Models of Geometric Surfaces

The Mathematical Institute has a large collection of historical mathematical models, designed and built over a hundred years ago. While the models retain their aesthetic appeal despite showing the scars of more than a century of use, their purpose can now appear obscure. Oxford undergraduate Adam Barker undertook a summer project (supervised by Sam Howison) to catalogue the models and put together these webpages describing the models with their history and underlying mathematics. The project was made possible with financial support from Christ Church, Oxford, which is gratefully acknowledged. We hope this web resource will make the material accessible to a wide audience.

Much of the content should be accessible to somebody about to start a mathematics degree, occasionally touching on more stretching topics (the harder material can be ignored). The aesthetic beauty of the models should be enjoyable for anyone with an interest in mathematics, art or history, regardless of your level of mathematical training.

Some theory relating to the models (e.g. projective geometry) is fairly standard undergraduate material, but some models exhibit curious properties which are not now commonly studied (e.g. Dupin Cyclides). The models give a snapshot of geometric concepts that were considered important over a hundred years ago. Looking back at them, we see how the study of the subject has since evolved. At the start of the 19th century, geometry was mostly still a visually understandable topic. In line with common intuition about geometry, familiar shapes, curves, surfaces, and other constructions were studied at the forefront of research. However, mathematicians such as Riemann developed more general and abstract structures which extended geometry beyond familiar 3-dimensional Euclidean space. This gave rise to modern differential geometry. Other mathematicians such as Hilbert and Noether developed the algebraic aspects of the subject, giving rise to what is now known as algebraic geometry. Both strands are central to modern mathematics and a host of applications in physics and elsewhere.

The site includes crash-courses in algebraic and differential geometry, with links to sites such as Wolfram MathWorld and Wikipedia. We suggest you start with a look at the Theory section, but it is also possible to jump straight in to look at the models - any unfamiliar terminology is linked back to Theory anyway. To explore the catalogue, just click the blue links on the left of this page. We hope you will enjoy the site!

Adam Barker and Sam Howison

Summer 2015

adam.barker@pgr.reading.ac.uk; howison@maths.ox.ac.uk