Academician Gábor Domokos, an architect and engineer with an unconventional approach, gave a lecture in Rényi titled Natural Tilings. The lecture will also be available in Hungarian at the Hungarian Academy of Sciences (MTA) on 3 November, as part of the Hungarian Science Festival.
In the continuous space and time that surround us, we navigate our world with the help of spatial and temporal patterns that fill space and time completely, without gaps or overlaps. How do the shapes and patterns that leave no empty spaces yet never overlap come into being? This is the question posed by Academician Gábor Domokos in his lecture entitled Natural Tilings. The lecture was delivered in English, and will soon also be available in Hungarian at the Hungarian Academy of Sciences on 3 November, as part of the Hungarian Science Festival.
Academician Gábor Domokos is a university professor at the Department of Mechanics, Materials and Structures, Morphology and Geometric Modelling Group, at the Faculty of Architecture of the Budapest University of Technology and Economics. His research focuses on geometry, mechanics, and the mathematical description of forms observed in nature. In 2006, together with Péter Várkonyi, he developed Gömböc, the first known homogeneous, convex body with just two equilibrium points: one stable and one unstable. Released from any position, it returns to its stable equilibrium. The discovery of Gömböc attracted international scientific attention and helped reveal new connections in the study of geometry and natural forms. The origins of Gömböc date back to 1995, when Domokos encountered a conjecture by Russian mathematician Vladimir Arnold, concerning the existence of a special body with an exceptionally small number of equilibrium positions. After more than a decade of research, Domokos and Várkonyi found the theoretically predicted body in 2006: Gömböc. Since 1998, Gábor Domokos has also been an honorary professor at Cornell University. He was elected a corresponding member of the Hungarian Academy of Sciences in 2004 and a full member in 2010. His work has been recognised with the Knight's Cross of the Order of Merit of the Republic of Hungary in 2007 and the Dénes Gabor Award in 2019. |
“I am an architectural engineer who took mathematics courses, but I have never worked in pure mathematics (fundamental research),” Gábor Domokos explains at the beginning of our conversation. He adds that he has co-authored numerous papers with mathematicians, including researchers at Rényi Institute. “I couldn’t do without mathematicians. I had been visiting Rényi Institute for years to work on joint projects. For example, as a student, I was asked by Domokos Szász and his colleagues to do programming, back in the days before modern computers.”
His present lecture focuses primarily on natural tilings of three-dimensional space. “We find beautiful examples in both the living and non-living worlds. One such example is the chambered shell of the Nautilus, in which the growth process creates new spaces that are free of gaps and corners.”
“The cracks in rocks, which often form regular geometric patterns, are equally fascinating, as are the intricate structures of human tissues, where cells and the spaces between them are arranged in relation to one another,” writes the BME professor in the abstract of his lecture.
The subject is so engaging and accessible that members of the Medve (Bear) Math Camp attended one of his lectures this summer and gave him a standing ovation. His lecture at Rényi Institute was aimed at students of Budapest Semesters in Mathematics.
“Space is all around us, but we can only imagine it if it is tiled with something. We cannot perceive space itself. Even a whitewashed wall acquires structure when we cover it with tiles. A crystal lattice, for example, gives structure to space. This provides us with a point of reference, because we humans live in space and time. These patterns act as crutches for our thinking, helping us make sense of space. We can imagine space most readily by picturing some kind of tiling within it.
A tiling consists of elements that fit together without overlapping or leaving gaps. If there are overlaps and/or gaps between them, we perceive this as a ‘disorder’. Tiling is already a fascinating subject in two dimensions: just think of the tiles lined up on a bathroom wall. Yet we understand the phenomenon far less well in three dimensions,” explains Gábor Domokos, pointing out that Plato and Aristotle were already contemplating and debating this subject thousands of years ago.
Their debate concerned whether the five Platonic solids: the cube, the tetrahedron, the octahedron, the dodecahedron and icosahedron could fill space. Plato believed they could, while Aristotle realised that not all Platonic solids were suitable for this purpose. Aristotle thought that some of them could not fit together without leaving gaps. The Platonic solids exemplify perfect geometric regularity, yet only one of the five, the cube, can tile space on its own. Aristotle believed that regular tetrahedra could also fill space. Interestingly, Domokos points out, this error by Aristotle contributed significantly, some 1,500 years later, to the development of the concept of solid angle. Without understanding this concept, Aristotle could hardly have solved the problem.
Another question is: now that we know regular tetrahedra cannot fill space completely, how large are the gaps left between them? One thing is certain: these debates helped stimulate human thought and advance our understanding.
Strict geometric rules govern the way space is filled. Fractured rocks, for example, are almost never cube-shaped, but when averaged out, the geometric theory of space-filling structures predicts that the average numbers of faces and vertices exactly match those of a cube. “We measured this on Hármashatár Hill (on the outskirts of Budapest), and, surprisingly, the averages agreed to several significant figures. This type of pattern occurs regularly in nature,” says Gábor Domokos. “We also see completely different types of space-filling structures. A room is usually polyhedral in shape, so we actually live in a polyhedral world. Architects design such forms either because these are the shapes we can imagine, or perhaps we can imagine these shapes precisely because architects design them.”
