## The mathematical monthly calendar : Otto Toeplitz, the dedicated teacher

When Otto Toeplitz was born in Wroclaw in 1881, his future career was almost predictable: both his grandfather Julius and his father Emil were active as mathematics teachers at grammar schools in Wroclaw and Lissa (Posen district), respectively; both had also published articles on mathematics education. In addition, Emil Toeplitz was known throughout the German Empire as the editor of the annual Philologenjahrbuch (Kunze’s Calendar), a directory of all teachers working at grammar schools and similar institutions that still exists today.

After passing the Abitur examination, Otto Toeplitz began to study mathematics at the University of Breslau. In 1905 he wrote his doctoral thesis on a topic in algebraic geometry ("On the transformation of multitudes of bilinear forms of infinitely many variables"). He then moved to Gottingen, where he habilitated in 1907 and became a private lecturer.

Inspired by David Hilbert, he worked intensively on the theory of integral equations, on which he wrote several papers, later also an encyclopedia contribution for the "Encyclopedia of Mathematical Sciences".

1911 Toeplitz published a paper on systems of equations whose coefficient matrix is symmetric to the secondary diagonal. So for finite systems, only at most 2*n* – 1 instead of *n* 2 different coefficients; the solution procedures simplify considerably. Matrices of this type are meanwhile called Toeplitz matrices.

### Beginning of career as professor

### The mathematical monthly calendar

Their scientific achievements are widely known, but who were the mathematical geniuses who had a lasting impact on our understanding of the world? For his students, Heinz Klaus Strick, former head of the Landrat-Lucas-Gymnasium in Leverkusen-Opladen, wrote the "mathematical monthly calendar" and supplemented it with appropriate stamps of the persons presented. All the exciting resumes, quirky portraits, and incredible stories behind the notable personalities can now also be found here.

In 1926, at the annual conference of the Society of German Natural Scientists and Physicians in Dusseldorf, Toeplitz gave a well-received lecture on the teaching of calculus, in which he advocated having students trace the historical development of calculus (the so-called genetic method): "Mathematics and mathematical thinking are not only part of a special science, but are also closely connected with our general culture and its historical development of mathematical thinking, a bridge can be found between the so-called arts and sciences and the seemingly so unhistorical exact sciences . Our main goal is to build such a bridge. Not for the sake of history, but for the sake of the genesis of problems, facts and evidence, for the sake of the decisive turning points of this genesis. "

In order to implement this request, Toeplitz planned a two-volume work, but was not able to put it into practice. In 1949, the materials compiled by Gottfried Koethe appear posthumously as a book entitled "The Development of the Infinitesimal Calculus – a Genetic Approach", consisting of the three chapters "The Essence of the Infinite Process", "The Definite Integral", "Differential and Integral Calculus". Due to the lack of previous knowledge of the first-year students, Toeplitz recommended that the concept of limit values should only be defined exactly at a later point in time, and that integral calculus should be treated before differential calculus – in accordance with the historical development (Archimedes, Cavalieri, Fermat, Saint Vincent, …). The book ends with remarks on Kepler’s laws.

His great interest in historical contexts led to the founding of the journal "Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik" in 1926 (together with Otto Neugebauer and Julius Stenzel).

In 1928 Toeplitz accepted a call to Bonn, where he had better working opportunities than in Kiel and where a larger number of students were also enrolled. At the University of Bonn he became friends with Felix Hausdorff.

Together with his assistant Gottfried Kothe he developed his own theory of infinite-dimensional spaces, because the theory of the Polish mathematician Stefan Banach seemed too abstract to him.

Toeplitz also had a lively exchange with the Munster professor Heinrich Behnke; from 1932 on, the Mathematisch-Physikalische Semesterberichte (Mathematical-Physical Semester Reports) were published, which – to this day – are directed in particular at mathematics teachers.

### Escape from Nazi Germany

After the Law for the Restoration of the Professional Civil Service of 1933 came into force, the professor of Jewish origin could initially continue his teaching activities, as an exemption was granted to those persons who had already been active as university teachers before 1914. This arrangement was abolished in 1935 by the Nuremberg Laws, and Toeplitz was retired against his will.

