
Looking at these extraordinarily strange and beautiful objects—mathematical models of complex surfaces—it’s not surprising to learn that they have had a profound influence on modern art. But the exact nature of that influence is still barely understood. My intention here is simply to sketch out a few of the details, pending further research.
The first thing to note is the date. The frontispiece above comes from a catalogue published in 1881 by Brill of Darmstadt. That puts them right at the beginning of the craze for mathematical models in university teaching. A brief potted history would take us back first to the 18th century and the models issued to illustrate Books XI and XII of Euclid’s Elements. Then, in the 19th-century, to the use of three-dimensional models in various fields, especially botany and anatomy—a way of standardizing methods in disparate departments.
The most important and widely documented development in late 19th-century mathematical modelling was James Clerk Maxwell’s 1874 thermodynamic surface. Maxwell, a master of communicating concepts visually, sculpted a 3-D graph to show a fictitious subtstance’s volume, entropy, and energy:

The left-hand chunk = solid; the bulbous right-hand segment = gas. This is half-way between a teaching model and an instrument of research; after constructing the model Maxwell plotted on the lines of equal pressure and temperature by positioning the model so that sunlight would glance off it through the day.
Now, I don’t know the link between Maxwell’s teaching methods and the general success of mathematical models, but I do know that the two things were linked explicitly in the 1929 Encyclopaedia Britannica, which contains an authoritative account of the various kinds and uses of models. But by then models were long past their high-water mark, so to speak, with manufacture petering out after 1910.
The neat coincidence here is that the early success of models coincides with the period when the sculptor Naum Gabo was at university in Münich. Going back to a previous post, I talked about Alfred Barr’s famous 1936 diagram:

The diagram’s ‘conclusion’ is what I’m interested in here. Barr divides the modern art-world into: “non-geometrical” or “geometrical”. In England in the late 1930s this meant either surrealism or constructivism.
The latter was a large and deeply influential movement. Artists who went on to become famous for free-flowing, organically inspired or even nostalgic forms—John Piper, Barbara Hepworth, Henry Moore—for a time espoused a harshly geometrical abstraction:



Piper, Hepworth, Moore. Hepworth’s could be sold to schools for lessons in solid geometry, but the forms are basic, and the influence hard to read. In fact, it seems to me that the more bizarre shapes issued by Brill affected artists only in a general way, perhaps via architecture:

(The interlocking ramp system in Berthold Lubetkin’s famous Penguin Pool at London Zoo, built in 1934.)
A much more direct and documented relationship holds between on specific aspect of mathematical modelling and sculpture in the period, and it has to do with string. Look again at that Moore, a sphere with interlaced surfaces represented by strings. And compare (click to enlarge):

This spectacular array was published issued by the firm Teubner in 1912, as the ‘H. Wiener and P. Treutlein collection’. It’s not hard to see why it has such an influence. Hepworth, Moore, Naum Gabo and even Man Ray all made documented trips to see the models wherever they could. Here is Moore:
Undoubtedly the source of my stringed figures was the Science Museum. Whilst a student at the Royal College of Art, I became involved in machine art, which in those days had its place in modern art. [...] I was fascinated by the mathematical models I saw there, which had been made to illustrate the difference of the form that is half-way between a square and a circle. One model had a square stone end with twenty holes along each side making eighty holes in all. Through these holes strings were threaded and led to a circle with the same number of holes at the other end. A plane interposed through the middle shows the form that is halfway between a square and a circle. One end could also be twisted to produce forms that would be terribly difficult to draw on a flat surface. It wasn’t the scientific study of these models but the ability to look through the strings as with a bird cage and to see one form within another which excited me.
I’m yet to identify precisely the model Moore is talking about, though I hope to soon, and also to plot the way he used the form. A more direct and well-known link comes from Gabo, who probably introduced these models to English artists around the time he moved to this country in 1936. Here’s a diagram from the 1929 Encyclopaedia entry I mentioned above:

Gabo traced this into his notebook sometime around 1936, and a year later made this:

And can we go back a stage further and ask: why did Gabo take such an interest in mathematical models? Well, in May 1936 the model in the Encyclopaedia Britannica, which was and hopefully still is held at the Institut Henri Poincaré, was exhibited as an artwork at a surrealist exhibition in Paris. (Man Ray, no less, produced a photo series of the models that was published in Cahiers d’art.) So Gabo, a keen student of the international avant-garde and a former engineering student in Münich, found a personal coherence in the use of these instruments and tied it to a progressive and scientistic world-conception. The result of all this was the great manifesto Circle of 1937, but that’s for another blog post.
Returning to Barr’s diagram yet again, does this prove it to be all too neat? Perhaps not. After all, the surrealists exhibited an actual model, attempting to show that even a mathematician could produce art. The constructivists, led by Gabo, were quite clear that only a mathematician could furnish the materials for art. Here, uniting the two strands(!) at last, are a few more objets trouvés:

I.e. more from Teubner (englarges), and below from the Encyclopaedia Brtannica:

