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AND John Sharp shares his enthusiasm for mathematics at the London Knowledge Lab.

Brock Craft Louise Mabbs Mike Field Natalie Dower Natalie Dower

20 Mathematics Teaching 226 Journal of the Association of Teachers of Mathematics © ATM 2012 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions.

MAThEMATICS AND ART

o many people, mathematics means broadly: art implies visual art (, a dull and tedious experience in drawing, , computer graphics, T school. It is, however, very much video), , , textile art, ingrained in our culture particularly in art. literature/ (and others), and In MT193, with Phillip, we reported on the mathematics implies both mathematics Bridges Conferences on Mathematics and as a discipline and the related disciplines Art and invited readers to join us for the in science and engineering for which London conference in August 2006 for mathematics is an essential means of which we were the local organisers at the expression and communication. The full Institute of Education. The conference was list of seminars is available on our website very successful and has continued to grow www.lkl.ac.uk/events/maths-art visiting Donostia (San Sebastian) in Spain, We ask each speaker to provide some Escher’s birthplace, Leeuwarden in holland, background to the talk and an image which Pécs in hungary and Coimbra in Portugal. we put on the website as a poster. You can We wanted to keep some of the richness see the range and content on the website of ideas and images in London and so at and we suggest you look at the posters for the beginning of 2007 we began a series a detailed look at the contents of the talks. of monthly meetings, the LKL Maths-Art We have taken video of most talks and Seminars at the London Knowledge Lab are in the process of uploading them which is a joint venture between the to YouTube. We see these as a Institute of Education and Birkbeck valuable archive to show the College. contribution mathematics Bridges has many papers makes to art and vice and workshops being versa. The posters also presented over four days. contain links to websites This does not give much by many of the speakers, so time for discussion when they are a resource in their own they are presented as talks; right. The examples have been there were over 100 this year. We chosen as a representative sample, wanted more audience participation, one which has been hard to make. which is why we run seminars. This has been very successful. Talks generally A rough analysis of the talks shows the last 45 minutes and often have to call a types of the forty or so talks we have had. halt to the discussion after a further hour. The groupings do not do justice to the The speakers gain valuable feedback content because the goes beyond and possible new ways of exploration the simplicity of the group names and some and the audience have both a chance to talks fit across more than one subject group: contribute and probe more deeply into Tilings obviously feature in the subjects, the ideas presented. Some speakers but they are not just obvious descriptions are very experienced and a few have of groups. Edmund harriss, never spoken in public before and are took us on a tour of Penrose and other slightly apprehensive. Because they are aperiodic tilings. Brian Wichmann is building seminars, we chair them and guide them a database of mainly Islamic tilings and he if necessary, and because the audience is told us how he analyses and draws them. very supportive and warm, every seminar The database is a feast of images. Richard has a very relaxed atmosphere. The henry showed us Islamic tilings from of the seminars has brought audiences from the point of view as well as many a wide range of experience usually over 30 images from his travels. Cameron Browne and we now get people asking to speak in is another who talked about Truchet addition to our contacts through Bridges and tilings and how he developed them in his the art and mathematics world in London. computer artwork on Truchet curves and The seminars are explorations of the surfaces another instance where you need connections between ‘mathematics’ and to explore his website. We also learnt about ‘art’, where both terms are understood Robert Reid’s unusual tilings.

January 2012 www.atm.org.uk 21 Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY © ATM 2012 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions.

