The *Plus* teacher packages are designed to give teachers (and students) easy access to *Plus* content on a particular subject area. Most *Plus* articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world,
find maths in unusual places, and delve into mathematical history and philosophy. We therefore hope that our teacher packages provide an ideal resource for students working on projects and teachers wanting to offer their students a deeper insight into the world of maths.

## Maths and art

Maths is all about patterns and rhythms, so it's no surprise that there's plenty of maths in art. Whether it's the visual arts or the performing arts, maths can be a tool, an inspiration, or simply something that's naturally contained within the structures and patterns. This teacher package brings together all *Plus* content on maths and art, grouped into the following categories.

- Maths and the visual arts
- Maths and design
- Maths and music
- Maths and film
- Maths and theatre
- Maths and writing
- Try it yourself with NRICH

### Maths and the visual arts

String Theory, Duality and Art: how the Higgs boson and Turner Prize collide — On the face of it, an artist and a theoretical physicist might seem an unlikely pairing. But Turner Prize-winning sculptor Grenville Davey and string theorist David Berman's collaboration is producing beautiful, thought-provoking work inspired by the fundamental structure of the Universe. Julia Hawkins interviewed them to find out more about how the Higgs boson and T-duality are giving rise to art. (*Physics, string theory*)

Understanding uncertainty: Pure randomness in art — This article is based on a talk the author gave at the recent John Cage exhibition in the Kettles Yard gallery in Cambridge. Cage is perhaps best known for his avant-garde music, particularly his silent 1952 composition 4′33″ but also for his use of randomness in *aleatory music*. But Cage also used randomness in his art.
*(Randomness, Sudoku)*

Maths and art: the whistlestop tour — Our brains have evolved to take pleasure in rhythm, structure and pattern. Since these topics are fundamentally mathematical, it should be no surprise that mathematical methods can illuminate our aesthetic sense. *(Geometry, the Golden Ratio, tessellation, polyhedra)*

Getting into the picture — Imagine stepping inside your favourite painting, walking around the light-filled music room of Vermeer's *The Music Lesson* or exploring the chapel in the *Trinity* painted by Masaccio in the 15th century. Using the mathematics of perspective, researchers are now able to produce three-dimensional
reconstructions of the scenes depicted in these works. *(Linear algebra, matrices, perspective)*

Universal pictures — At first glance, differential equations don't have much to do with beauty or art. But a careful look at nature reveals that the patterns they describe are everywhere, from branching rivers to the behaviour of crowds. This article is also accompanied by a podcast. *(Partial
differential equations, diffusion)*

Extracting beauty from chaos — Fractal images based on the mathematics of chaos are very striking. This article explains what the much misunderstood phenomenon of chaos really is, and how you can iterate functions to produce marvellous images of chaos from simple mathematics. *(Fractals, iterated functions, parabola)*

Restoring profanity — In 1979 decorating work in a house in Vienna revealed a set of medieval frescoes depicting a cycle of songs by a 13th century poet, who was particularly fond of satirising the erotic relationships between knights and peasant maidens. The frescoes are of great historical significance, but they are badly damaged. This
article explores how mathematicians use the heat equation to fill in the gaps. *(Partial differential equations, diffusion, the heat equation)*

Fractal expressionism — In the late 1940s, American painter Jackson Pollock dripped paint from a can onto vast canvases rolled out across the floor of his barn. This article explains that Pollock's patterns are really fractals — the fingerprint of nature. *(Fractals, logarithms)*

The artist's fractal fingerprint — A news story exploring how the fractal structures in Pollock's paintings can be used to identify the artist, both to prevent forgery and to understand why we feel drawn to certain works of art.

Rollicking pollock debate — A further news item revealing new research into the fractal structures in Pollock's paintings.

Digital art — Computer-generated art is on the rise, and with it comes a further blurring of the boundaries between maths and art. This article looks at some examples.

