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Hyperbolic Geometry

The above picture (1996-8) is by Tony Robbin. It is in a private collection in New York.

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The above animation is by Nick Mee. It shows a three-dimensional projection of the edges of a 24-Cell. The 24 Cell is composed of twenty-four regular octahedra. It is one of the six regular four-dimensional polytopes.


Thirty years ago when I formulated my aesthetic goals of depicting an abstract space that would embody our contemporary experience, I settled on the idea of using patterns to represent space. The problem for the artist who wants to depict space is that space is what is not there. For instance, if telephone poles or railroad tracks were used to depict space the result would be a picture of telephone poles and railroad tracks rather than a picture of space itself.

The Islamic tradition of pattern making, which revels in the intricate symmetries and implicit structures of space, is ideally suited to my artistic purpose: it is possible to see patterns as markers for space alone. No doubt I was drawn to this tradition having grown up in Iran.

In a typical painting of mine many spaces have been superimposed: A two-dimensional Persian pattern has been curved onto a dome form. A honeycomb of rhombic dodecahedra has been projected onto a hyperbolically curved surface. A field of octahedra and also a field of dodecahedra have been projected onto two separate perspective planes. A honeycomb of octahedra and cuboctahedra has been placed on a conical surface. Two different views of a quasicrystal lattice have been interwoven. Many spaces in the same place at the same time.

After a visit to Scott Carter at the Mathematics Department of The University of Southern Alabama in Mobile in 2000, I began to think of these paintings as four-dimensional knot diagrams.

My flowing hyperplanes, which have not only thickness but an internal structure as well, are like the sheets of Carter’s topology that flow through a space of four dimensions. The hyperplanes braid in ways that are impossible in three dimensions, but are the natural consequences of projecting higher dimensional structures into lower dimensional spaces, and locations that are singular (where the three-dimensional manifolds appear to intersect) are artifacts of the projection - in their own space the hyperplane slices pass behind one another.

Many physicists seriously consider that our three-dimensional space is but a projection of higher dimensional space, so that projective geometry should guide our understanding of the Universe. Thus, art that deals with the visualization of higher dimensions engages the thought of our time.

Text by Tony Robbin.