At this year’s Institutional Scientific Students’ Associations Conference at the Budapest University of Technology and Economics, five presentations were related to Morphodynamics: **Máté Szondi** (1st Prize + Pro Progressio Special Prize), **Gergő Almádi** (1st Prize), **Gergő Almádi and Eszter Ferencz** (3rd Prize), **Kinga Kocsis** (Special award of the Department of Geometry and Morphology + Special award for presentation), **Emese Sarolta Encz and Gergely Barta** (Special award of the Department of Geometry and Morphology).

*1st Prize + Pro Progressio Special Prize*

Applying a shape evolution equation to compute membrane surfaces

**Máté Szondi**

supervisor: András Sipos

In Mathematics, the exact description, study, and construction of smooth surfaces, which satisfy given constraints, are of central importance in differential geometry. For the first time, in 1768, J. L. Lagrange gave the differential equation that the shape of soap membranes with fixed boundaries must fulfill while working on the classical methods in the calculus of variations. The first solutions of this equation were discovered a couple of years later. G. Monge proved that membranes attached to fixed boundaries with a minimal surface area, called minimal surfaces, are characterized by zero mean curvature at all surface points. As their generalization, we consider surfaces where the Gauss and the mean curvatures satisfy a given linear relation at each surface point. J. Weingarten was the first to study these surfaces in the middle of the 19th century.

This work demonstrates that, on the one hand, the stationary solutions of the unidirectional Bloore model, which are associated with the outcomes of some natural shape evolution processes (e.g., the surface of the rocks in a river channel), are Weingarten surfaces and hence some surfaces in nature have close-to-Weingarten properties. On the other hand, as minimal surfaces are related to the membrane solutions of self-stressed structures, the Weingarten surfaces can also be associated with membrane surfaces under specific distributions of the loads. We find that some natural shapes can be used as membrane shells in structural design.

Based on the analogy mentioned above, I introduce a new form-finding method, which determines the minimal surface, or the Weingarten surface in general, via the simulation of the Bloore flow with Dirichlet boundary conditions. I point out that for a rectangular domain, the Dirichlet boundaries might be incompatible with the final solution, which provides a mathematical explanation of frustrated boundaries in the mechanics of thin shells. The simulation of the complete non-linear Bloore operator is not always convergent because of the strong non-linearity. Still, if it is convergent, then we get the solution of the non-linear elliptical Monge-Ampère partial differential equation.

*1st Prize*

On Equilibria of Tetrahedra

**Gergő Almádi**

supervisors: Krisztina Regős, Gábor Domokos

The complete content of the following TDK paper is the exact same as of an article, which has been published in The Mathematical Intelligencer, and is the result of the collaboration of Gergő Almádi, Robert J. MacG. Dawson, Gábor Domokos and Krisztina Regős. The article is available via the following link: https://doi.org/10.1007/s00283-023-10294-2

A solid body, resting on a plane under gravitational force, must have at least one stable equilibrium point on its boundary (minimizing the distance from the center of mass) and one unstable equilibrium point (maximizing the same distance.) A body with a unique stable (resp. unstable) equilibrium is monostable (resp. mono-unstable.) If it has at least one of these properties it is monostatic; if both, mono-monostatic.

Conway and Guy showed that a homogeneous polyhedron can be monostable, but that a homogeneous tetrahedron has at least two stable equilibrium points. The same idea has been used to prove that a homogeneous tetrahedron has at least two unstable equilibria. Conway also claimed that an inhomogeneous tetrahedron may be monostable. Here we give a formal proof of this statement, and show that all monostatic tetrahedra have exactly 4 equilibria. We also show that certain patterns of obtuse dihedral (resp. face) angles are equivalent to the existence of a monostable (resp. mono-unstable) weighting.

Our results imply that mono-monostatic tetrahedra do not exist. In contrast, we show that for any other legal number of faces, edges, and vertices there is a mono-monostatic polyhedron with that face vector.

*3rd Prize*

The time evolution of crack networks, or, the Gilbert-Piaffe

**Gergő Almádi, Eszter Ferencz**

supervisors: Gábor Domokos, Ferenc Kun

While in everyday life we often encounter the phenomena of cracking,it is very difficult either to control these processes or predict their potential occurrence. Since a fracturing process is an irreversible phenomenon and the formation of any new crack depends on the previously formed crack network,it is essential to study the development of fracture patterns as temporal processes to gain a deeper understanding of the phenomenon.Despite how fundamental this question may seem,the literature has only recently begun to address its description.One of the most significant experiments conducted on this area was documented by Nakahara and his co-authors in 2018.

The first step towards the mathematical description of fracturing processes is the (static) geometric description of crack patterns.The geometric model presented in the publications [2, 3] and illustrated with examples in the publication [4] summarizes the previous results in this field, putting them into a unified framework using the theory of convex mosaics. Within this model, a crack network can be identified with a symbolic point of the so called symbolic plane, spanned by the network`s two characteristic combinatorial averages n ̅*,v ̅*. Building on this model, we may ask the following question: if a crack network evolves over time due to various physical processes, how can the motion of its symbolic point be described based on knowledge of these physical processes?This question is raised in article [5] and answered in article [6].The latter presents a general system of ordinary differential equations that can be applied to a wide range of cracking processes,after defining a so-called fundamental table which describes the physics of the specific process.

