Henry Segerman: Material Harmony in Mathematics

2024 Author: Seth Attwood | [email protected]. Last modified: 2023-12-16 15:55

According to legend, Pythagoras was the first to discover that two equally stretched strings emit a pleasant sound if their lengths are related as small whole numbers. Since then, people have been fascinated by the mysterious connection between beauty and mathematics, a completely material harmony of forms, vibrations, symmetry - and a perfect abstraction of numbers and relationships.

This connection is ephemeral, but tangible; it is not for nothing that artists have been using the laws of geometry for many years and are inspired by mathematical laws. Henry Segerman found it difficult to abandon this source of ideas: after all, he is a mathematician by vocation and by profession.

Klein bottle “By mentally gluing the edges of two Mobius strips,” says Henry Segerman, “you can get a Klein bottle, which also has one surface. Here we see a Klein bottle made from Mobius strips with a round edge.

Rather, how it might look in three-dimensional space. Since the original “round” Mobius strips go to infinity, then such a Klein bottle will continue to infinity twice and cross itself, which can be seen in the sculpture. An enlarged copy of this sculpture adorns the Department of Mathematics and Statistics at the University of Melbourne.

Fractals

“I was born into a family of scientists, and I think that my interest in anything that requires advanced spatial thinking is related to this,” says Henry. Today he is already a graduate of the Oxford graduate and doctoral studies at Stanford Universities, and holds the position of Associate Professor at the University of Oklahoma.

But a successful scientific career is only one side of his multifaceted personality: more than 12 years ago, the mathematician began organizing art events … in the virtual world of Second Life.

This three-dimensional simulator with elements of a social network was then very popular, allowing users not only to communicate with each other, but also to equip their virtual "avatars" and areas for entertainment, work, etc.

Name: Henry Segerman

Born in 1979

Education: Stanford University

City: Stillwater, USA

Motto: "Take just one idea, but show it as clearly as possible."

Segerman came here, armed with formulas and numbers, and arranged his virtual world in a mathematical way, filling it with unprecedented fractal figures, spirals and even tesseracts, four-dimensional hypercubes. “The result is a projection of a four-dimensional hypercube in the three-dimensional universe of Second Life - which itself is a projection of a three-dimensional virtual world onto a two-dimensional, flat screen,” notes the artist.

Hilbert's curve: a continuous line fills the space of a cube, never interrupting or intersecting with itself.

Hilbert curves are fractal structures, and if you zoom in, you can see that parts of this curve follow the shape of the whole. “I have seen them thousands of times in illustrations and computer models, but when I first took such a 3D sculpture in my hands, I immediately noticed that it was also springy,” says Segerman. "The physical embodiment of mathematical concepts is always surprising with something."

However, he liked working with material sculptures much more. “There are huge amounts of information circulating around us all the time,” says Segerman. - Fortunately, the real world has a very large bandwidth, which is not yet available on the Web.

Give a person a finished thing, an integral form - and he will immediately perceive it in all its complexity, without waiting for loading. So, since 2009, Segerman has created a little over a hundred sculptures, and each of them is a visual and, as far as possible, exact physical embodiment of abstract mathematical concepts and laws.

Polyhedra

The evolution of Segerman's artistic experiments with 3D printing is strangely repeating the evolution of mathematical ideas. Among his first experiments were the classical Platonic solids, a set of five symmetrical figures, folded in regular triangles, pentagons and squares. They were followed by semi-regular polyhedra - 13 Archimedean solids, whose faces are formed by unequal regular polygons.

Stanford Rabbit 3D model created in 1994. Made up of nearly 70,000 triangles, it serves as a simple and popular test of the performance of software algorithms. For example, on a rabbit, you can test the efficiency of data compression or surface smoothing for computer graphics.

Therefore, for specialists, this form is the same as the phrase "Eat some more of these soft French rolls" for those who like to play with computer fonts. The Stanford Bunny sculpture is the same model, the surface of which is paved with the letters of the word bunny.

Already these simple forms, having migrated from two-dimensional illustrations and the ideal world of imagination to three-dimensional reality, evoke inner admiration for their laconic and perfect beauty. “The relationship between mathematical beauty and the beauty of visual or sound works of art seems very fragile to me.