The power of origami — We've all heard of origami. It's all about making paper birds and pretty boxes, and is really just a game invented by Japanese kids, right? Prepare to be surprised with this journey through origami, maths and science. *(Euclidean geometry, trigonometry)*

Imaging maths: unfolding polyhedra — Not only are paper models of geometric shapes beautiful and intriguing, but they also allow us to visualise and understand some important geometric constructions. Find out more about the gentle art of paper folding. *(Geometry, Platonic solids, Archimedean solids)*

Imaging maths: Inside the Klein bottle — This article takes you on an illustrated tour of an extraordinary geometric construction: the Klein bottle. *(Topology, Klein bottle, Möbius strip)*

Still life — Sometimes the best way to express a scientific idea is through an image that grabs the eye and invites viewers to wonder what they're seeing. This news story explores the mathematical winning entries of the 2006 Science and Engineering Visualisation Challenge. *(Geometry, topology, Klein bottle, Euclidean and
non-Euclidean geometry)*

Secrets from a bathroom floor — Tilings have adorned buildings from ancient Rome to the Islamic world, from Victorian England to colonial Mexico. But while it sometimes seems free from worldly limitations, tiling is a very precise art, where not much can be left to chance. We can push and turn and wiggle, but if the maths is not right, it
isn't going to tile. *(Geometry, tiling, tessellation, polygons)*

The trouble with five — Squares do it, triangles do it, even hexagons do it — but pentagons don't. They just won't fit together to tile a flat surface. So are there any tilings based on fiveness? This article takes us through the five-fold tiling problem and uncovers some interesting designs in the process. *(Geometry, tiling, symmetry,
aperiodic tilings)*

Career interview: Exhibition curator — Exhibition design is not a career that the mathematically inclined tend to think about, let alone pursue. Barry Phipps is the first interdisciplinary fellow with the Kettle's Yard gallery in Cambridge. His remit is to develop projects of an interdisciplinary nature — "to find the common ground between things." This interview is accompanied by a podcast.

*Plus* podcast: Cosmic imagery — From the complexity of the snowflake, to the London tube map and the spiralling Andromeda galaxy, imagery has always been a vitally important ingredient of science. In this podcast, *Plus* talks to John Barrow, professor of mathematics at Cambridge University and author of the new book Cosmic Imagery, about
the images that have changed science, and how we have viewed science, over the centuries. *(Geometry, fractals, complex numbers, graphs, statistics, normal distribution, probability and chance, maps)*

This podcast is also available in an enhanced version, which shows all the images mentioned in this podcast as you listen. You can view the enhanced podcast in your browser, or download the MP4 file to watch on your computer or for playing on your MP4 player (for example iPod).

### Maths and design

Perfect buildings: the maths of modern architecture — This article explores the maths behind some of the London's landmark buildings. *(Mathematical modelling, geometry, parametric equations)*

Bridges, string art and Bézier curves — The Jerusalem Chords Bridge, Israel, was built to make way for the city's light rail train system. Its design took into consideration more than just utility — it is a work of art, designed as a monument. Its beauty rests not only in the visual appearance of its criss-cross cables, but also in the mathematics that lies behind it. (*Geometry, coordinate geometry*)

Career interview: furniture design — Two designers tell us how they took the long way round to design, and how the maths and science they took in on the way helps them with their work today. *(Physics, engineering, geometry)*

Career interview: fashion design — Sandy Black, Professor of Fashion and Textile Design, has combined her love of art and design with her love of mathematics in her career as a knitwear designer. Sandy talks to *Plus* about the mathematics in fashion, knitting, and how science and fashion could make the world a better place. *(Ratios,
dimensions, tiling, numeracy, topology)*

### Maths and music

Fractal music — Fractals are a treat for your eyes, but what about your ears? Dmitry Kormann, a composer/keyboardist from São Paulo, Brazil, explains how he integrates fractal-like patterns in the very structure of his music, to obtain beautiful results. *(Fractals)*

What makes an object into a musical instrument? — Many things make a noise when you hit them, but not many are commonly used to play music — why is that? Jim Woodhouse looks at harmonic and not so harmonic frequencies and at how percussion instruments are tuned. *(ratios, waves, harmonics)*

The magical mathematics of music — According to Shakespeare, music is the food of love. But this article follows Galileo's observation that the entire universe is written in the language of mathematics — and that includes music. *(Waves, frequency, ratios)*

Why is the violin so hard to play? — As anyone starting out knows, the violin is a difficult instrument. It takes time before the novice player can expect to produce a musical note at the desired pitch, instead of a whistle, screech or graunch. This article explains why.*(Waveforms, frequency)*