Toeplitz, a Jew who until then had hardly practiced, took over as head of the Jewish community in Bonn. He founded a school for Jewish children, where he also took over teaching himself.

As head of the university department in the Reichsvertretung der Juden in Deutschland, he arranged scholarships for particularly gifted Jewish students and organized their departure for the USA.

When the number of suicides in his environment increased dramatically and he himself also felt unable to cope with the pressure from the National Socialists, he emigrated to the British Mandate territory of Palestine in February 1939. There he immediately participated in the construction of the Jewish university on Mount Scopus in Jerusalem; but a year after his arrival he fell seriously ill and died.

### Popular teacher

Toeplitz was described by his students – as well as by his colleagues – as a friendly and helpful person, who always took time for the others. In the course of his work as a university lecturer he supervised a total of twelve doctoral students, including three women.

In 1930, a collection of popular topics from his lectures was published, the book "Von Zahlen und Figuren – Proben mathematischen Denkens fur Liebhaber der Mathematik" (Of Numbers and Figures – Samples of Mathematical Thinking for Lovers of Mathematics), which is still worth reading today (in cooperation with Hans Rademacher, professor in Breslau; Rademacher, a pacifist, had to emigrate already in 1934).

In 22 sections the two authors try to "break through the partition" by which the non-mathematicians are separated from the world of mathematicians.

At the beginning, they spread out the ingenious idea of the Euclidean proof, which explains why there are infinitely many prime numbers. Next, they explain aspects of how to develop an optimal line layout in a tram-rail network. In the third section it is proved that among all the *n*-vertices inscribed in a circle, which has the largest area on a regular basis.

The proof of the irrationality of√2 is followed by two illustrative proofs that among all triangles which can be inscribed in a triangle, the triangle of the altitude footpoints has the smallest circumference.

In chapter eight Toeplitz goes into Cantor’s considerations of the power of sets and addresses the continuum hypothesis. The following chapter deals with the cuts on the straight circular cone, followed by a section on Waring’s problem for *n* = 2, 3, 4: "Every natural number can be written as a sum of at most *g(n)* Powers with exponent *n* where *g*(2) = 4, *g*(3) = 9, *g*(4) = 19.

In section ten Toeplitz deals with double points of closed, self-interpenetrating curves, in chapter eleven it is shown that the decomposition of natural numbers into prime factors is unique at infinity.

In chapter twelve he introduces the four-color problem and Euler’s polyhedron theorem.

Chapter 13 is devoted to the statement of Fermat’s conjecture (still unproven in 1930); first, it is explained how, in the case of *n* = 2 all Pythagorean number triples *x* 2 + *y* 2 = *z* 2 finds.

The next section deals with the question, how large the radius of a circle must be chosen at most, in which all points of a given cluster of points lie.

The 15. Section deals with the approximation of irrational numbers by rational numbers.

In chapter 16, the authors investigate straight line guidance by joint mechanisms; in the next section, they explain what Euclid and Euler found out about the perfect numbers. Then it is described why for a given circumference the circle is the figure of largest area (proof idea according to Jacob Steiner).

Chapter 19 deals with periodic decimal fractions, chapter 20 with curves of constant width. The penultimate chapter is devoted to the question of constructability with compass and ruler, and in which constructions one can dispense with the compass or with the ruler.

Finally, Toeplitz deals once again with prime numbers and their growth. It turns out that 30 is the largest number for which it is true that all numbers below it, which are not divisors of it, are prime numbers.

Otto Toeplitz made a conjecture in 1911, which has been proven for many types of curves, but not yet in general: "In every closed Jordan curve C (i.e. a continuous, non-intersecting plane curve) one can inscribe a square".

Examples: If C is a triangle, then depending on whether it is an acute triangle, a right triangle, or an obtuse triangle, you can draw one, two, or three squares, respectively, with their vertices on the perimeter line.

If C is a circle or a square, then you can draw in an infinite number of squares, if C is a regular hexagon, then three squares, if C is an oval, then one square.