Mathematics in Art: Communicating Mathematics to the Public Through Art

Mairi Walker November 25, 2011

A report written for the Lord Rootes Memorial Fund trustees in accordance with funding regulations

1 Abstract A common perspective among people of all ages is that mathematics is useless and boring. Over the course of my mathematics degree I have come across many areas of the subject which have shown me the converse, and this is something I wish to communicate to the general public. Being a keen artist, I decided to investigate the feasibility of the novel idea of communicating mathematics through art. After researching thoroughly the links between mathematics and art, I produced a series of six mathematically-related paintings and drawings which I exhibited, accompanied by explanatory posters, in the foyer of the Zeeman building, University of Warwick from 17th to 23rd October 2011. I also condensed my exhibition into six A2-sized posters which I distributed to schools, colleges and education centres across the UK. Feedback gathered at all stages of the project showed that taking an interdisciplinary approach to mathematics teaching can be effective and inspiring. By providing people with a novel setting for mathematics, and by showing them a variety of topics, the project allowed them to develop their interest in, and knowledge of, mathematics, sometimes without even realising it. I concluded that although teaching mathematics through art would be of limited use in preparation for exam-based assessment, it provides people with a wider mathematical background, and encourages general learning. Acknowledgements I would like to thank the Lord Rootes Memorial Fund, for without their financial backing this project could never have been completed in the form that it is in today. Thanks also to the University of Warwick Engineering Department for the use of their orinting facilities, and to the schools, colleges and education centres that kindly displayed my posters.

2 Contents

1 Introduction 5 1.1 Origins of the Report ...... 5 1.2 Project Aims ...... 5 1.3 Mathematics in Art: The Basics ...... 6 1.4 Project Outline ...... 6 1.5 Limitations ...... 7 1.6 Structure of the Report ...... 9

2 Project Methodology 10 2.1 Mathematics and Art ...... 10 2.1.1 Researching Mathematics and Art ...... 10 2.1.2 The Links Between Mathematics and Art ...... 11 2.2 Creating Mathematical Art ...... 12 2.2.1 Searching for Design Ideas ...... 13 2.2.2 Design Development ...... 14 2.3 Communicating Mathematics Through Art ...... 14 2.3.1 Communicating Mathematics ...... 14 2.3.2 Example: Hendrix ...... 15 2.3.3 Communicating the Project ...... 16

3 Analysis 18 3.1 Summary ...... 18 3.2 Findings and Analysis ...... 18 3.3 Conclusions ...... 19 3.4 Recommendations ...... 20 3.5 Personal Reflections ...... 20

Bibliography 22

A School Letter 23

B Questionnaire 24

C Financial Statement 25

D Exhibition Poster 27

E Sketchbook Extracts and Final Art Work 28

3 List of Figures

1 The first four stages of constructing the Peano Curve...... 16 2 Exhibiting in the Zeeman Building ...... 17 3 My exhibition poster...... 27 4 Finding suitable starting points...... 28 5 Studying fluid flow from an artistic perspective...... 29 6 Taking inspiration from the work of the graphic artist M. C. Escher. 29 7 Investigating mathematical drawing techniques ...... 30 8 Developing designs based around knot theory and crystal structure. 30 9 Generating fractal images by computer...... 31 10 Crystal, acrylic paint on canvas paper...... 32 11 Fractal, ballpoint and pencil on paper...... 32 12 Hendrix, acrylic paint and wax on canvas paper...... 33 13 Landscape, oil paint on canvas paper...... 34 14 Lily Pond, pencil on paper...... 34 15 Splash, oil paint on canvas paper...... 35

4 1 Introduction 1.1 Origins of the Report A common perspective among people of all ages is that mathematics is useless and boring. Over the course of my mathematics degree I have come across many areas of the subject which have shown me the converse, and this is something that I have always wished to communicate to the general public. A major catalyst was my participation in a three week teaching placement at a local secondary school in the spring of 2010. I noticed a distinct apathy amongst teenage pupils when it came to maths, and was inspired to do something that would show these youngsters that there is a lot more to mathematics than the content of the GCSE syllabus. Around about this same time I had just completed an essay detailing the links between art, nature and fractals, which are irregular-looking images with structure at arbitrarily small scales (see section 1.3). Given the graphical na- ture of this branch of mathematics, it seemed to me that the obvious way to communicate it would be through art, but I didn’t see why this method couldn’t be a possiblity for communicating mathematics in general. Slowly the idea for my project took form. After a small amount of research I found that there were links to art within almost every type of mathematics that I encountered. Having always been a keen artist, I looked into ways of putting these links and ideas into paintings and drawings. Accompanied by a suitable information sheet, I felt these works could potentially introduce members of the general public to university-level mathematics. At this stage, however, my ideas were limited to what I could produce using art materials I already had, and communication of my ideas would be very difficult without financial backing. I applied to the Lord Rootes Memorial Fund in January 2011, and fortunately I gained the funding necessary to mould my project into the form it is in today.

1.2 Project Aims When formulating my project I had three main outcomes in mind: • To teach mathematics to the general public through art. • To raise awareness of a variety of mathematical topics: the mathematics seen in schools is only a fraction of the mathematics out there. • To promote an interest in mathematics, and particularly to raise the as- pirations of secondary school pupils. I also, however, had some more personal aims. Since this project stemmed from my interest in the relationship between mathematics and art, this is something I hoped to gain a more thorough understanding of over the course of my project. By researching areas of mathematics that I had previously not studied, I hoped to find new links with art. I also hoped that the art work I produced would be exhibitable fine art: art in itself, not just art demonstrating maths. To summarise, I aimed: • To conduct and consolidate academic research into the topic ’mathematics and art’.

5 • To produce a range of high quality and original art work.

