Breaking a granite slab produces jagged fragments. The fractal geometry of the granite's fractured edges bears witness to the raw stone's natural history and contrasts sharply with the classical geometry of its manufactured facade.
In working with stone, a sculptor inevitably confronts breakage. When mathematician and sculptor Nat Friedman first looked at the shards of a broken granite slab, he saw something more than an unfortunate accident.
Friedman had become aware of how twentieth-century sculptors such as Henry Moore (see "Composition in Form and Space") had opened up the solid form to create space so they could revel in the interplay of form and light created by the opening.
Friedman envisioned how he could create space by leaving gaps between the fragments when he reassembled a broken slab. He then developed a technique for making prints from the granite assemblage.
A fractal stone print by Nat Friedman.
The resulting images, made with deep-blue or black ink on sheets of thin, porous paper, leave a vivid impression of jagged lightning bolts slashing across an otherwise dark sky. Some resemble drawings of crumpled coastlines edging unknown landmasses on an explorer's map. The stark contrasts in form and color tease the eye.
"Visual thinking leads to seeing that mathematical forms can also generate art forms," Friedman maintained.
Trefoil Knot Minimal Surface by Nat Friedman. Carved from limestone, this form is based on the shape of the soap-film minimal surface on a wire bent into a trefoil knot.
An artist can look at a mathematical shape and envision unlimited possibilities, even from a shape as seemingly simple as a tetrahedron, a trefoil knot, a Möbius strip, or a fractal surface. An artist can transform a mathematical idea into an evocative artwork.
Spiral Möbius by Nat Friedman.
To sculptor Arthur Silverman, for example, tetrahedrons were very special. He spent more than three decades investigating variations of tetrahedral forms, creating sculptures displayed in public spaces in New Orleans and other cities from Florida to California.
One of the more intriguing of Silverman's tetrahedral creations was an ensemble of sculptures he called Attitudes. The six pieces were spread across a grassy area at the Elysian Fields Sculpture Park in New Orleans.
Attitudes by Arthur Silverman.
All the pieces had the same geometry. Each one was made up of two identical tetrahedrons, having faces in the form of tall isosceles triangles that were welded together to form a single object. In the park, each piece had a strikingly different orientation.
To Nat Friedman, Silverman's creation was an example of a hypersculpture. Its ensemble arrangement represented a way of seeing a three-dimensional form from many different viewpoints at once.
To see every part of a two-dimensional painting, you have to step away from it in the third dimension. To see a three-dimensional sculpture in its totality, you need a way to slip into the fourth dimension. Friedman called this hypothetical process "hyperseeing."
A hypersculpture consisting of a set of several related sculptures provides one way to approximate that experience. Silverman's Attitudes, for instance, presents multiple views of an object from a single viewpoint, because copies of the same object lie in different orientations.
Another set of related sculptures is Rashomon by Charles Ginnever. The basic piece is an angular steel framework that can stand stably in fifteen different orientations. In each position, it looks startlingly different and tells a unique story.
A metal model demonstrates one of the stable positions available for Charles Ginnever's sculpture Rashomon.
By having several copies of the same sculpture in the same setting, each positioned differently, the artist can tell a remarkably complex, multidimensional tale.
Hyperseeing is easiest when a sculpture is highly symmetrical. For example, if the front and back views of a sculpture are identical, you can readily reconstruct from one view what the entire sculpture looks like. Simplest of all, a featureless sphere can be understood with just one glance.
Another strategy to encourage hyperseeing is to make the form at least partly transparent or to create a ribbed structure, as seen in many sculptures by Ginnever and Charles Perry.
Such approaches are somewhat reminiscent of the X-ray and time-lapse presentations of such artists as Pablo Picasso and Marcel Duchamp, who used multiple, fractured images of the same object in their paintings to convey a sense of three-dimensional space and time in a two-dimensional medium.
The result, Friedman said, can be bewildering to our conventionally conditioned eyes.
Experience with hypersculptures, like those created by Silverman and Ginnever, increases appreciation of sculpture in general, Friedman contended. "You learn to look harder and more closely from all angles."
When you hypersee a three-dimensional object, you begin to visualize the form from all sides, including the top—a view of a sculpture that is often neglected or unavailable.
Interestingly, the mathematical field of knot theory provides an ideal source of three-dimensional shapes that have no preferred front, back, top, or bottom. As open forms, mathematical knots are wonderful subjects on which to practice hyperseeing, Friedman said.
In effect, you can see all the points of a knot from any one view except for a finite number of points where the strand crosses itself.
In mathematics, a knot is a curve that winds through itself in three-dimensional space and catches its own tail to form a loop. Typically, mathematicians examine two-dimensional shadows cast by knots rather than actual three-dimensional knots.
Even the most tangled configuration can be shown as a continuous loop whose shadow sprawls across a flat surface. In drawings of knots, tiny breaks in the lines are often used to signify underpasses or overpasses.
Nat Friedman often used copper tubing to create three-dimensional models or sculptures of knots. The continuous loop of such a three-dimensional mathematical knot casts an intriguing shadow, which varies in interesting ways as the orientation of the knot is changed.
However, just as a suspended wire frame caught in a breeze on a sunny day, casts an ever-changing shadow on the ground, so a rigid knot illuminated from different angles can display different projections for a given knot. Usually, that most useful case is the shadow (or diagram) that has the smallest possible number of crossings.
Truly appreciating a knot, however, requires having a three-dimensional model of it. "You make a knot sculpture," Friedman insisted. He used strips of aluminum foil and embedded wires, plastic aquarium tubing, and copper pipe for his artful experiments.
"A knot can look completely different when viewed from different directions," Friedman noted. At the same time, the infinitely malleable shape of a given knot allows you to create innumerable space-form variants of the same knot, which also can serve as raw material for artistic creativity.
A physical model of a mathematical knot can have any one of an infinite variety of possible forms. The artist can select a configuration that has symmetries or other features that may make it particularly pleasing to the eye.
A knotted loop also can serve as the edge of a surface. An example of such a surface appears when a wire model of a knot is dipped in a soap solution and emerges with a soap film clinging to the wire.
Mathematically the result is known as a minimal surface—the surface of least possible area that spans the looped wire. If the wire is in the form of a circle, the resulting minimal surface is a flat disk.
Experiments show that the minimal surfaces associated with different knotted loops can take on a variety of shapes. Some of these surfaces feature the peculiar one-sidedness characteristic of Möbius strips.
Filling in a knot produces interesting surfaces that share properties of mathematical shapes such as Möbius strips, as shown in these models created by Nat Friedman.
"The operative word that unifies art and mathematics is seeing," Friedman wrote in his paper "Hyperseeing, Hypersculptures, and Space Curves." "More precisely, art and mathematics are both about seeing relationships. One can see certain mathematical forms as art forms, and creativity is about seeing from a new viewpoint."
See "Nat Friedman (1938-2020)."
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