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Interview with László Fejes Tóth
The following is an English translation to a conversation between István Hargittai and László Fejes Tóth. The Hungarian text appears in Hungarian Science (3) (2005), 318. A slightly shorter version of the same conversation is given in G Polinszky and M Valiskó (eds.), Fejes Tóth 100. Száz éve született Dr Fejes Tóth László (Veszprémi Vegyészekért AlapítványVeszprém, 2016).
We are capable of creating an infinite number of new universes, whose laws we have control over, but into which we cannot gain entry.
H S M Coxeter reports this quote by László Fejes Tóth and calls him "the geometer of the twentieth century."
László Tóth Fejes (born 1915, Szeged) is a mathematician, member of the Hungarian Academy of Sciences (since 1962), former director of the Mathematical Research Institute (1970-1983), Kossuth Prize winner (1957) and State Prize winner (1973).
The following conversation between István Hargittai and László Fejes Tóth was recorded at Fejes Tóth's home in Budapest on 22 October 1999.
István Hargittai: What is it about your career and work that you most fondly remember?
László Fejes Tóth: 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 have often had the experience that my professor, Lipót Fejér, emphasised that one must be amazed at certain things. I was amazed when I first understood the Taylor series of the sine function, and I have often been amazed since then, for example, by how sparse the dense forest of rational points on the number line really is.
When I was a first-year student, I had a serious result that I would like to share. 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 at a distance from the centre? Fourier solved this problem using a series that is similar to the ordinary Fourier series and has the form , where the numbers are not integers but the roots of some transcendental equation. 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. This was my first serious mathematical result.
How did you come up with this problem?
I found this problem myself through my reading. At the same time, I was already working on geometry, primarily the approximation of curves by polygons, more precisely the question of what order of magnitude a curve can be approximated by -sided polygons. The area that I later worked on throughout my life, i.e. packing and covering issues, was brought to my attention by a very nice colleague of mine, Dezső Lázár. He asked how to place points, say in a square or on a circular surface, in such a way that the minimum distance between them is maximum. This depends so much on the shape of the region in which we want to place these points that we can only talk about approximate solutions here. Only in very special cases can we find an exact solution. However, we can examine what happens if the number of points in question is very large? This asymptotic version of the problem is equivalent to how to place circles of the same radius as densely as possible on the plane. I solved this problem, but I didn't know that much earlier, Axel Thue, the great Norwegian number theorist, had solved it at the turn of the century. This inspired me to investigate this area in the future.
Let me tell you about Dezső Lázár, that when I came to Cluj-Napoca, he was also working there, he was a teacher at the Jewish High School. Later, he was called up for labour service, they picked up mines with him, he got a thigh wound, and they left the poor thing to bleed to death. When he was on labour service, we were in close contact with his family. We visited his wife and two small children many times. His wife told me what happened to him. I don't remember exactly when this was, because the years are merging in my memory. His wife was a very educated, beautiful woman, and even today I am horrified to think that this woman was dragged away in a cattle car and, after much suffering, ended up in a gas chamber in Auschwitz with her two small children.
So much for how I came to study mathematics. Mathematics has filled my whole life and has given me endless enjoyment and happiness.
How far did you take the problem of placing circles?
I have come the furthest in generalising the problem of the densest circle packing. Others also consider this to be one of my essential results. I have shown that the density of any arrangement of centrally symmetric, congruent, convex plates cannot exceed the density of the densest lattice packing. Let us now consider that there are two ways to arrive at regular solids. For example, I can make a requirement that all faces of a polyhedron be regular, and that the shapes of the edges that meet at the vertices be regular. These are two requirements, and I can examine which solids satisfy these requirements. But I can also arrive at regular solids in another way. For example, in the following way. I want to place 12 points on the surface of a sphere so that the minimum distance between them is maximum. Then I automatically get a regular icosahedron.
If you have 12 points.
Yes, if I have 12 points. And if I have six points, I get a regular octahedron.
And in the case of four points, I get a regular tetrahedron. All of this plays a big role in chemistry as well.
In stereochemistry the location of electron pairs in the valence shell of the central atom of a molecule can be determined using the same approach. The solution is not always as clear as, for example, in the case of four or six points. In the case of five points, when asked which shape gives the maximum of the minimum distances, the trigonal bipyramid is only slightly better than the tetragonal pyramid. A similar approach can also be used to determine the interaction energies. A Canadian professor developed a very popular model from this, and we wrote a textbook on its application together.
Tibor Tarnai, who is an architect, wrote an article with a chemist inspired by me.
Consider the following problem. Let there be a circles of radius on a sphere. How should these be arranged so that they cover the largest possible part of the sphere's surface? The solution depends on the radius of the circles. If the radius is such that exactly circles can fit on the given sphere, then I get the densest arrangement of circles. However, if the radius of the circles is a little larger, then the circles overlap, but the question remains how to arrange the circles so that they cover the largest possible part of the sphere. Finally, when the circles are large enough that circles can completely cover the sphere, then I get the sparsest circle coverage. In the case of intermediate values of the circle radii, the arrangement of the centres of the circles changes in a very interesting way. Tibor Tarnai and his co-author studied this problem for perhaps ten circles. An exact solution is not expected, but they made good guesses on a computer. I know that this area has chemical applications. Fullerenes are also related to this. I have read about this.