1.3 Mathematics in Art: The Basics The mathematics taught in schools is a very small and unrepresentative fraction of the mathematics known to mankind, with much focus on simple applications and little abstraction [2]. Generally speaking, mathematics is the systematic study of quantity, structure, space and change: the language of the world around us [1]. It is a vast subject area which can be divided, roughly, into two major sections: pure mathematic, and applied mathematics. Generally speaking, pure mathematics builds up, abstractly, the tools needed in applied mathematics and so it is rarely encountered by the general public, except in elementary secondary school algebra and geometry. Pure mathematics often studies the general properties of abstract concepts, with little practical motivation. Applied mathematics is easier to see around us: it includes the study of fluids, the study of population growth and disease spread, and statistics. Applied mathematics is used in many professions, and is essential knowledge for scientists and engineers. The relationship between mathematics and art can be traced as far back as the times of the ancient Egyptians and Greeks who incorporated the Golden Ratio, a ratio said to give aesthetically pleasing proportions, into such monu- ments as the Great Pyramid of Giza, the Parthenon and the Colosseum [3]. The Renaissance saw a rebirth of classical Greek and Roman ideals, and the study of mathematics was seen as relevent in the understanding of nature and art. Painters wished to depict three-dimensional space on two-dimensional canvas, and so the geometry of projections was essential. Geometry and proportion remained the only mathematical concepts com- monly used by artists until the 20th century, when there was an explosion of activity by both artists interested in mathematics, and mathematicians inter- ested in art. The famed graphic artist Maurits Cornelis Escher showed an intuitive understanding of non-Euclidean geometries - the geometries of worlds that have slightly different properties to ours - which link to the study of groups of equations. Various artists, such as Helaman Ferguson and John Robinson, were inspired by developments in topology, which is the study of the properties of shapes that are unaffected by squishing and stretching (such as number of holes). Another major development in the linking of mathematics and art came when the ideas of fractal geometry surfaced in the 1970s. The term ’fractal’ was coined by Benoit Mandelbrot in 1975 to describe a rough or fragmented geometric shape that can be split into parts, each of which is (at least approx- imately) a reduced-size copy of the whole [4]. Despite being highly geometric objects, fractals can surface in many ways, although often from the iteration of mathematics equations. More recently, many computer programs have be- come available that allow the user to create complex fractal images, spurring the current ’fractal art’ movement.

1.4 Project Outline I began work on the project shortly after the end of my third year exams in June 2011. The first few weeks were spent mainly sourcing links between math- ematics and art, and gathering resources. I used many books and websites in

6 my research, but also had to equip myself with the programming skills needed for later in the project. In July I moved on to researching the mathematics more thoroughly; it is impossible to communicate mathematics without having a full understanding of it, so it was important to spend time doing this. Around this time I also began sourcing the space and equipment needed to hold my exhibition, and began collating my research in my sketchbook (see Appendix E for extracts) and updating my website. By the end of July I had completed my necessary background research. I spent two days in London visiting galleries and museums, searching for artistic inspiration and gaining valuable historical insight into my subject. In August I visited family in Scotland and siezed the opportunity to visit Satrosphere, Scotland’s first science education centre, which taught me a lot about com- municating science. I also met with Professor Frederik Glasser from Aberdeen University to discuss applications of mathematics within chemistry. On returning to england I began the design development stage, although I was simultaneously conducting further research into both mathematics and art. I chose around ten mathematical topics that I believed had potential, and developed a couple of artistic ideas for each. From these I chose, aided by the advice of a range of people, six designs that I felt displayed the widest range of mathematics and art techniques, and prepared my final design ideas. By now I had confirmed exhibition space in the foyer of the Zeeman building, University of Warwick, for the week beginning the 17th October, so had a strict deadline to work to. Throughout September I balanced creating the art with writing and design- ing the information posters. Throughout this stage I continued to seek advice from others around me with regards to both the quality of the art and the suc- cess of my mathematical explanations. I wrote to schools in the Warwickshire area asking if they would like to display a set of my posters, and arranged fram- ing and display board hire for the exhibition. I also produced a questionnaire to help me get meaningful feedback on my project. In the weeks leading up to my exhibition, I displayed posters (see Appendix D) advertising my project around the Warwick university campus and my local town, Leamington Spa. I successfully opened my exhibition on Monday 17th October, 2011, displaying my framed art work along with posters describing the mathematics behind the art. I left a cardboard box, and copies of my questionnaire, for members of the public to leave comments. I printed A2 poster versions of my project through the university’s engineering department, and posted these, along with questionnaires, to the relevent establishments. During October I had managed to spread the word about my project further afield that just to local schools, and in the end I managed to display my posters in schools and education centres all around the UK. My final step was to gather and analyse the feedback I recieved about my project.

1.5 Limitations The first limitation I encountered was with regards to funding. Unfortunately I did not get quite as much funding as I had originally planned for (see Appendix C), meaning I would have to rebudget. Managing to obtain a free copy of the MATLab software allowed me to immediately reduce my costs by nearly £60,

