Already in the last years of my secondary school studies, in the seventh or eighth grade at the time, I was more or less familiar with calculus. Not with the rigour that is taught at university, but I knew the gist of it, and that fascinated me. ... I was amazed when I first understood the Taylor series of the sine function ...

Fourier used a method to describe the cooling of a sphere. Imagine an iron sphere whose temperature depends only on the radius from the centre, and we know this relationship. What happens if we immerse this sphere in zero-degree water? What will the temperature be after time t at a distance r from the centre? Fourier solved this problem using a series that is similar to the ordinary Fourier series ... Fourier's solution met the requirements of the time, but it was not exact in the sense of modern mathematics, since it presupposed the convergence of the series. Cauchy solved the problem using an even more general series, the exponential series named after him, and I dealt with his solution. Cauchy's proof was also incomplete, because he did not deal with the issue of convergence either. Picard, the leading French mathematician of the time, proved the convergence of Cauchy's exponential series, but only in the case when the generating function has a bounded change. I managed to generalise this to a large extent by proving that this series is convergent or divergent depending on whether the Fourier series of the function is convergent or divergent.

It was not so much concrete mathematics, but the ability to become fascinated by the beauty of new discoveries is what he learned from Fejér.

Let K be a convex closed plane curve, and P a closed polygon lying in the same plane. Let A be the set of points that lie in one, and only one, of the two regions bounded by K and P. The author defines the area measure of A as the deviation between P and K and poses the question: What distinguishes the n-gon that has the smallest deviation from K? Analogous questions are also addressed in three-dimensional space. Here, instead of n-gons, polyhedra with a given area are considered.

Cauchy investigated certain generalisations of the Fourier series using the residue method, and Picard proved a convergence theorem for such expansions. Here, under more general assumptions, the equiconvergence of the Cauchy and Fourier expansions of a function f(x) is proved.

I met my wife [Izabella Újhelyi] at university, she studied chemistry, and after graduation she had a good job as a chemist. ... I want to say that our marriage was successful, a very good marriage. She gave birth to three beautiful, healthy, talented, and, what I think is even more important, honest children. She gave up her own career, and she did this consciously, because she thought that as a mother and as a wife she could do much more for the family than if she stayed at work and earned something. She helped my work in everything, created and gave everything so that I could live completely for mathematics. We always lived modestly, we never had great demands.

I was called up several times, and when I was on the verge of collapse and had to go back in, a kind doctor gave me a paper saying "currently unfit". I had calcified nodules in my lungs that didn't cause any problems, but on that basis he could issue me this paper.

The simplest question was how to place as many one-forint coins as possible on a "large" table? The statement that, in the best arrangement, all one-forint coins, apart from the coins placed on the edge of the table, are touched by six others, was first proven by Axel Thue, a Norwegian number theorist, in 1892. The proof is by no means self-evident. In space, we can ask, at most, how many spheres of diameter one unit can fit in a huge container? Although some aspects of the problem were also addressed by Minkowski and Hilbert, for half a century it was mainly addressed by crystallographers and physicists. Their interest stemmed from the belief that atoms or molecules of a homogeneous material placed under high pressure would automatically arrange themselves into the densest crystal structure. The complete description of possible regular crystal structures was essentially completed by the end of the 19th century. Thus, in principle - through extremely lengthy calculations - the densest regular "lattice-like" arrangement for molecules (or other objects) of arbitrary shape could be determined with great accuracy. But what if the densest arrangement is not regular? This question did not even arise for most researchers at that time. László Fejes Tóth suspected that it was a deep mathematical problem, closely related to important areas of classical analysis, approximation theory and algebra. The study of non-lattice structures and quasicrystals has now become an independent discipline. It is impossible to overestimate the personal role of László Fejes Tóth in this process. Even before the end of the war, he devoted dozens of fundamental papers to such questions.

