Péter Gábor Szabó

Global Optimization in Geometry — Circle Packing into the Square

The present review paper summarizes the research work done mostly by the authors on packing equal circles in the unit square in the last years.

Packing up to 200 Equal Circles in a Square

The Hungarian mathematician Farkas Bolyai (1775–1856) published in his principal work (‘Tentamen’, 1832–33 [Bol04]) a dense regular packing of equal circles in an equilateral triangle (see Fig. 1). He defined an infinite packing series and investigated the limit of vacuitas (in Latin, the gap in the triangle outside the circles). It is interesting that these packings are not always optimal in spite of the fact that they are based on hexagonal grid packings. Bolyai probably was the first author in the mathematical literature who studied the density of a series of packing circles in a bounded shape.

Packing Equal Circles in a Square II. — New Results for up to 100 Circles Using the TAMSASS-PECS Algorithm

In this work we propose a new stochastic optimization algorithm for solving the problem of optimal packing of n non-overlapping equal circles in a square. It will be shown that our procedure can find most of the optimal solutions for all the problems previously solved and reported in the literature. Results obtained by our algorithm for up to 100 circles are given in relevant numerical and graphical form. For n = 32, 37, 47, 62 and 72 the algorithm has obtained better solutions than those reported on in the literature on packing. In addition, forty new and unpublished packing results are reported on. The arrangements obtained were validated by interval arithmetic computations.

Packing Equal Circles in a Square I. — Problem Setting and Bounds for Optimal Solutions

In the paper, a short review of the problem of finding the densest packing of n equal circles in a square is made. There will be new lower bounds for this problem defined on the basis of regular arrangements. Also, there will new upper bounds be established based on the computation of the areas of circle and minimum gap between circles and between circles and sides of the square. The paper also contains all the known exact values of optimal packings and the corresponding minimal polynomials.

On the roots of the trinomial equation

Here we summarize the works of the Hungarian mathematician Jenő Egerváry (1891–1958) on the trinomial equations. We present some of his ideas and methods with examples. Some earlier results in the history of mathematics in Hungary about the trinomial equations are also discussed.

On the Role of External Constraints in a Spatially Extended Evolutionary Prisoner's Dilemma Game

We study the emergency of mutual cooperation in evolutionary prisoner’s dilemma games when the players are located on a square lattice. The players can choose one of the three strategies: cooperation (C), defection (D) or “tit for tat” (T), and their total payoffs come from games with the nearest neighbors. During the random sequential updates the players adopt one of their neighboring strategies if the chosen neighbor has higher payoff. We compare the effect of two types of external constraints added to the Darwinian evolutionary processes. In both cases the strategy of a randomly chosen player is replaced with probability P by another strategy. In the first case, the strategy is replaced by a randomly chosen one among the two others, while in the second case the new strategy is always C. Using generalized mean-field approximations and Monte Carlo simulations the strategy concentrations are evaluated in the stationary state for different strength of external constraints characterized by the probability P.