7 but other cuts would be harder to make. The part of my budget that it was most feasible to cut was publicity and printing, so I decided to make more use of the university’s printing facilities. I managed to avoid going to a commercial printer, using printers in the university library for the majority of my needs. The university’s engineering department kindly allowed me to use their plotter to print my school posters. This cost £30 less than I had originally planned, despite upgrading from A3 size to A2. Other than this, however, my project was limited by the time I had to spend on it, and my skills and an artist and a mathematician. I was not completely happy with all of the art work I produced - an artist never generally is - but I believe the work was of a high enough quality to be considered art in its own right, and I certainly recieved many compliments on it. Similarly I feel that my mathematical skills did not limit me, and I was very pleased with the mathematical content of my project. The amount of time I had to spend on my project, however, was a great limitation. This was largely due to the breadth of the subjects concerned: although I do feel that I managed to satisfy all my aims in the time I had available, there is vastly more material that I could have covered in most stages of my project. Only being able to choose six mathematical topics, for example, limited my project - I could have continued with thousands if I had had the time - and the fact that mathematics is spanned by a continuum of topics made it difficult to decide exactly what to include in my information sheets. By making informed choices, however, I feel I managed to cover an excellent variety of topics and chose the most interesting and relevent topics to talk about. Lack of time also prevented me from completing some of the secondary aspects to this project. I didn’t get the chance to write a full-scale academic report on mathematics and art, although this is something I could still do now if I found the time, as I have done the necessary research. I was unable to attend one of the London Knowledge Lab’s Maths-Art public seminars as they only run during the University of London term times, but I feel I found enough related information elsewhere for this to not be a problem. Another difficulty was the communication of mathematics; it was much harder than I thought it would be to put myself in the shoes of someone who hasn’t studied university-level maths for three years. Visiting Satrosphere cer- tainly helped me in this respect, as did practicing explaining the mathematics to younger relatives, friends, and members of my local library. The fact that I had spent time thoroughly learning the mathematics behind my work certainly made it easier for me to create my posters, and from the feedback I recieved I feel that my explanations were suitably clear. The other difficulty I faced when creating the posters was condensing my research into a reasonable length. This meant that I could not include everything that I wished to in my explanatory text. I managed to include the necessary information to explain the mathematics behind each piece of art work in context, but could not include much further information which was a bit disappointing. I did, however, reference some outside sources so that people could, if they so wished, read up more themselves. My final major constraint concerns the writing of this report. It was my dream to write an academic report on the links between mathematics and are. The main aim of this project, however, regarded the communication of mathe- matics, and so it was more important for me to analyse my success in terms of

8 this, and I felt that for clarity’s sake I should stick to one aim for this report. I have, however, included a brief summary of the links between mathematics and art that I found. Please forgive any technical terminology: there was simply not enough room in this paper to clarify everything in generally-understandable terms.

1.6 Structure of the Report The primary aim of this report is to determine the feasibility of using art as a learning tool in mathematics, using my project as a case study. In order to do this I will detail the methodology of my project, and gauge the project’s success in terms of satisfying my aforementioned project aims. In order to fully understand the project’s progression it is useful to understand the links between mathematics and art from an academic point of view. As mentioned in section 1.5, it is not possible, in this space, to detail completely these links - a whole book would be needed for that - but I aim to consolidate the links between mathematics and art that I have studied. Along the way i will refer to my personal aims, as listed in section 1.2, but these will not be a focal point of this report. The report begins with the above introductory section, which aims to give a background of the project and an outline of what this report contains. The main body of the report, section 2, details my journey, from the prelim- inary mathematical research in the library to collecting feedback on my project. I hope to show how I came to the various decisions I had to make throughout the project, and give an idea what it was like for me during the course of the project, as well as show what was actually contained in my work. After a brief summary I then begin my analysis. I aim to show the success of both my project, and the use of art as a learning tool in mathematics. I provide recommendations of what I would advise to anyone wishing to undertake a similar project, but also what I would do to expand on this project. I finish with some personal reflections. There is a full bibliography and appendices containing extracts from my sketchbook, photographs of my paintings, a financial statement, my letter to schools and my questionnaire at the very end of this report.

9 2 Project Methodology

The course of my project can be divided roughly into three sections: my research into mathematics and art, my designing of mathematical art, and the commu- nication of my project. The interdisciplinary nature of the project meant that I was rarely focused on just one of these aspects, and I had to continually refer to my project aims to keep me on track. For the purpose of this report, however, it is appropriate for me to recount my methodology in the following three sections, although it is important to note that I also had ongoing tasks involving the or- ganisation of exhibition space and equipment, sourcing inexpensive materials and printing, and promoting my project.

2.1 Mathematics and Art This first section describes my research into mathematics and art, culminating with a summary of my research.

2.1.1 Researching Mathematics and Art In order to successfully communicate mathematics through art it was essential for me to research thoroughly the links between the two disciplines. To engage the public in my project I would need to find links that would allow for an interesting and appropriately-leveled explanation, but I would also need to be able to produce a beautiful piece of art work. Research into the links between mathematics and art was also of great personal gain to me as this is a subject I have been very interested in for quite a while. As with all research I began by sourcing information on the subject matter. There is relatively little literature relating the subjects of mathematics and art, and what there is generally focuses on a few elementary links within geometry. Since my project aimed to introduce to the public a variety of mathematical topics, I wanted to avoid using ones that are seen at school, and hence this literature was only really relevent to my personal research and not so much to the continuation of my project. That said, there were two books that I found very useful: Pickover’s The Pattern Book [5] and the popular science book Indra’s Pearls [6]. This latter book, being an excellent example of high-level mathematics written for a general audience, was more useful to me when it came to the communication of my project (see section 2.3), but The Pattern Book was invaluable to me throughout the research stage of my project. It consists of a range of images created by mathematicians and scientists around the world, and gives a brief description of the scientific content, providing a starting point for further research. A particular highlight of the book is that it has sections containing mathematical images in art and nature as well as in pure mathematics, and it provided much inspiration for me. Aside from a handful of books, however, my background research was gen- erally conducted through the internet. Despite the links between mathematics and art not being terribly obvious to the majority of the population, there is a multitude of websites devoted to this subject. Many of these give examples, both historic and more recent, both by professionals and amateurs, and I began to build up a picture of the relationship between mathematics and art. Using

10 the internet also allowed me to find information on various exhibits, conferences and journals relating to mathematics and art. It was now time for me to move forwards and begin conducting my own research. I had two choices as to how I went about this. I could begin with a mathematical topic and see how it could be expressed through art, or I could search for the mathematics within art. Given that an aim of my project was to communicate mathematics through art, and that I needed to produce original art work, I settled with the former method, although the latter was also a useful source of inspiration. In this stage of my research I still used books and the internet for inspiration, but focused on finding and constructing my own links between mathematics and art. I continued finding new links, refining my ideas, and hence expanding my research, throughout the course of my project, even once I had begun creating my art work. In particular, I was inspired by my visit to London: I found relevent exhibits in the Tate Modern, the Science Museum, the Victoria and Albert and the Design Museum. I collated my research in a sketchbook, dedicating sections to each mathe- matical starting point (see Appendix E). I compiled basic informaton on the historical context and as the mathematics involved, but I also began experi- menting with ways in which to express this artistically.