We started from the assumption that the planar arrangement of circles cannot be denser than the densest lattice arrangement.
This is true not only for circles, but also for any centrally symmetric convex disks.
There is still debate about the densest spatial arrangement.
A few years ago, the American mathematician Thomas Hales solved the problem, using a computer, but exactly. The proof is extremely difficult, there are thousands of cases to be examined that have to be solved separately. I am not familiar with Hales' work, but the first strategy that could be used to approach this problem at all was given by me, in my book Lagerungen, and there are several references to it. A Chinese mathematician living in America, Wu-Yi Hsiang, recently tried in a similar way, but he was left with several details to work out, and Hales is considered the first to solve the problem of the densest packing of spheres.
But no one has checked it yet. Checking it would be a huge job. They just accept it intuitively and wait for someone to find a simpler proof.
Computers can help a lot in finding simpler solutions.
You mentioned that you became interested in math in high school. Where did you go?
To the Széchenyi István Reálgimnázium, here in Budapest near Népliget. I was born in Szeged and was five years old when my parents moved to Pest. My father was a railwayman, a cashier at the Keleti Railway Station. He was probably fifty when he obtained his doctorate in law. My mother was a teacher at a girls' high school, specialising in Hungarian-German. I can't say anything about my future career, but I like to talk about my children. I have two sons and a daughter. My daughter is a psychologist. My older son is a mathematician, and he has achieved very good results in my footsteps, he is a doctor of mathematical sciences. My younger son is a professor of physiology at Dartmouth University in New Hampshire. This is a very well-known university in the USA. He works well, he has many publications related to the kidney.
How did the name Fejes Tóth come about?
At school I was still László Tóth. There were four Tóths in one class. The many Tóths were terrible, and we wanted some distinction. We had a nickname, which was common among peasants. My paternal grandfather was Fejes Tóth, but Fejes was only a nickname. My paternal grandmother's family had the same nickname, who were also Tóths, but they were Tücskös Tóth. At that time, there was a ministerial decree that did not allow half-Hungarian, half-German names, such as Iványi-Grünwald. Based on this, a stupid official did not allow us to change our name to Fejes Tóth either. That's how I graduated from university as László Fejes. After the war, our name officially became Fejes Tóth. So I had two name changes in total, first from Tóth to Fejes, and then from Fejes to Fejes Tóth.
You have already mentioned Lipót Fejér among your professors. Did you have any other famous professors?
There was no one else besides him. Let me tell you about Lipót Fejér, that he kept a small list of his doctoral students. He didn't have many students, maybe ten or fifteen, but he was very proud that all of them became professors at foreign universities. I also became a doctor under Fejér, which was an honour for me. However, I must say that the result of my dissertation, which I have already mentioned about Cauchy series, was not inspired by him. However, Fejér was later also interested in my work. This special one of the Cauchy series, which I later dealt with in more depth, Fejér called the Fourier series of cooling.
I understand that you were in the military for two years, between 1939 and 1941. How did it happen that you didn't have to serve in the military after that?
I was called up several more 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. Later such papers didn't count anymore, but on that basis, when my unit was taken to Germany, I didn't go with them, but hid here at home. This was at the end of 1944, and that's how I survived the war. I was in this house with my mother, this house had been hit nine times at that time. You could see the sky from the basement.
You entered the University of Cluj in 1941. Was that your first job?
Yes, and I stayed there until 1944. I came back when we had already heard Stalin's statement on the radio that the annexed territories would have to be returned to Romania. I didn't want to live in Romania, everything tied me to Hungary. When I came back, I went to teach at a high school. I was happy to get a job. I taught at the Árpád High School.
Then, in 1946, you were hired as a private tutor at the University of Budapest. Had you already completed your habilitation by then?
Yes, but I was still employed at Árpád High School. From there I moved to Veszprém. I taught at the University of Chemical Industry of Veszprém for fifteen years. I continued my research and eventually returned to Budapest, to the Mathematical Research Institute.
Is Hungarian mathematics still as good today as it was in the past?
Maybe even better. In the past, Fejér and Riesz were outstanding in world terms. There were also many great mathematicians, Gábor Szegő, György Pólya, Ottó Szász and many others. They went abroad because there were not enough places in Hungarian universities.
Your CV includes many foreign universities where you taught, Freiburg, Wisconsin, Ohio, Salzburg. Have you ever wanted to stay abroad?
There was one time when I would have liked to have stayed abroad. I received a very favourable, permanent job offer from the University of Zurich. I would have liked to have accepted this, but in the end I stayed home. I only wanted to go legally. The Canton of Zurich also initiated my departure diplomatically, but the Hungarian government at the time did not agree. I have thought about it a lot since then, and my wife in particular cannot forgive me for not allowing it. The standard of living we would have enjoyed there cannot be compared to what we have here. In the end, I don't really regret it, because we managed to create a mathematical environment and a school here in Hungary that perhaps wouldn't have been possible in Zurich.
