In this essay, before I start anything, I must first clarify that I deeply consider mathematics as a subject that has had a great influence on the artist and his masterpieces, therefore I already alarm you that throughout my essay I will talk about Escher’s work and try to persuade you that there has been a considerable integration of the subject matter with his very artworks. In order to make you understand my objective, I have gathered some of his work, then selected a few, which I found had more mathematical elements, then with a decreased amount of drawings to work with, I would be able to study all components and show you that there has been a great influence of maths on him. I believe these images without the existence of any mathematical aspect would not be able to be fully accomplished.

Elements like: symmetry (reflection also included), pattern/tessellation (repetition), transformation, crystallography, “impossible shapes”, proportion and the ‘Fibonacci Sequence’ or the ‘Golden Ratio’. Are suggested to be present in M.C. Escher’s artworks, these I believe have been responsible to create the effect they create on the viewer, which is wonder and marvellous of the impressive art that cannot belong to the real world. Later on I will mention and try to explain these components, so that a random person may understand fluently my line of thought, as these elements are particularly specific to the subject studied and might need a little knowledge in order to comprehend them.

First of all, I will talk about Topology, a characteristic of maths that when used on regular images is responsible to create the ‘impossible shapes’, these are named this way because they are only possible to be created as pictures or as images but not as 3D shapes due to their structure: they sometimes have no real linking lines. Topology is mainly the study of a space that stays invariant even under continuous variations i.e. deformations on something, which do not create ‘holes’ or ‘tears’. When used it might not be noticed when looking at it superficially, as this technique is quite discrete and more concentration might be necessary.

This may be found in some of Escher’s great works like: Belvedere, Impossible Triangle, Impossible Crate, Ascending and Descending, Waterfall, Cube with Ribbons. The drawings sited actually fit into one group as they contain very like characteristics. These images also contain more elements apart from Topology: Geometry and perspective for instance have been explicitly present in most of his works apart from many other characteristics of minor importance, which will be discussed later on.

Looking closely at Belvedere, it might be said that at a first glance a person may not notice its incredibility. Though when analysing the picture for a longer time, the person might realise how impossible it would be to construct this building: if looking at each floor at a time, it is not possible to sense the transformation occurred from one floor to the other, though when staring at the whole building at once, it seems as if an imaginary line was placed between both floors, it is possible to understand the complexity of the image. In this artwork a simple displacement of the columns has lead to unreality.

Topology, a technique used in this particular drawing, is responsible for the image and its suffered distortions, though one of its characteristics is not having the appearing transformations i.e. the picture has not suffered any ‘tears’ or holes’ as if they only appeared realistic but could not be accomplished in a real world, which comes to be differentiated from the Print Gallery or Balcony where the artist has caused explicit visual torsions into these artworks. This differentiation happens due to the use of rotation on the image rather than displacement on areas of the picture.

Apart from topology, geometry and proportion are involved. Geometry has been needed in order to create the construction itself as it is based on parallelograms (the floors, walls and balcony) and circles (the roof, balcony – a mixture of both circle and rectangle). The parallelograms are ‘transformed’ into rectangles or rectangular objects, this is due to perspective and the direction to which it points.

Though there are much more to it, for instance pattern made by: squares (actually rhombus, but appears as squares due to perspective) to make a squared floor and the balcony, which is composed of rectangles and half circles (these two shapes together make up the actual spacing between each of the supports on the balcony). But what is really amazing is that Escher was able to separate the image into 2 halves. The top half, where it seemed to be viewed from left to right, and the bottom half, which seems to be viewed from right to left, and on the middle the transformation occurring, which gathers both views into one in order to make the illusion.

Moving on to another interesting topic that acquires geometry, Escher’s spheres have been responsible for much criticism and value. A few of the works like: Three Spheres, Hand with Reflecting Sphere and Concentric Rinds use what mathematicians call perfect shapes. I have taken Hand with Reflecting Sphere, a well- known and criticised artwork, into consideration to be analysed.

Symmetry as said before, also plays its part when talking about M.C Escher’s spheres. This is because the sphere fits into numerous types of symmetry i.e. it can be divided into different equal parts into various ways: a simple reflection of its half, rotational symmetry as well as combinations of symmetry (there are 17 types of symmetry: some combinations of the three- rotation, reflection and translation). If looking closely at the Hand with Reflecting Sphere it is possible to realise that the centre of the person’s head has been drawn on the centre of the sphere.

Artistically analysing this image, I may say the artist has drawn the image with the purpose of valuing an egocentric idea, when positioning his head (what is valued as a symbol of uniqueness or self image) in the absolute centre (a circle or sphere are the only shapes with one exact centre). Though when mathematically analysing his work, I must say there has been a study and complex measurements in order to accomplish such task, where various attempts or calculations were done in order to draw the whole image inside the sphere, starting from its centre.

Also, from the centre of the head a few other measurements were made:

* The absolute centre of the sphere is located on one of the three imaginary horizontal lines in which this image is divided into.

* The three parts in which this image is divided into is based upon the centre of the head: the distance from the centre of the head is twice to the bottom edge of the picture than it is to the top edge of the picture.

* The top, left and right edges of the image are equidistant to the centre of the sphere: the edge of the sphere is equidistant to the top, left and right edges of the picture.

In a way this artwork respects the ‘Golden Ratio’: a ratio created by the Greeks and believed to be the most aesthetic ratio. The rules of the golden proportion are: numbers related to the sequence 0,1,1,2,3,5,8,13…(the next number is always based upon the addition of the last two numbers) actually called ‘Fibonacci Sequence’ derived from the fact that most of the elements in nature grow this way i.e. 0 represents nothing, then 1 represents an individual and together with another individual there might be a newborn and so on. With this sequence the ‘Golden Ratio’ was created: 1/1, 2/1. 3/2, 5/3, 8/5 and so on, being that the ratio is based upon one of the numbers from the sequence as the denominator and the next number as the nominator, being that the greater the nominator and denominator, the nearest it gets to the ‘Golden Ratio’ (1,61804- approximated to 5 decimal places).