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Wild Sphere by Helaman Ferguson

Umbilic Torus NC (3,1) and Umbilic Torus NIST (3,1) by Helaman Ferguson. Each sculpture is the other sculpture inside out. Umbilic Torus NC is covered by a Peano space-filling curve.


There are artists who engage effectively with science. And there are scientists involved at a high level in various arts. But there are few who can proffer such a unified curriculum vitae in art and science as the mathematical sculptor and sculptural mathematician, Helaman Ferguson.

Originally apprenticed to his stonemason stepfather, Ferguson studied painting, sculpture and mathematics at Hamilton College, and subsequently gained a doctorate in mathematics from the University of Washington at Seattle. He taught mathematics for 17 years at Brigham Young University and was co-discoverer in the late seventies of the first generalised Euclidean algorithm. His sculptures are founded on advanced topological theorems of plastic geometry, working variations on forms with such picturesque names as Alexander's horned wild spheres, toruses with cross-caps, and Thurston's Hyperbolic Knotted Wye.

Such a literally calculated approach might lead us to expect that Ferguson's sculptures will assume the guise of cool abstractions in hard stone or intellectual conundrums in gleaming metal. Yet, in reality, they are beguilingly sensual and organic, subtly responsive to light and the mobile viewer, stimulating our tactile instincts in a remarkably seductive manner.

A characteristic example, on show at the National Science Foundation in Arlington, is Whaledream II, carved from a wonderful piece of Carrara marble. It assumes the form of a horned wild sphere in which the branched outer contour of the sphere's surface actively encompasses intricate spaces (in contrast to a tame sphere in which the insider of the surface simply surrounds a spherical space in the normal way). When we look at the complex mathematical form arising from this intellectual bedrock, we are presented not with a dry demonstration but with a mellifluous series of embracing curves which speak serenely of physical and emotional harmony. Cool calculation generates warm feelings. How can we explain the apparent paradox?

At one level, the reconciliation is affected by Ferguson's sensual intuitions as a direct carver and skilled modeller, a master-craftsman who delights in the inherent responsiveness of varied materials, alternatively resistant to stretching and compression. At the morphological level, however, there is no paradox at all. He has shown how topology stands alongside other geometries that generate forms analogous to those that are manifest in the natural world - alongside D'Arcy Thompson's more Euclidean formulas of growth and form, and more recently alongside the new mathematics of complexity, chaos theory, fractals and self-organised criticality, The repertoire of shapes is aligned generically with such natural structures as shells, horns, tendrils, and soft-bodied marine organisms. He is inventing a form of living engineering, which exists in parallel to the organic architecture of nature.

Ferguson recalls a recent demonstration of the probity of his structural principles: He points to a photograph in the NSF exhibition, which depicts "a negative gaussian curvature carving I did in 20 tons of snow at the ski resort in Breckenridge, Colorado [for an International Snow Sculpture Symposium]. I carved the snow walls down to about 4 inches thick. The next week Breckenridge had a heat wave and all the other sculptors' works sagged and imploded. Mine just got thinner and more gracile, retaining its form. The reason seems to be that on a negative gaussian curvature carving every point is a saddle point, every point is a keystone of a fabric of arches. The snow event was a proof of a simple but beautiful mathematical principle."

It is clear that we react intuitively at a very deep level to the morphological ‘rightness’ that is eloquently embodied in forms which are at once invitingly complex and elegant. Ferguson's creations work compellingly with our fundamental human instinct for the special kinds of mathematical order that are embedded in the world we see around us.

Text from ‘Mellifluous Mathematics’ by Martin Kemp