Maybe the artist even wants to draw things that aren't really there. In these cases, an understanding of how perspective works is required from the start. Consideration of perspective really began with the ancient Greeks, who showed some understanding of it in building their stage sets, yet there is no evidence that they understood the mathematics behind it - rather, they were doing what looked right to them. By the 13th Century Giotto understood that in a painting lines above the eye-level should slope downwards, and lines below the eye-level should slope upwards. However, it is Brunelleschi who is credited with first discovering the 'vanishing point' in 1413 - that is to say, the point in a painting towards which all lines which aren't parallel to the observer should slope. Although Brunelleschi was trained in mathematics and obviously understood the mathematical properties of the vanishing point, he never wrote them down. This was left to Alberti in around 1435, who wrote that perspective is completely mathematical and concerns the roots in nature from which this graceful and noble art arises. Alberti also chose not to give mathematical proofs, giving only a background to the principles of geometry and optics, and used triangles and pyramids to illustrate his ideas. Another big name in perspective was Piero della Francesca, who was both an artist and a mathematician. Although many works on perspective were written in the 15th Century, his On Perspective for Painting was the most mathematical. Leonardo da Vinci provided the illustrations for this work, and also studied perspective and geometry himself.

Da Vinci distinguished two different types of perspective: artificial perspective - which was the way that the painter projects onto a plane, when the plane itself may be seen foreshortened by an observer viewing at an angle - and natural perspective, which reproduces faithfully the relative size of objects depending on their distance. In natural perspective, da Vinci correctly claimed that objects will be the same size if they lie on a circle centred on the observer. After da Vinci, the only real breakthrough in perspective was in the use of conic sections in perspective, conic sections being the shapes you get by slicing through a cone at different angles. Circles viewed with perspective become conic sections, although it is almost impossible to attribute the discovery of this fact to one particular person - nevertheless, it is thought to have been realised around 1650. Since then, there haven't been any huge developments in the area - however, people have continued to play with perspective, and use it in new ways - such as Escher, famous for his innovative use of perspective to create his works of art.

Escher didn't just use perspective, however, he also used art to illustrate many complicated mathematical ideas. He created tiled patterns based on spherical, Euclidean, and hyperbolic spaces (these being different types of two-dimensional surfaces), and is famous for his use of symmetry and tessellation.

Mathematics in Literature

So far we have looked at mathematics in both music and art, and shown that mathematics and the arts are linked in both of these branches of the arts. Next, we come to look at mathematics in literature – is it possible to write a novel or a play with a mathematical link? Obviously we have mathematical textbooks, but these can hardly be called literature. Then we have books that, although not textbooks in the strictest sense of the word, are aimed at teaching maths, usually through making mathematics fun. These can't really be called literature either, however. Finally we have novels which use mathematics, or mathematicians, as leading ideas in the story. For example, novels such as those by Michael Crichton often have a mathematician as one of the main characters - a prime example is the character of Dr Ian Malcolm (mathematician and chaotician) in the novel Jurassic Park. This provides the main link between mathematics and literature – using mathematicians as characters in novels gives the author a character who is completely logical in all situations, which is the perception of mathematicians. The logical aspect of mathematicians is also apparent in cases where mathematicians have written novels, such as in the works of Lewis Carroll. Not forgetting, of course, Flatland by Edwin A Abbott. Other authors, such as Dostoyevsky, Tolstoy and H2G2="A1171595">Austen have used mathematical logic and concepts to illustrate ideas. The link between mathematics and literature obviously isn't as strong as the link between mathematics and music, or art, but it is still there.

Mathematics as Art

Mathematics as art? Surely some mistake? But no, even mathematics itself can be considered beautiful - every mathematician has encountered a proof that is simple, pleasing, and feels 'right', and which they would even say is artistic. Looking at, or working through, such a proof can give you the same feelings as looking at a work of art by a grand master, or hearing a symphony.

Conclusion

In conclusion, there is obviously a strong link between mathematics and the arts. Music, fine art, and literature wouldn't be the same without mathematics. From Mozart to Escher to Crichton, musicians, artists and novelists have used mathematics to highlight, improve and develop their work. This doesn't mean you need a degree in maths to play the piano, paint a picture, or write a novel - however, it does mean that an understanding of certain mathematical concepts can make you a better pianist, artist or author.

Related BBC Links

1The curriculum of medieval universities.2A chord is when two or more notes are played simultaneously.

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