We consider a sufficiently large cube of ordinary Euclidean space and place within it, in a certain arrangement, a number of equally sized spheres, approximately of radius 1. As a measure of the density of this ball arrangement, we define the total volume of the enclosed spheres divided by the volume of the given cube. We now further consider a sequence of cubes $W_{1}, W_{2},..., W_{i},...$, whose volumes increase without limit, and simultaneously with each cube $W_{i}$ the maximum value $D_{i}$ of the densities of all possible ball arrangements. The sequence $D_{1}, D_{2},..., D_{i}, ....$ then approaches a limit: $D_{i} \rightarrow D$. In this essay, we will give an estimate of this limiting density $D$.

You could see the sky from the basement.

... played table tennis with the best students during his breaks. In the summer, he took examinations on the top of an upturned boat on the beach in Balatonalmád, and in the autumn, he climbed the thirty-metre chimney of the university boiler house for a bet.

The book starts with a discussion of some elementary, and some not quite so elementary, results about convex bodies and polygons which are needed later. Then the different problems are discussed in the plane, on the surface of a sphere, and, after a section on polyhedra, in space. As the book proceeds, the author finds questions to which he has not yet succeeded in finding the answer (much of the work described in the book is due to the author); he does not hesitate to supply a conjecture. At the ends of the various sections the author gives collections of remarks containing references to many original papers concerned with the problems he discusses and with many interesting generalisations and variations.

The book is written in a very clear style; the sentences are short, making the task of the dictionary thumbing reader easy. The mathematics is based almost entirely on elementary geometry and should be understood by anyone with a knowledge of school mathematics, provided that they have a mature outlook on the subject. The problems considered are quite fascinating and are much nearer those of the ordinary world than is common in pure mathematics. The field is one in which there is scope for the amateur mathematician to make important contributions.

The results, books and problems raised by László Fejes Tóth have launched countless researchers in their careers all over the world and continue to have an impact to this day. His idea was the term "intuitive geometry", with which he addressed fundamental geometric questions that were understandable to the man in the street.

This book seems to have everything that could be desired in a mathematical monograph - a pleasant style, careful explanation of all technical terms used, a great variety of topics with a single unifying idea, a good bibliography and index, and many beautiful illustrations, including four plates and a pocketful of stereoscopic anaglyphs. Although the work is mostly concerned with geometry, it has connections with art, crystallography, biology, city planning, and the standardisation of industrial products.

We have always marvelled at the mathematical astuteness of the lowly honeybee. But has he been overrated? Is he really as efficient in the use of wax and space as history would have us believe? Is the sum total of wax used a minimum in this odd-shaped cell, whose top is hexagonal and whose bottom is three equal rhombi? The author proceeds to show us where the bees do well and where they do not.

The rhombic dodecahedron gives the best ratio of volume to surface while also acting as space fillers, but there is a better one; namely, the truncated octahedron. Now if the bees would stop at an optimal height as they build from bottom to top they then would have a better system; that is, the bisected rhombic dodecahedron. Better yet, take the regular octahedron and elongate a diagonal to form a 120-degree dihedral angle, then proceed to truncate some corners by planes perpendicular to diagonals and we have a new symmetric figure that fills space and uses less wax than the bees' system. The bottom is two hexagons and two rhombi rather than the three rhombi used by the bees.

This is a fascinating article; and, as you have already concluded, I have only touched on one phase. The author extends this to other illustrations with drawings and mathematical relations. Your students will enjoy this, and they may go out and teach the bees a more efficient system.

Today, the sculpture park is enriched with a bronze bust of one of the university's legal predecessors, the former renowned professor of the University of Chemical Industry of Veszprém, the internationally renowned father of discrete geometry, academician László Fejes Tóth. ... Respect for human greatness is as old as humanity: with the inauguration of the bust of László Fejes Tóth, the University of Pannonia welcomes its former renowned professor, who enriched its intellectual heritage, into the hall of immortals.