Conversations in a foreign language

Topology and art were firmly connected as I was growing up. My artist uncle’s painting of an Alexander horn sphere hung on the dining room wall. My topologist father would delight in showing my siblings and me the strange qualities of the Mobius strip and of, what was to us, the very bizarre Klein bottle and, of course, the famous doughnut-coffee mug. His enthusiasm for these surfaces found room inmy imagination and they have been nestled in there, embedded, ever since. So too the tantalizing and challenging mystery of 4-space, the mathematician’s fourth dimension. If ever I had any doubt that topology and art were natural bedfellows, over here in London, Britain’s flagship modern art gallery, Tate Modern, held a symposium called simply; ‘Topology’. (November 2011 – June 2011). For themost part I work in 2-D and use traditional materials. I draw on fine papers with pencil, charcoal and pastel. I have several rules I impose on myself as I work. I set out a composition using the renaissance practice of rebatement. This is the geometric division of the canvas used to steer the viewer’s eye to all areas of the image and to direct the main focus on the critical parts of the narrative (Bouleu, 1963). I remain faithful to the topological rules of no tearing, cutting or intersecting of the surfaces and no puncturing either, however representing surfaces that intersect in 3-space but do not intersect in 4-space in a 2-dimensional image is a challenge to say the least. Some topological surfaces can engender so many ideas that I have to be firm and temper the wanderings of my imagination. The constraints of the topological surface in the question, the medium in use and staying true to my original inspiration present exactly the kind of challenges that I delight in. For many people even the mention of mathematics is off-putting and mathematical art can seem an oxymoron but there are ways to circumvent this reluctance. Beauty and grace are alluring and can be powerfully persuasive and with topologically derived art there is no need to apply these qualities superficially. They are built-in. As the coffee mug can morph into a doughnut, the Hopf link, two simple interlinked rings, can morph into multiple forms. In the drawing ‘Conversations in a Foreign Language; Three Solid Arguments’ (Figure 1) three solid forms bounded byHopf links are each presented from five different viewpoints. (The models for these forms are three small clay