After all, many people are acutely aware of one form of this beauty, completely not understanding the other. Mathematical ideas can be translated into visible or vocal forms, but not all, and not nearly as easily as it might seem,”adds Segerman.

Soon, more and more complex forms followed the classical figures, up to those that Archimedes or Pythagoras could hardly have thought of - regular polyhedra that fill Lobachevsky's hyperbolic space without an interval.

Such figures with incredible names like "tetrahedral honeycomb of order 6" or "hexagonal mosaic honeycomb" cannot be imagined without having a visual picture at hand. Or - one of the sculptures by Segerman, which represent them in our usual three-dimensional Euclidean space.

Platonic solids: a tetrahedron, octahedron and icosahedron folded in regular triangles, as well as a cube and an icosahedron consisting of squares based on pentagons.

Plato himself associated them with four elements: "smooth" octahedral particles, in his opinion, folded air, "fluid" icosahedrons - water, "dense" cubes - earth, and sharp and "thorny" tretrahedrons - fire. The fifth element, the dodecahedron, was considered by the philosopher to be a particle of the world of ideas.

The artist's work begins with a 3D model, which he builds in the professional Rhinoceros package. By and large, this is how it ends: the production of sculptures itself, printing the model on a 3D printer, Henry simply orders through Shapeways, a large online community of 3D printing enthusiasts, and receives a finished object made of plastic or steel-bronze-based metal matrix composites. “It's very easy,” he says. “You just upload a model to the site, click the Add to Cart button, place an order, and in a couple of weeks it will be delivered to you by mail.”

Figure Eight Complement Imagine tying a knot inside a solid and then removing it; the remaining cavity is called the complement of the node. This model shows the addition of one of the simplest knots, the figure eight.

beauty

Ultimately, the evolution of Segerman's mathematical sculptures takes us into the complex and mesmerizing field of topology. This branch of mathematics studies the properties and deformations of flat surfaces and spaces of different dimensions, and their broader characteristics are important for it than for classical geometry.

Here, a cube can be easily turned into a ball, like plasticine, and a cup with a handle can be rolled into a donut without breaking anything important in them - a well-known example embodied in Segerman's elegant Topological Joke.

The tesseract is a four-dimensional cube: just as a square can be obtained by displacing a segment perpendicular to it at a distance equal to its length, a cube can be obtained by similarly copying a square in three dimensions, and by moving a cube in the fourth, we will "draw" a tesseract, or hypercube. It will have 16 vertices and 24 faces, the projections of which into our three-dimensional space look little like a regular three-dimensional cube.

“In mathematics, the aesthetic sense is very important, mathematicians love“beautiful”theorems, - the artist argues. - It is difficult to determine what exactly this beauty consists in, as, indeed, in other cases. But I would say that the beauty of the theorem is in its simplicity, which allows you to understand something, to see some simple connections that previously seemed incredibly complex.

At the heart of mathematical beauty can be pure, effective minimalism - and a surprised exclamation of "Aha!" ". The deep beauty of mathematics can be as daunting as the icy eternity of the Snow Queen's palace. However, all this cold harmony invariably reflects the inner orderliness and regularity of the Universe in which we live. Mathematics is just a language that unmistakably fits this elegant and complex world.

Paradoxically, it contains physical correspondences and applications for almost any statement in the language of mathematical formulas and relations. Even the most abstract and "artificial" constructions will sooner or later find an application in the real world.

A topological joke: from a certain point of view, the surfaces of a circle and a donut are "the same", or, more precisely, they are homeomorphic, since they are able to transform into one another without breaks and glues, due to gradual deformation.

Euclidean geometry became a reflection of the classical stationary world, differential calculus came in handy for Newtonian physics. The incredible Riemannian metric, as it turned out, is necessary to describe Einstein's unstable universe, and multidimensional hyperbolic spaces have found application in string theory.

In this strange correspondence of abstract calculations and numbers to the foundations of our reality, perhaps, lies the secret of the beauty that we necessarily feel behind all the cold calculations of mathematicians.