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
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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.
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Contents
1 Introduction
1.1 Origins of the Report . . . . . . .
1.2 Project Aims . . . . . . . . . . .
1.3 Mathematics in Art: The Basics
1.4 Project Outline . . . . . . . . . .
1.5 Limitations . . . . . . . . . . . .
1.6 Structure of the Report . . . . .
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2 Project Methodology
2.1 Mathematics and Art . . . . . . . . . . . . . . .
2.1.1 Researching Mathematics and Art . . . .
2.1.2 The Links Between Mathematics and Art
2.2 Creating Mathematical Art . . . . . . . . . . . .
2.2.1 Searching for Design Ideas . . . . . . . . .
2.2.2 Design Development . . . . . . . . . . . .
2.3 Communicating Mathematics Through Art . . .
2.3.1 Communicating Mathematics . . . . . . .
2.3.2 Example: Hendrix . . . . . . . . . . . . .
2.3.3 Communicating the Project . . . . . . . .
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3 Analysis
3.1 Summary . . . . . . .
3.2 Findings and Analysis
3.3 Conclusions . . . . . .
3.4 Recommendations . .
3.5 Personal Reflections .
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Bibliography
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A School Letter
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B Questionnaire
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C Financial Statement
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D Exhibition Poster
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E Sketchbook Extracts and Final Art Work
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List of Figures
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The first four stages of constructing the Peano Curve. . . . . . .
Exhibiting in the Zeeman Building . . . . . . . . . . . . . . . . .
My exhibition poster. . . . . . . . . . . . . . . . . . . . . . . . .
Finding suitable starting points. . . . . . . . . . . . . . . . . . .
Studying fluid flow from an artistic perspective. . . . . . . . . . .
Taking inspiration from the work of the graphic artist M. C. Escher.
Investigating mathematical drawing techniques . . . . . . . . . .
Developing designs based around knot theory and crystal structure.
Generating fractal images by computer. . . . . . . . . . . . . . .
Crystal, acrylic paint on canvas paper. . . . . . . . . . . . . . . .
Fractal, ballpoint and pencil on paper. . . . . . . . . . . . . . . .
Hendrix, acrylic paint and wax on canvas paper. . . . . . . . . .
Landscape, oil paint on canvas paper. . . . . . . . . . . . . . . . .
Lily Pond, pencil on paper. . . . . . . . . . . . . . . . . . . . . .
Splash, oil paint on canvas paper. . . . . . . . . . . . . . . . . . .
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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 nature 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 aspirations 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’.
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• 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 monuments 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 commonly used by artists until the 20th century, when there was an explosion of
activity by both artists interested in mathematics, and mathematicians interested 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 approximately) 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 become 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 mathematics 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 communicating 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 designing 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 success 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 framing 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 certainly 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 mathematics, 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 preliminary 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.
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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 communication 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 organisation 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 generally 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 mathematical starting point (see Appendix E). I compiled basic informaton on the
historical context and as the mathematics involved, but I also began experimenting 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 mathematics 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 mathematics 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 inspira11
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 applications, but often when describing symmetries. It is hence not that surprising
that we meet the subject quite regularly when looking for links between mathematics 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 simple graph functions to space-filling curves. The latter relatates to fractal dimension 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 everything 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: fractal 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 iteration 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 development 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 mathematical 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, particularly 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 mathematics in architecture, looking, for example, at the structure of the Eden Project
biodomes.
I found the most inspiration within geometry, looking at geometric sculptures 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 specifically 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 people’s views. Wherever possible I asked people, such as school pupils, family,
friends and public library users, their opinions on both my choice of mathematical 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 decided 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 mathematical 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 comprehension 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 accurate, 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 throughout 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 outreach 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 Warwick 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 provide 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 aspirations 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 understand 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 quality 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 impressed 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 feedback 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 representative range of people. Although the questionnaire featured many open
questions, which can result in people not putting in the effort to answer properly, 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 successfully communicate mathematics through art, at least to the people who responded 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 wellwritten 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 general 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 preparation 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 establishments 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 timemanagement. 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 Science Available at: http://academics.adelphi.edu/artsci/math/ [Accessed
22/11/11]
[2] QCDA
The
National
Curriculum
http://curriculum.qcda.gov.uk/ [Accessed 22/11/11]
Available
at:
[3] Seghers, M.J. The Golden Proportion and Beauty Plastic and Reconstructive 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 Scientific, 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 [Accessed July-September 2011]
[8] Context
Free
Art
Context
Free
Available
http://www.contextfreeart.org/ [Accessed August 2011]
at:
[9] Taylor and Francis Group Journal of Mathematics and the Arts Available 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 mathematics, 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 particular 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 education 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