In our work we set up a model-hypothesis that the mosaic observed in experiment is combinatorically equivalent to a convex mosaic, specifically the Gilbert mosaic.The Gilbert mosaic is a mathematical model in which cracks start from randomly chosen points on the plane and subsequently propagate in two opposite directions at uniform speed along a line defined by a randomly chosen angle until they reach another crack, where they stop at a so-called irregular (“T”) node of degree n*=2.

We identified the fundamental table describing the evolution of the Gilbert mosaic and used it to define the model-specific version of the general equation presented in the article [4].The model-specific system has one global attractor (n ̅*,v ̅*)=(2,4).The evolution of the Gilbert mosaic inherently carries one of the most interesting phenomena observed in the experiment, namely that it only generates irregular nodes with the degree n*=2. Since n*=2 is true only for a single point in the domain of convex mosaics on the symbolic plane, we can conclude that the time evolution of the Gilbert mosaic on the symbolic plane is a stationary process („piaffe”),as the the mosaic’s position on the plane will differ from the fixed point only because of the finite size of the mosaic.

Since the theory in [4] deals with the time evolution of infinite mosaics, we developed a numerical model that operates on finite mosaics for the purpose of precisely matching the experiment. The model has two important free parameters: the time delay between the initiation of each crack and the interval allowed for the random angle determining the line.By choosing these two parameters appropriately, we have achieved that the time evolution of the Gilbert mosaic reproduces not only the average cell number v ̅* measured in the experimental mosaic, but also the evolution of the total cell degree distribution with surprising accuracy.

To our knowledge,this is the first time that a dynamic geometric model based on a convex mosaics has been used to describe the evolution of a physical crack network.

*Special award of the Department of Geometry and Morphology + Special award for presentation*

Thousand tonnes of marble, or space-time journey to Michelangelo’s marble blocks

**Kinga Kocsis**

supervisor: Gábor Domokos

After Michelangelo Buonarroti – painter, sculptor, architect – designed the façade of the church of San Lorenzo in Florence at the request of Pope Leo X, the Pope ordered him to obtain the marble needed for the construction from the quarries around Carrara. The extraction and transport of the marble presented the great artist with serious engineering challenges (including road construction), and he spent two years on the task. In his honour, a quarry in the area is now known as Cava Michelangelo. Raw blocks of marble played an important role in Michelangelo’s life. As he put it, “The marble not yet carved can hold the form of every thought the greatest artist has.”. My purpose was to get as close as possible in space and in time to the blocks of marble that Michelangelo transported from Carrara to Florence. I travelled the spatial part of this journey by car, and during a week-long field survey at the Cava Michelangelo quarry I documented in detail the planar crack pattern of a 7.33×6.14m2 vertical marble wall and the adjacent 8.36*6.14m2 horizontal plateau. To facilitate time travel, I resorted to the tools of convex geometry. By using the respective combinatorial averages of nodal and cells degrees of the two measured crack networks, I identified the two corresponding points of images on the symbolic plane. Further on, I used the two planar crack networks to partially reconstruct the spatial crack newtork of the marble block bounded by the wall and the plateau. This block, needless to say, no longer exists but its weight exceeds a thousand tonnes. Based on my partial model I was still able to determine the geometry of one particular block and I could also derive estimates on the average size of the marble blocks.

*Special award of the Department of Geometry and Morphology*

The phenomenon of corner cracks in planar patterns

**Emese Sarolta Encz, Gergely Barta**

supervisor: Gábor Domokos

The geometry of crack patterns contains valuable information about the physical phenomenon leading to the formation of the pattern: using the respective average degrees n ̅^*,v ̅^* of the nodes and cells we can unambiguously match the mosaic to a point on a symbolic plane and make interesting comparisons between natural mosaics using this geometric model. Recent research has also shown that crack patterns also contain information that can be used to model the process in time.

The dynamical evolution model set up in the [5] is based ont he assumption that an infinite random sequence of spatially localized, temporally discrete elementary steps generate the evolution of the infinite mosaic. The model of [5] allows, in principle, an arbitrary number of such steps of arbitrary geometry, but only two such steps are discussed in detail: during step R0 a new crack connects one interior point of two different edges of a cell, creating two new “T” nodes, while during step R1 the “T” nodes in the network are transformed into “Y” nodes. Even the publication [3] does not show any mosaic whose formation cannot be explained by the two steps described above.

However, as basic as the R0 and R1 steps seem to be, there are still crack networks for that we may need other elementary steps, if we intend to describe them with the model of [5]. In our work, we investigated fatigue cracking patterns in asphalt pavements. This task is physically interesting [6], but, in addition, in our paper we point out a geometrical speciality: to describe asphalt mosaics, in addition to the two elementary steps indicated (R0, R1), at least two additional elementary steps are needed: in step R2 a cell is cracked in two cells by connecting the interior point of an edge with a vertex, and in step R3 the crack connects the two vertices of the cell. Although the relative frequency of the indicated cracks in the mosaics we studied is relatively low, we nevertheless found such a configuration in almost all patterns.

Eleven crack patterns were examined, their geographical locations recorded and we matched the mosaics with the adequate points of the symbolic plane. The latter diagram was compared with an analogous diagram from a study [7] based on the surface pattern of masonry. We also tried to understand the evolution and development of the crack network over time. To this end, assuming a correlation between the time of crack formation and crack length, we separated older (primary) and newer (secondary) cracks and also looked at how the new crack fits with the old pattern.