Figureight Knot Complement vii/CMI (Clay Mathematics Award)
Helaman Ferguson

“We are living in a golden age of science and a golden age of art, and I like to celebrate that.” -Helaman Ferguson

Back in March, I attended a talk by mathematician/sculptor Helaman Ferguson. He’s one of the so-called algorists, artists who create art based on algorithms of their own devising (Ferguson is a co-creator of the PSLQ algorithm, among others). The Clay Mathematics Institute describes the algorithm used to create the sculpture at the top of this post, Figureight Knot Complement vii, thus:

The mathematical object which the sculpture represents is the orbifold X given as a quotient of three-dimensional hyperbolic space by a discrete group action, as described by the following equations, permanently inscribed on the larger granite sculpture:


To which I say both “what?” and “ouch, my brain!” (If I ever want more pain like that, I’ll try to understand the mathematical innovations for which the award’s recipients have been honored). However, appreciating the sculpture’s beauty – both visual and tactile – requires no math at all. At his talk, Ferguson passed around a similar bronze, lustrous and jewel-like, that fit the hand like an innovative ergonomic grip. The pleasure of these sculptures lies in exploring their surfaces with both eye and hand, while trying to simultaneously wrap your brain around what they represent conceptually.

Figureight Knot Complement Vi
Helaman Ferguson

When he spoke, Ferguson had just returned from Macalester College, where he installed his piece Invisible Handshake, a carved piece of diorite weighing in excess of 6,000 lbs.

Invisible Handshake, 2008
Helaman Ferguson
Macalester College, MN
Photo by Stan Wagon

Invisible Handshake is topologically equivalent to two hands not quite touching. Ferguson likes to call this piece a product of his “negative-Gaussian-curvature phase.” It’s all concave, saddle-shaped surfaces:

A surface has negative curvature at a point if the surface curves away from the tangent plane in two different directions. The classic example is a saddle, which can be found on your body in the space between your thumb and forefinger, or along the inside of your neck. Any point on the inside of a torus has negative curvature because there are planar cuts that yield curves that bend in opposite directions with respect to the tangent plane at the point. Negative curvature — the saddle shape — arises spontaneously as nature tries to minimize energy. (source)

According to Ferguson, the creation of this sculpture was “between me, the rock, and my diamond chainsaw,” but it’s a little more complicated than that. He first models the sculpture mathematically using imaging software, then slowly carves away the extraneous stone with diamond chainsaw, drills and sandpaper. He even has a CRADA (confidential research agreement) with NIST for the metrology tools he uses to accurately convert a computer model into a carving.

Figure-8 Esker on Double Torus
alabaster, 6″
Helaman Ferguson

Ferguson generally prefers to work in durable materials like stone and bronze – he noted with some glee that Macalester College’s diorite sculpture would easily outlast Macalester College. But during his interactive talk, he had his audience work in paper. He instructed us to twist paper towels together, producing a roomful of miniature tripods similar to his piece Borromeans with Feet I:


To approximate the sculpture, you twist a paper towel into a rod, then bend it in half at the center to make a loop, and twist the two trailing ends together in the opposite direction from the original twisting motion. The tension between the two twists holds the shape, which is like a needle with a large eye or loop at one end. Then you repeat this process twice more, passing the second loop over the first, and running the third loop through the others. (Confused? Don’t worry about it – even a room full of science/art lovers had some trouble with the process.)

The Borromean rings thus created are groupings of three circles (or, in this case, paper towel loops) entangled in such a way that removing one releases the other two to separate. Ferguson said that these represented triads of entangled photons. I have to take his word on that – he didn’t explain further and I know nothing about physics! To me, though, the twists were reminiscent of DNA – and therefore reminded me of case-parent triads (parent/child groupings which provide useful data for identifying genetic markers associated with a disease or syndrome). This is generally true of Ferguson’s sculptures: to different people they evoke very different concepts, from DNA to desert dunes to the curve of a neckline. And to mathematicians, they evoke, well, really cool math.

Figure-Eight Knot Complement II
Helaman Ferguson
(from Mathematics by Experiment, Borwein and Bailey)

Eightfold Way
marble and serpentine
Helaman Ferguson
Berkeley, CA
via Graham’s flickrstream

Hyperbolic Five
Helaman Ferguson


“Helaman Ferguson: Mathematics in Stone and Bronze” by Claire Ferguson

Science profile (subscription only)


  1. #1 Anon
    May 6, 2008

    Two things:

    One, that is very very very cool!

    Two, “Diamond Chainsaws” would be a great name for a band.

  2. #2 Gray Gaffer
    May 9, 2008

    Did you not think Bathsheba’s work to be sufficiently on-topic for this post? 3D, from mathematical descriptions, but available at prices ordinary folks can afford.

  3. #3 Gray Gaffer
    May 9, 2008

    Oh – seems that my earlier comment just got lost, sorry, I thought you had moderated it out.

    I urge you to visit She does some most excellent work, including several molecules (DNA, Hemoglobin, etc) laser etched inside optically perfect glass blocks, and will do them to order if you give her the right file format. Stock ones are about $80. Also has astronomical subjects, from the nearby stars to the large scale structure. And a stunning Calabi-Yau Manifold.

  4. #4 John Ohab
    May 14, 2008

    One of them looks like an ampersand.

  5. #5 Jessica Palmer
    May 14, 2008

    Gray Gaffer – those scultures by Bathsheba are charming. I’ll see if I can remember to write about them closer to the holidays when I do my science nerd gift list.

  6. #6 Mike from Ottawa
    May 17, 2008

    I stumbled across an ad for Bathsheba’s sculptures in the back of a SciAm a couple of years ago and now have more than a dozen of them (mostly the small ones that are around $100 each). I’m a hopeless addict. I cart them around with me all the time.

    Not very ‘bio’, but a marvelous wedding of art, CAD and rapid prototyping to produce reasonably priced pieces. I especially like that she does not do limited editions.

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