2.1.2 The Links Between Mathematics and Art Despite seeming such disjoint subjects, mathematics and art have a long historic relationship dating back to the times of the ancient Egyptians and Greeks. Emphasis was put on the study of geometry and proportion by artists striving for beuty and perfection, and this was particularly noticable in architecture. The Golden Ratio, and shapes derived from it, particularly feature in ancient art and architecture is; the Great Pyramid, the Parthenon and the Great Mosque of Kairouan are examples that still stand today [3]. For centuries the links between mathematics and art were restricted to those within geometry, and this is partly due to that being the main branch of mathe- matics being studied. Even now, the majority of literature relating mathematics and art is based on geometry, possibly because these are the most obvious links. This is not to say that geometry cannot be artistically inspiring, however; even in the least mathematical art aspects of geometry can be seen, whether it’s simply the canvas proportions or the use of projective geometry. While there is a bit of geometry in every piece of art, there is also a lot of art explicitly centred around geometric ideas, from models of the platonic solids, to intricate harmonograph drawings. The popularity of geometry in the 18th and 19th centuries inspired the creation of a variety of mathematical drawing tools, one of which survives today as a popular children’s toy: the Spirograph. Geometric pattern is also often a focal point in a lot of tribal or cultural art, such as Islamic tiling patterns, Aboriginal art, and Japanese diaper ornaments [5]. Since there is a lot of documentation on the links between geometry in math- ematics and geometry in art, I will not linger too long upon the subject. It is important to note, however, that the discovery of non-Euclidean geometry in the early 19th century provided inspiration, and posed a challenge, for artists around the globe. Non-Euclidean geometry was to be a major source of inspira-

11 tion for the Dutch graphic artist M. C. Escher, and more recently it has sparked computer scientists to develop algorithms to create hyperbolic tesselations. The study of tesselations links us to group theory and topology. Abstract algebra, and in particular group theory, surfaces in many appli- cations, but often when describing symmetries. It is hence not that surprising that we meet the subject quite regularly when looking for links between mathe- matics and art. In particular, groups of isometries are essential when describing tesselations in any sort of geometry, and group theory is used to describe the structure of crystals, a common feature of jewellery. Group theory in relation to hyperbolic tesselations leads us to toplogy. We have the famous example of the coffee cup merging into a donut, but the links between topology go further than that. The relatively recently developed knot theory has applications in partical physics, but can also be used to describe the beautifully intricate Celtic knotwork of the 8th and 9th centuries that is so commonly seen in jewellery. The pure branch of analysis also houses much artistic inspiration, from sim- ple graph functions to space-filling curves. The latter relatates to fractal di- mension and density, which in turn links to the idea of creating an image out of small blocks containing images of different densities in order to produce different shades, not unlike images constructed from hundreds of tiny photographs. Within applied mathematics, the links between mathematics and art become more numerous but less well-defined. Given the continuity of the two disciplines it becomes hard to say what is or isn’t maths or art. Mathematics can be used to create graphical models of almost everything, as can be seen from looking at computer or film graphics, and mathematical models can be created to represent the beautiful things around us. Fluid dynamics is a particularly interesting example, as it models almost ev- erything around us; it can be found in the splash of a raindrop or the dissapation of smoke. Similarly mechanics is essential in product design and architecture, and both of these have artistic aspects. The images produced when analysing applied mathematics problems can also be seen as art in themselves. Graphs and bifurcation diagrams can give complex patterns, and fractals can be found in strange attractors. This leads us to the most recent development in mathematics and art: frac- tal geometry. Fractals come in many forms, and are often found in nature; perhaps this is why they are so often the subject of art. The simplest show self-similarity and are more geometric, whereas fractals generated from the iter- ation of complex systems in the branch of dynamical systems can be incredibly complex. In conclusion, art can be found in almost every branch of mathematics, both coming from the pure mathematics itself, or showing up in applications. It is an aim of my project to demonstrate as much of this as possible within my project.

2.2 Creating Mathematical Art As mentioned in section 2.1.1, I began my first artistic experiments during the course of my research. Although by the end of July I already had several ideas in mind for my final art pieces, it was important for me to keep an open mind during the designing phase because of the importance of my design choices: when choosing which ideas to consider for the design phase I had severak things

12 to bear in mind if I wanted to satisfy my project aims. I would have to choose a mathematical topic that was relatively unheard of, yet could be explained in terms suitable for secondary school pupils. In order to engage the reader the topic would have to be interesting, so having real-life applications would be a plus. It was also essential that a high-quality, original artistic design could be produced, so I could only choose mathematical topics for which I had a couple of basic design ideas in mind. Extracts from my sketchbook showing my various stages of deisgn develop- ment can be seen in Appendix E.

2.2.1 Searching for Design Ideas I decided that my best course of action would be to take an appropriate math- ematical starting point and see what I could develop artistically. I began with the idea of fluid dynamics, looking both at actual fluid flows and images found when representing fluid flow mathematically. I took particular interest in smoke patterns, soap films and splashing liquids. I then moved onto tesselations, par- ticularly those in the work of the mathematical graphic artist M. C. Escher. I focused on his tesselation patterns where the tiles get smaller and smaller as they move outwards (or inwards) within the picture, but also studied his so-called ’impossible structures’. My next inspiration came from the branch of knot theory. I looked at Celtic knotwork patterns, producing a mathematical algorithm for constructing these, and looking at examples in jewellery. I then moved my focus to mathemat- ics in architecture, looking, for example, at the structure of the Eden Project biodomes. I found the most inspiration within geometry, looking at geometric sculp- tures and patterns with varying amounts of mathematics behind them. This led me to look at historic mathematical drawing techniques, and experiment with the pupular toy Spirograph. This geometry led me to focus more specifi- cally on spirals, and hence the golden ratio. The ideas I produced within this section ranged from geometric mathematics-based spirals, to sea shells. I found many spirals in sculpture, particularly in ironwork, but also in the work of Art Nouveau artists. Spirals are often a feature of fractal art, and this led me to experiment with fractals. I used Context Free [8] and MATLab to produce a range of fractal images, and found much inspiration in The Pattern Book [5]. I also found examples of fractal images within quilting and needlework which struck me as a novel idea. Since fractals are frequently found in nature, I had the idea of computer-generating small fractal images and ’stamping’ them repeatedly to create pictures. This idea of creating images out of hundreds of smaller ones led me to think about fractal dimension and the possibility of creating an image out of a line. This in turn led me to look for art within pure analysis, which focused mainly on curves and surfaces. I then moved on to a completely different topic, looking at crystals and their structure. I studied various forms of crystals, from snowflakes to diamonds, before looking at modelling atomic structure with beads. I then looked at magnifications of crystal slices, and mathematical images showing crystal and atomic structure. The final topic for my initial design ideas looked at mathematics in cultural

13 art. In particular, I studied Persian and Islamic patterns, and Japanese Diaper Patterns, looking at these as applications of geometry and tesselations.

2.2.2 Design Development When developing my artistic designs I drew heavily upon the resources I had collected during my research. My visit to the London galleries, as well as other galleries I had the opportunity to visit over the summer, were a major source of inspiration, as was the aforementioned Pattern Book [5]. I also found online coverage of various mathematical art exhibitions, for example one organised by the American Mathematical Society [7]. The hours I spent learning to program MATLab paid off as I produced inspiring images for myself, and the computer program Context Free [8] was invaluable. Throughout this process my decisions were heavily influenced by other peo- ple’s views. Wherever possible I asked people, such as school pupils, family, friends and public library users, their opinions on both my choice of mathemat- ical topics and artistic design. I like to think it was a series of collaborative decisions that shaped my project into the form it is in now. By beginning my artistic exploration early on I gave myself time to develop my artistic skills, so I did not not need to spend extra time preparing to create my final pieces of art. After some extra design development and a draft of each, I produced six pieces of art work, as shown in figures 10 to 15.

2.3 Communicating Mathematics Through Art This final section of my methodology covers both the communication of my project in general, and the communication of the mathematics within it. This was, in some ways, the most important part of my project, given my aim to communicate mathematics through art. The majority of the work towards this aspect of my project was undertaken towards the end, during September and October, but it relied on ideas that I had developed throughout the course of the project.

2.3.1 Communicating Mathematics Once I had chosen the mathematical topics that I was going to cover, and de- cided on the exact piece of art work I was going to use, my next step was to collate the information that I would need when writing the explanatory text. I aimed to include, for each piece of art, a brief description of the related math- ematical topic, including historical context and relevence to modern culture, and an explanation of how the mathematics links in with the picture. I would include any relevent background information about both the mathematics and art, and would provide references to further information. Once I had a thorough idea of what I wished to include in the explanatory text, I had the task of not only writing this in accessible terms, but also of reducing it to a suitable quantity. In terms of writing mathematics for a general audience, my trip to the science education centre Satrosphere in Aberdeen was invaluable, as I saw a wide variety of science applications written in accessible terms. I began to get an idea of what terminology and concepts are appropriate to include, and also of what length of text is suitable for such a project. Here

14 the book Indra’s Pearls [6], amongst other science books aimed at a general audience, was also useful. The journal of Mathematics and the Arts [9] was an excellent resource demonstrating how to make descriptions accessible yet professional. Once I had drafted my text, I had the tough job of condensing it so that it would fit clearly onto a sheet of A3 paper. Admittedly, this is still a lot of text to read, especially for teenagers, but by breaking the text up into sections and using diagrams in explanations I managed to keep it at a suitable comprehen- sion level. Condensing my work, unfortunately, meant that I could not include everything that I wished to say. I focused on giving a more general, although ac- curate, mathematical background and limiting the artistic explanations, whilst still keeping everything linked together.

2.3.2 Example: Hendrix The following shows the explanatory text for the Hendrix painting (Figure 12), before adaption into poster format. This painting shows the iconic image of Jimi Hendrix, the legendary guitarist of the swinging 60s. Hendrix was around at a time of social revolution and this was a major influence in the fusion of visual art and music that was seen through- out the decade. Consequently artists and musicians worked closely together to produce innovative art work, whether for promotional purposes or as album art work. The image shown in this picture is one that was used in many pieces of art designed for Hendrix, each displaying the ideals of 60s art in its own unique way: the transition between the popular art of the 50s and psychedelia of the 70s is ever present in Hendrix posters. You might have seen lines in geometry and thought that they are about as simple and boring as maths gets, but in fact mathematicians have spent hundreds of years rigorously studying them. Lines are studied in a branch of mathematics called analysis, which is used to prove all the complicated equations used by scientists and engineers. The origins of analysis date back to the times of the ancient Greeks, although the rigorous study of lines didnt come into fashion until the 17th Century. For the next 200 years or so mathematicians looked at their nice little well-behaved lines and thought they knew pretty much everything about them. Then along came Karl Weierstrass, a German analysist. Hed found a line that twists and curves so much that wherever you are on it, you cant tell which direction youre going in. For years mathematicians had been saying lines like this couldnt exist, but instead of getting excited about the new discovery, they decided to make life easier for themselves and sweep this monster away under the carpet. Not long later, however, a guy called Peano discovered a line that covers every single point in a square. Now known as the Peano space-filling curve, this line is constructed in stages, as you can see in the diagram below. If you carry on this process forever, you cover every single point in the square! I was looking at Peanos curve one day when I had the idea for this painting. Notice that as you go through the stages of construction of the Peano curve in the above diagram, the square appears to get darker. Its quite simple really: you can take a black and white photo, replace darker areas of the photo with later stages of the Peano curve, and lighter areas with earlier stages. Join the curves all up and the overall picture will look quite a lot like the original photo,