Music and Euclid's algorithm — What does a mathematician from the 3rd century BC have to do with tuning musical instruments in 17th century Europe? This article tells us about one of the more unusual places you might find Euclid's algorithm being used. *(Highest common factor, Euclid's algorithm, frequency, ratios)*

Sine language — As an electronic musician Oli Freke has always been fascinated by sine waves, so much so that he's created a song based on them. In this article he explores his song, touching on ancient Greek mythology, strange piano tunings and Johann Sebastian Bach. *(Sine waves, waveforms, frequency, ratios)*

Self-similar syncopations — This article investigates the links between music and mathematics, throwing in limericks, Fibonacci and Scott Joplin along the way. *(Fibonacci series, symmetry, pattern)*

Lost in music — These are great times if you're a music lover. The World Wide Web opens the door to a near infinite volume of music, and new technology allows you to store a sizeable chunk of it on your hard drive or carry it around with you in a little box. The problem is that choice is time-consuming. This article explores the maths that allows you to interact with your music collection.

Career interview: Audio software engineer — Skot McDonald talks to *Plus* about how he uses mathematics to understand music, and how he managed to combine his passions for music and computing to create a successful career. *(Sine wave, frequency, waveforms)*

Career interview: computer music researcher — Teaching a machine to *understand* music is an incredibly difficult task, but teaching a machine to *compose* music is quite another matter, and the wonderful world of mathematical patterns proves to be a gold mine. Nick Collins talks to *Plus* about his artificial musician. *(Waves,
frequency, sinusoidal waves, phase, algorithmic composition)*

### Maths and film

Maths goes to the movies — Computer generated movies and electronic games: find out about the mathematics and engineering behind them. *(Geometry, linear algebra, vectors, complex numbers)*

It's all in the detail — The computer animation used in movies and games is now so lifelike, it is very hard to believe that you are actually watching a surface built from simple shapes of triangles. Find out how mathematics is used to help bring these models to life. *(Geometry, linear algebra, vector, dot product)*

Career interview: Visual effects director — Alexis Wajsbrot is a visual effects specialist who has worked on a number of high-profile films including Harry Potter and the Half-blood Prince, Harry Potter and the Order of Phoenix, Tim Burton's Sweeney Todd, and also on some of those visually stunning commercials you see while waiting for your film to
start. *(3D coordinates, diffusion, mathematical models, physics)*

Maths, madness and movies — In the movies mathematicians are mostly mad. Since here at *Plus* we firmly believe in our sanity, we're puzzled as to why. So we charged one of our authors with the unenviable task of sifting through five well-known maths movies and speculate towards an answer. This article is accompanied by a podcast. *(Pi, mathematical proof, game theory)*

Did a philosopher kill WALL-E? — Artificial intelligence works fine in the movies, but can it ever become reality? *(Logic)*

### Maths and theatre

A disappearing number — Mathematics took to the stage with *A disappearing number*, a work by *Complicite*, inspired by the mathematical collaboration of Hardy and Ramanujan. This article explains some of the maths. This article is accompanied by a podcast. *(Infinite series,
convergence, partitions, patterns)*

Career interview: Actor and mathematician — Victoria Gould has always known she would be an actor, and went straight from studying arts at school to running her own theatre company. But she eventually had to come clean about her guilty secret — she loves maths — and has since managed to combine a career as a research mathematician and teacher with a
successful acting career on television and in theatre. She tells *Plus* why she needs to use both sides of her brain. This interview is accompanied by a podcast. *(Physics, partitions, geometry, linear algebra)*

### Maths and writing

Career interview: Maths editor and fiction writer — Plus bumped into an old friend at the International Congress of Mathematicians this year: Keith Mansfield is the author of the Johnny Mackintosh series and commissioning editor for mathematics at Oxford University Press at the same time. In this interview he tells us how his he's built a career around his two talents, writing and maths.

### Try it yourself with NRICH

Our sister site NRICH published a collection of maths and art resources for all Key Stages from 1 to 5 in June 2010.

Also, there is a collection of archived project work resources on maths and art & design created to accompany past video-conferences run by the Motivate project (now ended).