MAThEMATICS AND ART

Polyhedra appeared in a number of guises. in inspiration, particularly the Dudeny Nick Sayers talked about his artworks dissection of an to based on polyhedra, including sculpting a square. Justin Mullins’ work is called with coffee stirrers and building Geodesic “Mathematical photography” and sees Shelters from Estate Agent Boards. Patricia equations as beautiful in their own right. Wackrill described how the of bubble Linda Karshan uses numerical in packing was used in designing the Watercube grids. Raymond Brownell uses permutations Aquatics Centre for the Beijing Olympics. and combinations and Latin squares and Chaos appeared in a number expressed through colour which open up of guises. Michael Field described how he different ways of looking at mathematics in has used to create stunning an ordered way. Chris Gough’s work also artwork. Daniel White uses the Mandelbrot uses colour as a means of expression, set in his artwork and he has extended this to using mathematics based on random rules. three dimensions with the . Using stochastic methods in a different Music has not been as prominent as way, Simon Schofield makes interesting we would like. Paul Prudence, uses and beautiful textures in his artworks. computation involving the mathematics Roy Osborne who writes and lectures on of feedback in his “Sonified” generative colour theory was another artist showing artwork. Meurig Beynon explored how , grids and shape can be and in communicated through colour. relation to music and creativity. We also Sculpture can also be kinetic. Tom had a joint meeting with the Computer Wilkinson’s talk, entitled ‘Energy - A source Society which included the group Slub who of Inspiration’ uses wind among other use programming to compose music on the means in his work. Bálint Bolygó’s art fly as a kind of game. ‘Tracing, motion and ’ has some Surfaces link mathematics and art in many basis in harmonographs with swinging ways. Tony Wills and John Sharp talked pendulums, as well as using light to draw about developable surfaces and D-Forms. with lasers. Kate Mackrell, in a more Gregory Epps took a different look at educational slant showed how to use developable surfaces describing how he Dynamic Geometry software, particularly has taken curved folding from craft to a Geometer’s Sketchpad to either create art practical method of design. or understand it. Fiction and mathematics was surveyed by Architecture is notably mathematical these Tony Mann, covering a surprisingly wide days. Brady Peters and xavier De Kestelier range of aspects. from Foster and Partners described the work Textiles and mathematics appeared in of Foster and Partners’ specialist modelling different ways. Penelope Woolfitt discussed group and how they use digital design and how geometry is used in the design of generative geometry in architecture. This is Asian Trousers. Indu Choraria who knits also dynamic in its methods. high quality fashion garments showed how Mathematical ideas generating art featured surfaces and other aspects of mathematics in a different way from artists using is important in design. Louise Mabbs quilts mathematics. Whereas most of the artists uses mathematical sequences series as described above came to mathematics well as other geometrical ideas in her work. after working as artists, some speakers Mary harris, described the mathematics came from a mathematical or scientific behind some traditional textiles crafts background. This demonstrates the Bridges related to her work on Creative Threads and LKL Maths-Art idea of art inspiring in the 1980s which will be familiar to mathematics and vice versa. So Alan ATM members. Sutcliffe uses circle theorems on circle Artists who use mathematics have featured packing to create his art and has inspired in a number of talks. Gary Woodley, from the his artist wife Nicola to paint using them. Slade School of Art, has used 3-D Drawing: Brock Craft’s “Computer generated art using Modelling and Projection in his artworks. context-free grammars” showed work which Natalie Dower has taken mathematics as is very in using mathematical

22 Mathematics Teaching 226 Journal of the Association of Teachers of Mathematics Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY © ATM 2012 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions.

MAThEMATICS AND ART

logic. In complete contrast, David the medical field. The artist Clive head, Singmaster talked about the Three Rabbits in a discussion with Michael Paraskos, mathematical puzzle with its history showed how his could manipulate and appearance in art dating right back perspective to show a wide field of vision to 600AD across various cultures. Daniel and how mathematics could change our Piker, with a background in architecture, view of the world; we were privileged to called his talk “Intuitive Geometry”. The learn about this about a year before his work of many artists at Bridges falls under successful exhibition at the National Gallery. this umbrella; it’s just that they don’t use is an obvious choice for an overlap the same tools and their mathematical between mathematics and art. We had a language is their art. Daniel, however, very popular session give by Mark Bolitho understand the mathematics, but employs and the British Origami Society. This it in an artistic/architectural direction. his gave many insights into recent ideas of work covered symmetry, tensigrity, mathematical origami and was, of course, and transformations showing him as a true very hands-on. Our pre-Christmas session man. Mark Stock researches in 2010 was another hands-on one with the mathematics of fluid mechanics and small workshops of a range of paper-folding uses vortices to create his artwork because activities given by presenters John Wootton, he sees the of the mathematics. Tim Rowett, Tony Wills, Richard Ahrens and The Philosophy of mathematics and art has John Sharp. We hope to continue this idea. been covered by a number of sessions. Anamorphic art is a subject that brought the We had a very successful discussion two of us together. See “how to Anamorph” event with artists and on the mathematics of creating plane comparing the two disciplines as creative anamorphic images, by John Sharp in MT activities, with much audience participation. 217. We held a special seminar on this to Ernest Edmonds with ‘The Art of Logic’ complement a study day - partly organised compared logic and constructivism both by John Sharp - at the National Gallery in from an artistic and philosophical point of December 2008. view and included the way this applies to , particularly generative art. Conclusions Simon Morgan looked at the theoretical The range of topics and the distinguished side of art, covering and the speakers we have had show that Gestalt theory of and then mathematics is not a dry subject and has showed how mathematics could be applied made, and continues to make, a very to the analysis of images particularly in significant contribution to our culture.

John Sharp is Visiting Fellow at the Institute of Education and London Knowledge Laboratory.

Where to find out more We recommend that you visit our website at http:// www.lkl.ac.uk/events/maths-art/index.html, not only to find out more details about the talks but also about the speakers and gain links to their work. If you are in London when the talks are on, do join in and if you wish to give a talk, we welcome any offers. We have an archive of videos off the talks which we are putting on YouTube at http://www.youtube. com/user/LKLMathsArt. You should also visit the Bridges site at http://www.bridgesmathart.org. Look at the Galleries to see some stunning images of mathematics and art. Better still join the conference and meet some very interesting people.

go to www.atm.org.uk/mt226 for details of the artists.

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