15 Figure 1: The first four stages of constructing the Peano Curve.

except that its made from one line. I didnt base my line on the Peano curve in the end, but it gives a similar effect. Going back to the topic of mathematical analysis, Peanos curve wasnt liked very much. It didnt seem right at the time that a one-dimensional line could look exactly like a two-dimensional square. It wasnt long, however, before more curious lines with weird properties were discovered, and eventually mathematicians had to acknowledge their existence. So a lot of study has gone into trying to find and classify strange lines, and many more space-filling curves have been discovered. Lines are an example of a mathematical function. A mathematical function between two groups of objects takes each object in the first group and associates it with an object in the second group. As an example, if you have 7 pairs of socks then you can create a function that associates each pair of socks with a day of the week. Similarly if you take every number between 0 and 1, you can map them onto points on a sheet of paper, making a function. If you choose these points on your paper so that for numbers very close together, the points they map to are very close together, then we get a line! I pointed out earlier that Peanos curve is a one-dimensional line creating a two-dimensional shape. So what about this painting? Is it one-dimensional, since it is created from a line, or is it two-dimensional since it creates a two-dimensional picture? The answer is that the dimension is somewhere in between! Mathematicians have many different ways to define what exactly a dimension is, but one way to look at our problem is to think about how much of the page is the line taking up. To do this we can choose a small number, say x, and count how many squares of side length x are needed to completely cover the line. Obviously as we make x smaller, the number of squares needed gets bigger, and the rate at which this happens determines what is called the box-counting dimension.

2.3.3 Communicating the Project The communication of my project consisted of a public exhibition and an out- reach program aimed at local schools. The exhibition, held from 17th to 23rd October 2011, displayed my framed art accompanied by explanatory text in the format of A3 posters. The War- wick Mathematics Institute kindly allowed me to display my work in the foyer of the Zeeman building, on a display board hired from Warwick Conferences. I advertised my exhibition through posters (see Appendix D) which I placed around the university campus and Leamington Spa town. I held a successful opening night on Monday 17th which gave me an ideal opportunity to collect feedback firsthand, and for the remainder of the week I collected people’s re-

16 sponses through a short questionnaire (see Appendix B) which people filled out and posted in a comments box.

Figure 2: Exhibiting in the Zeeman Building

The other apsect of the communication of my project involved combining my art work with the explanatory text to create A2 sized posters. I wrote to twelve secondary schools and colleges in the Warwickshire area, plus my own high school, in Spetember, outlining my project and inviting them to display a set of my posters (see Appendix A). Although I recieved only a handful of responses, those that did helped to spread the word about my project, and I recieved various emails from schools, colleges and education centres across the country asking if they could participate. I managed to print my posters near cost price through the Warwick University engineering department, and poster these, along with copies of my questionnaire, to the various participating establishments. By the end of October I had recieved the feedback needed to compile this report.

17 3 Analysis

In this section, after a brief summary, I will analyse the success of my project with regards to my initial aims, and use my findings to determine whether or not art is a suitable medium through which to communicate mathematics. I will reflect upon my personal experiences of working on the project, and pro- vide recommendations on how I would improve or further the project given the opportunity. I will also discuss the community benifits of my project.

3.1 Summary ”Mathematics in Art” is a project aiming to investigate the feasibility of using art as a learning tool in mathematics, whilst at the same time consolidating research into the links between the two disciplines. From a personal perspective I had a unique opportunity to research both the links between mathematics and art, and methods of communicating science to the general public. More generally speaking, however, the project developed a series of six pieces of art work, each demonstrating a mathematical topic or concept. These were exhibited, accompanied by explanatory text, in the foyer of the Zeeman building, University of Warwick, from the 17th to the 23rd October 2011. The art work and text were also combined into A2 educational posters which were sent to various learning establishments across the UK for display. As well as collecting people’s opinions of all aspects of my project throughout the research and design phases, members of the public viewing the exhibiton, or the posters, were invited to provide feedback through a short questionnaire.

3.2 Findings and Analysis To analyse my project I will use data from the questionnaire responses as well as general comments recieved from others and my own personal experiences and findings. My personal aims were to conduct and consolidate research into mathematics and art, and to produce high-quality, original art work reflecting this. I feel these aims were met. I conducted and consolidated a vast amount of research into the links between mathematics and art, and over the course of the project I developed an in-depth knowledge of many of these. I was also please with the quality of the art work I produced. I would count at least three of the pieces amongst my best works, and feedback from the questionnaire, and general comments recieved, showed that people were genuinly impressed with the art. I was particularly pleased to see a comment on the questionnaire describing the art as exhibitable in its own right. It was much harder, however, to determine the success of my project with respect the my three general aims, which were as follows: • To teach mathematics to the general public through art.

• To raise awareness of a variety of mathematical topics: the mathematics seen in schools is only a fraction of the mathematics out there. • To promote an interest in mathematics, and particularly to raise the as- pirations of secondary school pupils.

18 It is hard enough to say what it means to teach and learn without putting these concepts in such an unusual context. There is no objective way to measure the success of my project with respect to these aims, but I can use my feedback to make an analysis. The majority of responses from the questionnaires responded positively about the level of the text on the posters, with only a few saying they didn’t under- stand all the mathematics, so I believe that I successfully targeted the posters at the correct level. Almost every response claimed that new knowledge had been learnt, with over three quarters of the responses explicitely commenting that they had learnt new mathematics. This is a very pleasing result given my aim to teach mathematics through art. The questionnaire results also gave evidence that people were impressed with the variety of topics covered, and many people commented that they hadn’t realise there was so much to maths. It is clear then that my project successfully raised awareness of different areas of mathematics to those who viewed it. General comments showed people were particularly impressed with the qual- ity of the art work, and that they found the project interesting. This backs up firsthand comments from people at the opening night who were genuinly im- pressed with the project as a whole. In terms of negative feedback, the most common complaint was that the posters were a bit bland and academic, which is something I had feared. Other than a few typographical errors, there was not much complaint about the content of the explanatory text, except for a few people expressing a wish for more information about the art work, or references to further information. Overall, the project seems to have gone down well, with little negative feed- back and many compliments on the novelty of my idea. I feel that my project has met its aims.

