# Exotic mathematical surfaces.

Sliceforms – John Sharp

Mathematical surface models are sculptural visualisations of mathematical formulae or a representation of a mathematical concept. Surface modelling originated in the 19th century Germany, when there was a revival of geometric ideas in mathematics, which stemmed from the architectural and engineering movements of the time. Felix Klein, who gave his name to the famous paradoxical bottle, which has no edge, no inside or outside, set up a workshop in the 1870’s purely in the business of producing strange shapes and forms that wouldn’t look out of place in an art gallery.

The intersection of math and art is always interesting, and its one of the main inferences of this blog as you will know. Things have a special allure when they avoid easy categorization – and in the same way these models have the power to engage the viewer on an artistic level through their sculptural character, not least since they represent the abstract and render intangible mathematical concepts tangible. There is also playfulness about these shapes and geometrical forms; this quality was undoubtedly picked up by the Kindergarten education system.

Angela Vierling has a page with exhaustive links and annotations regarding mathematical models, she’s even made us a nice family tree of historical model makers.

The Surrealist and Constructivist art movements have also flirted with these kinds of models. – And here’s a good overview on this particular area. The Surrealists were particularly enchanted by Algebra and Trigonometry. Giorgio de Chirco, one of my favourites, made his well-known ‘metaphysical’ studies of mathematical instruments and ominous compositions of surfaces and spheres in dreamlike landscapes – some of them contained surface models. Later Escher experimented with tessellation and recursion infinitely to produce endless variations of math art.

My favourite models, aside from variations of the Klein Bottle and perhaps Polyhedra, are the Sliceforms. Invented by one Olaus Henrici at the end of the 19th century they are modelled using cross sections of quartic surfaces and are similar to a spheres but with cross sections which are ellipses, hyperbolae or parabolae. The Strange Surfaces exhibition at the Science Museum in London contains some Sliceforms from the 19th century to the present day and other interesting historical models of surfaces – check out your local museums for the equivalents. Unfortunately I recently lost my camera memory card containing many snaps of these beauties.

Not surprisingly Flickr is a great place to track down some recently made sliceforms – check out dsliceform’s pictures. And an apparent increase in origami practitioners has seen a revival in the interest of these shapes, good news!

Today artists use more sophisticated means to produce much more complex artefacts, but none the less mathematical at their core – the resultant works sit within the realm of science as much as art. Rapid Prototyping, mentioned here before, is a technology used to cut impossible-to-hand-craft-shapes, often using complex 3D models developed on computers. For artists using this technique check out George Hart & Bathsheba Grossma’s work for starters.

Behind the staple fodder of Generative Art lays a canon of mathematical formulae and methods from times past, equations that are borrowed with a rich history remaining relatively unknown. Recently Marius Watz, well known for his Processing work, took the ‘sketch’ into ‘r/l’ with some rapid prototyping work producing some excellent results.

Extras:

A Cabinet of Mathematical Curiosities

The Institute for Figuring – dedicated to the poetic and aesthetic dimensions of science, mathematics and the technical arts. Highly recommended.

The Wolfram site has near enough a complete list of major mathematical and topological surfaces for your inquisitive eyes

### 2 Responses to “Exotic mathematical surfaces.”

1. Roland Dobbins writes:

Why have you turned off full-text syndication? Full-text is really a big deal for those of us who read lots of feeds. Please re-enable full-text syndication, many thanks!

2. paul writes:

heh roland, I had no idea i had turned off full-text syndication, or if by default it was turned off. I shall investigate and try switch back on.