Mathematics In Art

Introduction

"Although I am absolutely without training or knowledge, I often seem to have more in common with mathematicians than my fellow artists"

This comment was made by M. C. Escher, an artist well-known for his work on impossible figures. He obviously thought that there was a link between his art work and mathematics, but is mathematics often used by artists? Is it essential to produce a good piece of art work?

First we are going to decide what it is we mean when we say art, or at least what is going to be covered in this essay. Then we are going to look at various instances where maths is used in art, such as the golden ratio, and fractals, and work by the artist Escher.

What is Art?

Art can mean different things to different people. Some people would argue that some forms of modern art by artists such as Damien Hurst isn’t actually art. This is commonly an area of discussion, and opinions can vary greatly. So, I feel it necessary to outline what I think art is before advancing any further.

The dictionary defines art as;
1. the creation of works of beauty or other special significance;
2. imaginative skill as applied to representations of the natural world or figments of the imagination.

What is art to me? Well, most people would agree that art is usually something of beauty, something which is aesthetically pleasing and stimulating to the eye and to the mind. Art to me can also be something which reflects life, and maybe nature. A good work of art should also make you think. Pre-concepts may also be challenged, and art doesn’t always have to be real. Some artists, such as Escher, have based their work on impossible figures. These use your perception, your mind may fill in gaps or assume certain aspects which aren’t there. Basically, I think art is something which someone finds beautiful, whoich means a wide variety of things may be viewed as art.

For this essay, I am going to concentrate on art done in two dimensions. Although they may not be viewed as art as they are done using computers, I am going to include fractals, as they can be used to help represent things in art such as water or clouds and trees, and are actually sold as art themselves.

Golden Ratio

The pentagram above was considered to be a symbol of health by Pythagoreans. The ratio AB:BC (and the ratio AB:AC) is 1.618:1. This ratio is equal to

The Greeks believed that a rectangle in the golden ratio is more aesthetically pleasing than any other. A rectangle of this kind is called a golden rectangle, and can be very useful.

Proportions of the golden rectangle have been used by many artists, in Greek art, and later works from Leonardo da Vinci and other Renaissance artists. Leonardo da Vinci's Madonna and Child with Saints is an example of this as he seems to have used the golden ratio to divide the painting into pleasing proportions, in which he has the most important aspects of the painting.

Leonardo da Vinci also illustrated Fra Luca Pacioli’s book De divina Proportione, which was published in 1509. In this Pacioli showed 13 properties of the golden ratio, but ended there as there were 13 present at the last supper, and so he felt he had to stop there.

The golden rectangle can be divided into a square and a smaller golden rectangle. This can be continually repeated, and by drawing a quadrant in each of the squares with the radius the length of one side of the square, can be used to draw an equiangular spiral.

This spiral is similar to itself, and is often seen in nature, in shells and in the arrangement of sunflower seeds, and so can be used to help represent these in art. Escher also used it in some of his work.

So, the golden ratio can be very useful in art, to find aesthetic proportions of a painting, and to represent various things in nature.

Escher

Escher used many mathematical ideas in his work, and once said he found that he had more in common with mathematicians than with other artists. Certainly mathematicians were interested in his work.

One mathematical idea he often used is tessellation, in fact, he found some tessellating shapes that mathematicians didn’t. He often developed this idea, though, making the object reduce in size. Using this, he tried to represent infinity by making the object appear to diminish into infinity. We can see this in his woodcut Smaller and Smaller. This woodcut depicts lizards crawling round the work, getting smaller towards the centre.

Escher also managed to tessellate on a circle, which many would think impossible. We can see this in all his circle limit woodcuts. In Circle Limit III, fish start at the edge, and become first larger, then smaller, following an arc back to the edge. As well as tessellation on a circle, Poincare hyperbolic geometry (where lines are arcs of circles) is used in this woodcut.

Escher also used similarities in a lot of his work, including some of his tessellation work. In many of his works, he used one or two figures, changing the size and translating them in such a way that they tessellated.

Cycles are also a common feature in Escher’s work. We can see this in a lot of his work, including Verbum. In this lithograph, animals (frogs, fish and birds) each in their own element (land, sea, air) evolve from triangular figures in the centre. He also alternated between colours, each animal appearing twice, once black with white background, and once white with black background. Verbum was chosen for the name because of its connection with the story of creation.

In his woodcut Metamorphose, Esher goes from the word ‘metamorphose’ placed horizontally and vertically to shapes which transform into a number of various animals, then eventually into a city, then a chessboard, before returning to the words ‘metamorphose’. Escher uses a variety of shapes including squares, hexagons, equilateral triangles and diamonds, which he transforms into animals, boats, and a city. This work uses ideas of tessellation, and takes it further, slowly changing them into other tessellating figures, then slowly changing them again into other shapes.

So Escher uses many mathematical ideas in his work, the main one being tessellation. He also uses a number of impossible figures, which are mathematical in themselves. In Waterfall, he uses the tri-bar three times, and in Ascending and Descending, he depicts a set of stairs which either continually go up or continually go down, which is impossible.

Fractal Geometry

A fractal is something which increases in detail the closer you look at it, and is infinitely complex. It is also a non-differentiable function. Using computer technology, fractals can be represented as pictures. Although some people wouldn’t consider computer generated pictures to be art, these fractals are sold as art, and have an aesthetic nature, so I am including them as art.

Mandelbrot was the one to give these the name fractal, and the Mandelbrot set is one of the most famous fractals. You can zoom into it an infinite number of times and it will still remain just as detailed, if not more so. Zooming in on the right area will produce another Mandelbrot set. This fractal is the one most usually sold as art. Below is an example of a blow-up of part of the Mandelbrot set.

Chaos theory is the underlying principles about fractals, and this can be used to help depict things such as the flow of water. To show fluid motion accurately is a challenge to any artist, but chaos theory may help them to understand it better, which in turn may help artists depict it better.

Fractals can therefore be sold as art, or used to help other artists show some things more accurately. Chaos theory, the underlying principles of fractals, can also be useful.

Conclusion

Maths and art are not often thought of as being connected, but in actual fact, maths can be used a lot in some areas of art. Even though art is more about how the individual views the world, in some work some kind of precision is needed to make things recognisable and realistic, and maths can help provide this.

Perspective has not been mentioned yet, but this is vital in some artists work, yet is a mathematical concept. Using this, the artist can represent distance. The golden ratio has proved to be very useful in deciding aesthetic proportions, and the equiangular spiral is also useful for showing some aspects in nature. Escher is one example of an artist who uses maths in his work. Leonardo da Vinci, and Pierro Della Francesca are others.

Fractals are a mathematical concept which are both sold directly as art, and can be used in the production of other art involving trees, clouds and coastlines, as well as many other features in nature.

So, maths is often used in art, and some knowledge of maths can help artists, especially those producing work representing nature.

Bibliography