The Mathematische Forschungsinstitut Oberwolfach invites you to have a look at mathematical creations “through the eyes of mathematics“ in a way you have perhaps never done before. The figures shown are taken from geometry, which is just one field of mathematics, though an important one. Maybe, the word geometry makes most of you think of triangle geometry from school days, but modern geometry is much more. It is of major current importance and, what is more, it is extremely attractive. Today, this attractiveness can be expressed in images generated by mathematical programs by means of computers such as was not possible some years ago.

In various galleries we show you amazingly beautiful images, which often obey quite simple formulas. Legends to the images indicate the origin and meaning of the images and suggest some mathematical concepts.

The interactive SURFER program allows playful contact with geometry, i.e. it encourages the users to create formulas themselves or to change already existing ones in a creative manner. It is surprising to see how amusing it is and how you begin after some time and pondering to understand how formulas and forms, i.e. algebra and geoemtry, are connected.

You certainly wonder what these images of surfaces and their singularities are good for, what they mean, whether they have applications or serve to describe real phenomena. This is true, indeed, but it is not at all simple to explain. Introductory articles by various mathematicians try to illustrate some of the real backgrounds or kindly provide a glimpse into the mathematical workshop and finally presents you with a current research problem.

Moreover, there are innumerable relationships of algebra and geometry to other sciences, but there are also quite specific applications in industry and economy. To describe these relations in detail would involve more than one exhibition. My own research interests that I pursue and teach at the University of Kaiserslautern are algebraic geometry, singularity theory and computer algebra - all present at the exhibition. If you are interested in the mathematical concepts or applications of this theory for cryptography or micro-electronics, please visit my homepage.

I wish you an inspiring and informative visit of the exhibition IMAGINARY, be it on the Internet or live at the exhibition. The big printed pictures are far more attractive and than the pictures on the screen. At the exhibition you can additionally see films, fly through pictures using a 3d mouse or watching them through 3d glasses. On a huge touch screen you can create own pictures interactively and take colour print-outs at home. I am sure, that your children, students and you will enjoy!

Gert-Martin Greuel
Technical University Kaiserslautern and
Director of the Mathematisches Forschungsinstitut Oberwolfach

The idea behind the IMAGINARY exhibition is – as the name suggests – to use the visual and aesthetic component of mathematics as an eye-catcher in order to explain the visitors the mathematical backgrounds in an interactive manner. The imaginary and unimaginable of mathematics is illustrated, it turns to images which you can generate yourself.

On the one hand, the exhibition consists of a gallery of interesting and beautiful mathematical images taken from algebraic geometry. They are presented at a size of 85 x 85 cm on acryl glass and can be gazed at in a walk-in aluminium cube, the IMAGINARY cube. Each image is provided with a table which explains mathematical properties and how the image was generated. The important elements of the images, such as singularities, are described. On the other hand, interactive installations at the exhibition invite the visitors to take mathematical artistic action themselves. For this purpose the Surfer program was designed to calculate, indicate and change algebraic surfaces in real time. Visitors can enter and change polynomial equations on a large touchscreen with their fingers, shift parameters, determine the colours of the surfaces and turn the figures as they like.

Herwig Hauser from Vienna presents a film showing mathematical fancy surfaces and artfully presented algebraic surfaces. His images form the core of the IMAGINARY Cube. The film and the images were already very successfully shown at the International Congress of Mathematicians in Madrid in 2006. Richard Palais and Luc Benard (USA, Canada) present elaborate visualizations, e.g. Wada Basins or Triple Periodic Surfaces. The awarded image with a collection of well-known surfaces on glass such as the Boy surface, is also on display. It won the Science Visualization Challenge, 2006, of the Nature Journal. Palais, for years, has been working on the 3D_XLPLORMATH program, a very comprehensive interactive program which aims at discovering mathematics from a three-dimensional view. Hermann Karcher from Bonn was among his team members. The program is presented, among others, with an introduction to be viewed with 3D glasses.

Jos Leys, Étienne Ghys and Aurélien Alvarez (Belgium, France) present the first part of their new DVD series DIMENSIONS, which explains the mathematical concepts on a generally understandable level. The first part which clearly presents the stereographic projection and offers a look into the fourth dimension can be seen in public for the first time. In addition, Ghys and Leys put spectacular images on display such as the hecatonicosachoron or the Anosov flow. Ulrich Pinkall and his group in Berlin participate with the interactive JREALITY program, which generates a virtual mathematical world where visitors can move freely, such as climbing on surfaces or falling down. Images of the group contain, among others, an artful torus or a tetranoid.

From Berlin comes the multiply awarded film MESH on discrete mathematics made by Konrad Polthier and Beau Janzen. Jürgen Richter-Gebert from Munich prepared a series of interactive applications for IMAGINARY based on the program CINDERELLA. They communicate various mathematical topics as simulation, chaos or symmetries in a playful way. Martin von Gagern’s program MORENAMENTS allows to paint symmetrical patterns in one of the 17 space groups in the Euclidian plane. The Surfer program based on the Surf program by Stephan Endrass was specifically developed for the IMAGINARY exhibition by Henning Meyer from Kaiserslautern and Christian Stussak from Halle assisted by Oliver Labs from Saarbrücken.

The companies Voxeljet Technology at Augsburg, and Alphaform at Feldkirchen are engaged, among others, in generative manufacturing of 3D models by selective gluing plastic powder (PMMA) or by means of stereolithography. Both companies have accepted the challenge to print in 3D a selection of algebraic surfaces of the exhibition and present them as sculptures. The challenge was to find appropriate model data records. Therefore, the Institute FORWISS of Passau University implemented various techniques for the exhibition to translate algebraic surfaces into printable data. Ten sculptures at a diameter of about 25 cm are exclusively displayed at the exhibition. They were produced for the exhibition by the two companies free of charge.

In order that the exhibition and also the contents continue to be used sustainably a didactic package together with manuals and software is established and is available for visitors and also schools. All are invited to also use the Surfer program from home and generate interesting surfaces. A mathematics-art-competition motivates the visitors to take action themselves also after the exhibition. The exhibition will continue to be open in 2009. Parts of the exhibition are to be installed in German museums permanently, among others, in M2 – Mathematik und Mineralien Erlebniswelt Oberwolfach / Oberwolfach mathematics and minerals world of experience – which will be opened in 2010 .

The IMAGINARY travelling exhibition is supported by the Federal Ministry of Education and Research. A number of private persons, firms, universities and institutions have campaigned for the exhibition to be implemented in the Year of Mathematics. Thank you very much to all who supported the exhibition!

The current exhibitions can be found here.