3.3 Conclusions Before making any conclusions, it is important to discuss the reliability of my data. Fortunately, I recieved questionnaire responses from people with a wide range of ages and education levels, so I feel that I have opinions from a rep- resentative range of people. Although the questionnaire featured many open questions, which can result in people not putting in the effort to answer prop- erly, I actually recieved a great deal of qualatitive comments. There is always risk of a biased selection of answers, since there is the worry that the people who filled out the questionnaire were simply those who were particularly impressed with it, and that those with more negative feedback wouldn’t bother. There is clear evidence, however, that my project has managed to success- fully communicate mathematics through art, at least to the people who re- sponded to my questionnaire. To determine exactly how effective art is at communicating mathematics would require much further study, but it seems as though the art functions perfectly for catching people’s eye, and that a well- written explanation holds peoples’ attention for long enough for them to absorb the information. Overall, I believe that although art can be an extremely valuable tool in teaching when accompanied by good explanation, and that encourages a gen- eral interest in learning. As the saying goes, a picture paints a thousand words, and I believe this is particularly true when communicating mathematics. By

19 providing people with a novel setting for mathematics, and by showing them a variety of topics, the project allowed them to develop their interest in, and knowledge of, mathematics, sometimes without even realising it. I conclude that although teaching mathematics through art would be of limited use in prepara- tion for exam-based assessment, it provides people with a wider mathematical background, and encourages general learning.

3.4 Recommendations I was extremely pleased with the outcome of my project, but appreciate that there is a lot that could have been done better, and there is a vast scope to further the project. As mentioned in section 1.5, the biggest constraint on the project was time. With more time, I feel I could improve slightly the quality of my art work, and improve the explanatory text significantly. I would rewrite this text to include more information on the art side of things, and have sections of more technical mathematics for those interested. Ideally I would have an interactive website presenting my project, and on this I could include much more detail on all aspects of the explanatory text, including text to suit different ability levels and applications allowing people to investigate the maths for themselves. I would also redesign my posters to make them look more exciting and less academic. With extra time I could also consolidate my research into mathematics and art more formally through an academic report, although as mentioned before, this is something I am planning to do when I have some free time. The biggest disappointment with my project was the lack of interest it sparked. Although I sent out more than ten copies of my posters, to estab- lishments all around the country, I feel that I could have done a better job of promoting the project. I should have sent letters to a wider range of schools and colleges, and sent these earlier. I should also have tried to get my project mentioned in local newspapers and any relevent journals. I also should have put up more posters advertising my exhibition. If I could further my project I would aim to display the exhibition in a location where more people would see it. I would ideally be able to place it in some sort of science museum or education centre. A copy of my posters is on display at the Satrosphere science education centre in Aberdeen, but this isn’t the same as displaying the original framed art work. I would love to produce similar posters showing an even wider range of mathematical topics, and maybe experiment with a wider range of media, such as sculpture.

3.5 Personal Reflections Throughout this project I had many opportunities to both contribute to society and make personal gain. Primarily I got a unique opportunity to pass on my knowledge of something I’m passionate about to members of the general public, and in a novel way. I believe that my project not only showed what a varied and exciting subject mathematics is, but encouraged people of all ages to take more of an interest in learning the subject. Even if no new mathematical knowledge was taken away from viewing my projects, I hope I have encouraged a more positive outlook on learning in general.

20 The interdisciplinary nature of this project gave me a chance to research and link two subjects of great interest to me. I hope to have inspired enthusiasts of both mathematics and art with my project, and also to have encouraged teachers to try more interdisciplinary teaching methods within the classroom. I have also developed a range of transferable skills over the course of this project; the project involved a lot of communication, organisation and time- management. In particular, organising the exhibition involved liaising with several departments within the university as well as outside contacts. I also had communication with many schools and education centres in the latter part of the project. The sheer amount of research I had to conduct, and the various project deadlines meant that good management of my time was essential. Finally, the project clearly developed my art skills, as well as giving me further experience of holding an exhibition. It also gave me experience over the whole design process, and challenged me to produce art work to a plan. Above all, though, this project has inspired me, both to learn and teach, and it may just be the first step in me realising my dream to open up the country’s first hands-on mathematics education centre.

21 References

[1] Adelphi University Department of Mathematics and Computer Sci- ence Available at: http://academics.adelphi.edu/artsci/math/ [Accessed 22/11/11]

[2] QCDA The National Curriculum Available at: http://curriculum.qcda.gov.uk/ [Accessed 22/11/11] [3] Seghers, M.J. The Golden Proportion and Beauty Plastic and Reconstruc- tive Surgery, Vol. 34, 1964

[4] Mandelbrot, B.B. The Fractal Geometry of Nature W.H. Freeman and Company, 1982 [5] Pickover, C.A. The Pattern Book: Fractals, Art and Nature World Scien- tific, 1995 [6] Mumford, D, Series, C and Wright, D Indra’s Pearls: The Vision of Felix Klein Cambridge University Press, 2002 [7] American Mathematical Society Mathematical Art Exhibition Available at: http://www.bridgesmathart.org/art-exhibits/jmm09/index.html [Ac- cessed July-September 2011]

[8] Context Free Art Context Free Available at: http://www.contextfreeart.org/ [Accessed August 2011] [9] Taylor and Francis Group Journal of Mathematics and the Arts Avail- able at: http://www.tandf.co.uk/journals/tmaa [Accessed July-September 2011]

22 A School Letter

The following shows the text from the letter that I sent to schools, advertising my project.