The lecture Natural Tilings presents some fundamental yet surprising ideas about the spatial arrangement of shapes and the rules governing the formation of structures observed in nature.
Nature has a particular affinity for so-called soft tilings, i. e. gap-free structures without corners that are possible only in three dimensions. It is no coincidence that they have existed for hundreds of millions of years. Their occurrence has also provided humans with valuable insights, later applied in the design of engineering structures.
“The cells that fill the interior of a muscle tissue look very different from the spaces inside a building. If there were corners in a tissue membrane, it could easily tear at those points. In two dimensions, however, it is impossible to eliminate corners entirely. Although architects often favour angular forms, engineers have learned from nature how to design corner-free structures that avoid stress concentrations. Submarine hulls and aircraft windows are good examples.”
“For centuries, neither people nor science studied corner-free tilings of space in any systematic way. The first geometric construction pointing towards this concept appeared in a work by Bernhard Riemann, published by one of his students after his death,” explains Gábor Domokos, recalling the historical background. “I have often wondered myself how Riemann came up with the idea. Even today, I cannot give a definitive answer,” he admits.
Yet corner-free tilings have also found their way into human art. According to Domokos, a famous and compelling example is the Laocoön Group, a marble sculpture depicting the Trojan priest Laocoön and his two sons caught in the deadly coils of serpents. The sculpture is housed in the Vatican Museums in Rome.
“Up close, we can see that the sculpture essentially consists of limbs and serpents, carved from a single block of marble. If we imagine adding further limbs and serpents, following the internal logic of the artwork, these imagined elements would fill the entire space, allowing us to reconstruct the original block of marble in its entirety. Through this work, the unknown ancient sculptor demonstrated that space can be filled not only with cubes but also with cylinder-like objects,” Gábor Domokos points out. “Michelangelo immediately recognised this. The influence of this idea can also be traced in his sculpture. He regarded this work as the pinnacle of ancient sculpture and, according to contemporary accounts, spoke about it extensively. Few people understood gap-free tilings of space as well as Michelangelo, because when carving a block of marble, you cannot afford to make a mistake and simply undo it. Riemann, too, may have seen the Laocoön Group, as his biography suggests. His intuition changed many of our longstanding ideas about space and how it can be filled.”
“I have devoted a great deal of thought to this subject, exploring both its geometric and historical background. I also ventured into geology and cosmology; these unusual and surprising examples had a profound impact on me,” Domokos continues. He explains that astronaut Tibor Kapu also experimented with a corner-free space-filling body, known as a soft cell, in space: he took a soft-cell framework aboard a spacecraft and filled it with water. “In the absence of gravity, water forms minimal surfaces. Riemann had the idea of creating minimal surfaces that fill space, with such a surface naturally defining soft cells.”
| Another major achievement of Gábor Domokos’s research is the discovery of a new, universal class of geometric forms known as soft cells, which he described in an international collaboration with Krisztina Regős, Ákos G. Horváth, and Alain Goriely. Soft cells are shapes that can fill space without gaps or sharp corners. Similar forms occur in many places in nature, including living tissues, marine snail shells, and zebra stripes. Their mathematical study may also help us understand how natural forms develop. Since then, the research has yielded further results. Using soft cells, the researchers have uncovered connections between the microstructures of living and non-living materials and the geometric models used to describe foams and certain polymers. |
“We accept that space can be divided up in ways that do not involve corners. We see no corners in the cosmos or on planets, and it is possible to construct geometric models that partition the entire cosmos without corners. Recognising these possibilities enriches our thinking and broadens our perspective,” adds Gábor Domokos.
His approach as an architectural engineer is decidedly unconventional, but his goals as a researcher and professor are very tangible. “I would like to inspire young people to explore the subject of three-dimensional tilings. Perhaps one of them will turn out to be a thinker of epoch-making talent, like Riemann. But you cannot nurture a Riemann out of nothing: it takes a culture and a community that, when a genius of this calibre is born into it once every few centuries, will at least recognise such an exceptional mind.”
“What I talked about in my lecture is not mathematical research conducted with artificial intelligence. Quite the opposite: I want to show that there is still mathematics beyond AI. Young people can and should be inspired. Even if artificial intelligence had existed in their time, I do not believe it would have made a meaningful contribution to the groundbreaking discoveries and radically new insights of Riemann or Michelangelo. AI cannot ‘substitute’ this unique, creative dimension of human thought, which, albeit on a more modest scale, works perfectly well in many people. We still know very little, and there are plenty of secrets of nature left to uncover, including the ones that can be described using the tools of geometry.”
“Science is not an industry whose goal is the continuous production of Q1 papers. That is precisely why it is so difficult to determine which research is worth funding. Although we do not always recognise it, nature offers us beautiful phenomena on a plate—phenomena that can be expressed in the language of mathematics and understood through mathematical description, and that may even inspire entirely new mathematical constructions. In this process, AI can provide only very limited assistance. Creative scientists and mathematicians will continue to be essential.”
The entire lecture can be watched HERE in English.
Cover: a portrait of prof.Gábor Domokos. Credit: BME