Did you have a relationship with Pál Erdős?
We have a joint paper, but we didn't really have a close relationship.
Were you involved with him as a director?
We had an institute seminar, at which Erdős also gave lectures many times.
What was the main reason why Erdős lived a nomadic life?
His personality was such that he loved to travel and share problems here and there.
Now they say that circumstances forced him to this lifestyle.
This was not the case, Erdős did not pursue this lifestyle out of necessity, and I have not encountered such an opinion. He would have been welcomed into permanent employment anywhere in the world.
Also at home?
Of course, here at home too. He was happy to come to Hungary, for a week or two, a month or two, but then he went to another part of the world, where he would also have been welcomed permanently.
Has he ever been offered a job back home, in Hungary?
He had a permanent paid position at the Mathematical Research Institute, and he always received his salary when he was at home. I don't remember the administrative details. When he came from America once in the 50s to a world mathematics congress, perhaps in Amsterdam, they told him that if he left, they wouldn't let him back. And that's what happened. After the congress, he didn't get an American visa for many years. It was really a constraint, but by then he was so famous that he would have been welcomed at any university in the world.
You worked for a long time, and Coxeter is still very active even at a very old age. And yet many people say that mathematicians are most productive when they are young.
One of the great English mathematicians of this century, Hardy, wrote in his book A Mathematician's Apology that mathematics is a young man's game. According to him, no great mathematical discovery has been made after the age of sixty. I think that is really true. Coxeter is indeed very active, but what he is doing these days is no longer a breakthrough in mathematics. I feel the same way. I have recently published a paper. I think this is the swan song.
This means that you were still creating when you were over eighty years old.
Over eighty, but of course that doesn't mean a breakthrough either. By the way, Coxeter and I have a very good joint paper, from a little earlier, when I spent half a semester with him in Toronto.
Could you tell us something about your interest in the history of science?
I enjoy reading writings on the history of science, but I have not bothered to write on such a topic myself. In my book Lagerungen, every chapter contains a historical overview. This book was published in German, Russian and Japanese, Regular Figures in English and German. Neither in Hungarian.
I would like to tell you something about myself that others have found characteristic. Some people are very good problem solvers. You may have heard of Szemerédi. He is a great person and can solve very difficult problems beautifully, but I understand that he does not raise problems himself. In this, in raising problems, I am considered very good. I will tell you something to characterise this. I have a hunch, I call it the loop hunch.
Consider the following problem. In -dimensional space, let us place unit spheres in such a way that the volume of their convex hull is minimal. The question is, in some sense, to place the spheres in the smallest possible space. In the plane, we are close to solving this problem; we know the best arrangement for an infinite number of values of . In three-dimensional space, the problem is hopelessly difficult to solve, and the situation is similar in four-dimensional space. However, my guess is that from the fifth dimension onward, the spheres must be placed so that their centres are along a straight line, two units apart, so that they touch each other. The convex hull of the spheres is then in the shape of a loop, which is why I call it the loop conjecture. This is one of my problems that has had a huge impact. In a short time, more than twenty articles have appeared on it, containing partial results. I originally published the problem in Periodica Matematica, but it was not considered there for a long time. Then there was a Coxeter symposium, perhaps on the occasion of his seventieth birthday. There I met the German mathematician Jörg Wills, who got the idea for this problem, and then he also spread it. The complete solution is still not available, although we now know that the conjecture is true in spaces of dimensions higher than 42.
Could you try to explain what you based your guess on?
The volume of this loop can also be calculated for the -dimensional case. From the fifth dimension onwards, the volume of the loop is significantly smaller than when I do not place the spheres like this, but rather extend them in all directions, apparently making better use of the possibilities that five-dimensional space offers. This inspired the conjecture.
Can you give an example of a time when you raised an issue and it was resolved?
We are in the Euclidean plane and let us consider circles, not necessarily congruent circles, but any circles. Let us examine arrangements of these circles in which the circles do not touch each other, and in which each circle is touched by exactly six other circles. I also introduced the concept of the homogeneity of circles, which characterised how much the smallest and largest circles differ from each other. In the case of very different circles, this homogeneity is small, and in the case of congruent circles, the homogeneity is 1. My guess was that the condition for such a six-neighbourly arrangement of circles is that the homogeneity of the circles is either zero or 1. This means that either circles of very different sizes are included in such an arrangement, or only congruent circles. This guess of mine was confirmed by Imre Bárány, Zoltán Füredi and János Pach.
I mentioned these examples to show that this is what they consider me good at, that I can raise problems.
There will soon be another conference in Budapest that Coxeter will attend.
I've heard about it, but I don't go to conferences or academic class meetings anymore. I just spend my days at home, with my small family.
One more thing. I saw in your interview with Coxeter that his wife is also mentioned. I would also like to say that I met my wife at university, she studied chemistry, and after graduation she had a good job as a chemist. Again, referring to the interview with Coxeter, I want to say that our marriage is also successful, a very good marriage. She gave birth to three beautiful, healthy, talented, and, what I think is even more, 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. That's what I wanted to say.
Last Updated March 2026