Mathematics in Art Project

Dear Sir/Madam, I am writing to you with regards to the above project. My name is Mairi Walker, a fourth year mathematics student at the University of Warwick with a keen interest in art. My project, funded by the Lord Rootes Memorial Fund, aims to communicate mathematics to the general public; in particular I would like to show high school and college pupils the beauty and relevance of mathe- matics, and to encourage them to continue the subject beyond GCSE. To do this I will be creating a series of 6 paintings and drawings, each showing a particular link with mathematics. I will create an information sheet for each piece of art work showing this link and explaining the mathematics behind it. This will all be exhibited in the Zeeman building at the University of Warwick campus from 17th to 23rd October, but I will also be combining my images and text into poster format. I will be offering sets of these posters to local schools and colleges, free of charge, around the time of my exhibition, and the purpose of this email is to see if you would be interested in displaying a set. The topics covered in the posters include fluid dynamics, symmetry groups, escape time fractals, mathematical modelling of nature, analysis and hyperbolic tessellations. These are topics not encountered until undergraduate level study at the earliest, but I have endeavoured to make the content of the posters accessible to a general audience. If you would like any further information do not hesitate to get in touch: my contact details are provided above. Thank you for your time.

Kind regards, Mairi Walker

23 B Questionnaire

I designed a questionnaire in order to obtain feedback on my project, in partic- ular with respect to the project aims. Copies of it were left with the exhibition alongside a comments box to post them in, and further copies were distributed, along with stamped addressed return envelopes, to the schools, colleges and ed- ucation centres that I sent the posters to. The questions on my questionnaire are as follows: • Do you think the text on the posters is at a suitable level? If not, how could this be improved?

• Do you think the range information included on the posters was good? If not, how could this be improved? • Do you think a good range of mathematical topics has been covered? If not, how could this be improved?

• Have you learnt anything new from the posters? • Has this project changed your opinion of mathematics? If so, how? • How do you think the posters could be improved? • Have you got any other comments on the posters or the project in general?

24 C Financial Statement

I requested £861 from the Lord Rootes Memorial Fund, and was awarded £700. This meant that I had to rebudget before I could begin my project.

Original proposed budget

(Research) MATLAB Student Version - £89 (as of 06/01/2010 approx £58) The Pattern Book by Clifford A. Pickover - £44 Reeves Hardback A3 Sketchbook - £14 Colour printing for resources - £5 Sketching pencils, pens and watercolours - £20 (not requested from fund) Subtotal - £141 Subtotal requested from fund - £121

(Trip to London galleries and museums) Train tickets Lowestoft to London Liverpool Street (off-peak return with railcard) - £30 Tube fares (2 day passes) - £10 Accommodation (one night, Travelodge) - £90 Subtotal requested from fund - £130

(Painting) 6 board canvases (16 x 12), or similar - £30 Acrylic paint (Daler-Rowney System 3 set) £25 Oil paint (Daler-Rowney Georgian selection set) - £30 Extra paints needed (i.e. other colours) - £15 Extra surfaces needed for prep work - £10 (not requested from fund) Brushes (Pro Arte Acrylic brush set) - £15 Low odour thinner and finishing solutions £5 (not requested from fund) Frames and mounts (local framer) - £180 Subtotal - £310 Subtotal requested from fund - £295

(Publicity and printing) 50 95x210mm Posters (Vistaprint) - £9 100 postcard invites (Vistaprint) - £8 100 95x210mm information flyers (Vistaprint) - £18 6 A3 Colour prints of information sheets (library printer) - £5 100 surveys (library printer) - £6 20 letters to schools (inc. envelopes and postage, library printer) - £10 6x20 286x439mm posters (Vistaprint) - £111 Website hosting for blog - £60 (not requested from fund) Subtotal - £227 Subtotal requested from fund - £167

(Exhibition) Exhibition panels and hanging hooks hire (accessdisplays.co.uk) for 5 days - £93

25 Hanging equipment (thread, tape etc, local framer) - £5 Opening night drinks and nibbles - £50 Subtotal requested from fund - £148

Total - £956 Total requested from fund - £861

Here follows my revised budget and financial statement, with budgeted amount in brackets.

02/06/11 - The Pattern Book - £43.30 (£44) 11/06/11 - Daler Rowney A4 Sketchbook - £10.25 (£12) 28/07/11 - Bus to London and train back - £22.90 (£30) 29/07/11 - Oyster card - £15 (£15) 29/07/11 - Travel Lodge - £59 (£70) 02/08/11 - Printing resources - £8 (£5) 22/08/11, 29/09/11 - Canvas, paints and brushes - £66.75, £27.66 (£100) 28/08/11 - Extra paints - £13.08 (£15) 27/09/11 - Printing advertisement posters, school letters, questionnaires and posting - £25 (£35) 07/10/11 - Framing - £216 (£180) 14/10/11 - Printing exhibition and school posters - £80 (£80) 17/10/11 - Hiring display boards and hanging work - £52.50 (£65) 17/10/11 - Opening night refreshments - £43.10 (£50) 24/10/11 - Buying poster tubes and posting posters - £25 (£0) Total - £707.54 (£700)

26 D Exhibition Poster

Figure 3: My exhibition poster.

27 E Sketchbook Extracts and Final Art Work

Figure 4: Finding suitable starting points.

28 Figure 5: Studying fluid flow from an artistic perspective.

Figure 6: Taking inspiration from the work of the graphic artist M. C. Escher.

29 Figure 7: Investigating mathematical drawing techniques

Figure 8: Developing designs based around knot theory and crystal structure.

30 Figure 9: Generating fractal images by computer.

31 Figure 10: Crystal, acrylic paint on canvas paper.

Figure 11: Fractal, ballpoint and pencil on paper.

32 Figure 12: Hendrix, acrylic paint and wax on canvas paper.

33 Figure 13: Landscape, oil paint on canvas paper.

Figure 14: Lily Pond, pencil on paper.

34 Figure 15: Splash, oil paint on canvas paper.

35