At this year’s Institutional Scientific Students’ Associations Conference at the Budapest University of Technology and Economics, 4 presentations were related to Morphodynamics: **Gergő Almádi** (1st Prize + Pro Progressio Special Prize), **Ágoston Szesztay** (1st Prize), **Klaudia Nagy** (Csonka Pál Special Prize), **Balázs Sárossi** (2nd Prize).

*1st Prize + Pro Progressio Special Prize*

The mechanical complexity of inhomogeneous polytopes – does a tetrahedron have a spirit?

**Gergő Almádi**

supervisors: Gábor Domokos, Krisztina Regős

Polytopes are geometric objects bounded by (d-k) dimensional linear manifolds ((k=0, 1, … (d-1)) in the d-dimensional Euclidean space. We call d = 2 dimensional polytopes polygons, d = 3 dimensional polytopes polyhedra. In this paper we discuss the two aforementioned cases.

Polytopes are called convex, if the line segment connecting any 2 arbitrary points of the polytope remains inside the polytope. Assuming inhomogenseous mass distribution, the center of mass of a polytope can be located anywhere inside the polytope.

In this paper, we introduce the term balancing class vector to describe the static behaviour of polygons. The balancing class vector of an N-sided polygon Pn is an N-dimensional, binary vector. The kth element of the balancing class vector is 1 if there exists Pn with exactly k stable equilibrium points, otherwise this element is 0. In case of triangles and quadrangles we prove that out of the 2^3 and 2^4 respective combinatorial possibilities only 2 and 5 cases can be realized, respectively, geometrically, and We list these vectors explicitly.

In case of polyhedra we introduce the term balancing class matrix, which, in case of a polyhedron Pf,v with F faces and V vertices is an FxV matrix containing binary elements. The (i,j) element of the balancing class matrix is 1 if there exists Pf,v with exactly S=i and U=j stable and unstable equilibrium points, respectively. Otherwise, the element (i,j) is zero.

Based on results computed in MATLAB we formulate some conjectures regarding the balancing class matrices of tetrahedra. Numerical analysis of 760000 tetrahedra located at the nodes of an orthogonal net in the 5 dimensional configurational space of all tetrahedra we identified less than 300 different balancing class matrices out of the 2^16=65536 combinatorial possibilities. Although we do not prove that no other matrices can exist, rather, we lay down some principles which exclude the existence of certain matrices. We especially emphasize the balancing class matrices of monostable and mono-unstable tetrahedra, and we draw a comparison between the balancing class matrices of homogeneous tetrahedra as well.

*1st Prize*

On the static equilibrium of iteratively truncated polyhedra

**Ágoston Szesztay**

supervisors: Zsolt Lángi, András Sipos

This paper investigates 3-dimensional polyhedra created by applying a so-called ‘chopping’ algorithm on an initial arbitrary convex polyhedron P_0, where by chopping we mean the removal of a vertex V by a random plane, which is constrained to intersect solely the edges starting at V.

This paper examines the equilibrium properties of these polyhedra. It is shown that

• if the initial polyhedron P_0 is a minimal polyhedron, i.e. every face, edge and vertex of P_0 contains an equilibrium point, then, using suitably chosen chopping planes, the constructed polyhedral remain minimal polyhedral,

• For any initial nondegenerate polyhedron P_0, using suitably chosen chopping planes, the numbers of the stable, unstable and saddle points of the constructed polyhedra do not change during the process.

During the first process the mechanical complexity C(P), i.e, the difference between the sum of the faces, edges, and vertices and the sum of the stable, unstable, and saddle type equilibrium points of P do not change. Meantime during the second process by the removal of each vertex the value of C(P) increases by at least 6.

*Csonka Pál Special Prize*

Deviation of contact forces as the measure of stability in historical and geometrical walls

**Klaudia Nagy**

supervisors: János Török, Sára Lévay

Walls are essential elements of spatial design. The stability of walls strongly depends on the arrangement of the building blocks. The builders have always tried to create the strongest possible structures: it shows in the bond structure of the bricks or stones.

In my former TDK study “Geometry of walls”. I exclusively studied the patterns showing on the faces of a wide range of historical walls with the help of the mean field theory of convex mosaics. The main result of the former study is that bonds in walls can be measured with vertex density (ρ), which can be a measure of stability in case of arbitrary wall structures and the definition of bonds.

However, this method cannot distinguish between cases where the vertex density remains the same, but the geometry varies and we know that it strongly influences stability. For example, all brick walls laid in ½ bond and ¼ bond have the vertex density of ρ=2,00 but their stability differs.

As an improvement on the mosaic theory, in my second TDK study “Study of the stability of walls based on the force-indeterminacy”I also studied the deviation in the force-network with contact dynamics simulations in brick walls. I have shown that the fluctuation in the force network is related to the stability of the wall therefore we found it as a good measure for wall structure stability.

The goal of this paper is to study further the measure of the stability, as well as to extend the method to study the stability of arbitrary and historical walls. So far I exclusively studied walls on horizontal planes, but now I study the force-network also in tilted walls. In contrast to my former examinations, not only the vertical but also the horizontal (friction) forces will influence the stability.

*2nd Prize*

Analytic and numerical study of the evolution of pebble size distribution

**Balázs Sárossi**

supervisor: János Török

Pebbles are usually a product of a fragmentation process in mountains. They have a wide initial size distribution and very rectangular shape. In coastal or fluvial environments pebbles abrade each other due to the waves or flow of water, resulting in the pebbles having a smooth, rounded shape, and in some environments they become uniform in size. On the other hand there are processes where pebbles keep their initial distribution, or even spread out more. There are also processes where one can find outliers, boulders ten times the size of the mean size of the pebbles. The whole process of abrasion is extremely complex, it is hard to describe the effects acting on the size and shape evolution, therefore we simplify the process into a single mathematical formulation of a kernel function.

In this thesis I will focus on the evolution of the pebble size distribution, which I will investigate using a binary collision model. These collisions are driven by the statistics of collisions, and by the environment that collides the pebbles. This may be different for different environments, therefore I explore a zoo of kernels motivated by physical or mathematical arguments.

For certain kernels one can describe the size evolution analytically, for others only the conditions of emerging outliers. I also used numerical simulations to investigate the cause of different effects.