Ludwig Schläfli

Ludwig Schläfli

Ludwig Schläfli (born January 15, 1814 in Grasswil, today in Seeberg , Canton Bern , † March 20, 1895 in Bern) was a Swiss mathematician who dealt with geometry and function theory. He played a key role in the development of the concept of dimension , which among other things plays a crucial role in physics . Although his ideas are now covered in every undergraduate degree in mathematics, Schläfli is rather unknown even among mathematicians.

Life

Youth and education

Ludwig Schläfli spent most of his life in Switzerland . He was born in Grasswil, his mother's hometown. Shortly afterwards, his family moved to nearby Burgdorf , where his father worked as a businessman . Ludwig was supposed to follow in his father's footsteps, but he wasn't made for practical work.

Due to his mathematical talent, he was given the opportunity to attend high school in Bern in 1829 . At that time he was already learning differential calculus from Abraham Gotthelf Kästner's Mathematical Beginnings of Analysis of the Infinite (1761). In 1831 he went to the academy in Bern for further training. In 1834 the academy became the new University of Bern , where he began studying theology .

Teaching

After graduating in 1836, he was appointed teacher in Thun . He pursued this occupation until 1847, spending his free time studying mathematics and botany and visiting the university in Bern once a week to continue studying theology.

The year 1843 marked a turning point in Schläfli's life. Schläfli had planned a visit to Berlin to get to know the mathematical community there, especially Jakob Steiner , a well-known Swiss mathematician. But unexpectedly Steiner came to Bern and met Schläfli. Steiner was not only impressed by Schläfli's mathematical knowledge, but also by his excellent language skills in Italian and French .

Steiner suggested to Schläfli that he support his Berlin colleagues Carl Gustav Jacob Jacobi , Peter Gustav Lejeune Dirichlet , Karl Wilhelm Borchardt and Steiner himself as interpreters on the upcoming trip to Italy . Steiner extolled this idea to his friends in the following way (which is an indication that Schläfli was a little clumsy in everyday matters):

... while he praised his newly recruited travel companion to his friends in Berlin that he was a rural mathematician near Bern, a donkey for the world, but that he learned languages ​​like child's play, they wanted to take him with them as an interpreter. [ADB]

Schläfli accompanied her to Italy and benefited greatly from the trip. During his six-month stay in Italy, Schläfli even translated some of the works of the other mathematicians into Italian.

Next life

Schläfli stayed in contact with Steiner until 1856. The prospects that were opened to him encouraged him to apply for a position at the University of Bern in 1847. He was appointed in 1848 lecturers appointed in 1853 as Associate Professor and in 1872 a full professor. Schläfli's teaching activity lasted until his retirement in 1891. Until his death in 1895 he devoted himself to studying Sanskrit and translating the Hindu script Rigveda into German .

Higher dimensions

Schläfli is one of the three founders of multidimensional geometry together with Arthur Cayley and Bernhard Riemann . By 1850, the general concept of Euclidean spaces was not yet developed - but linear equations in variables were already well understood. In the 1840s, William Rowan Hamilton developed his quaternions and John Thomas Graves and Cayley developed the octaves . These two systems worked with a basis of four or eight elements and suggested an interpretation analogous to the Cartesian coordinates of three-dimensional space. ${\displaystyle n}$

From 1850 to 1852 Schläfli worked on his main work Theory of Multiple Continuity , in which he founded the study of the linear geometry of dimensional space. He also defined the -dimensional sphere and calculated its volume. He decided to publish his work and sent it to the Academy in Vienna, but it was rejected because of its size. A second attempt in Berlin ended with the same result. Finally, in 1854, Schläfli was asked to write a shorter version, but he did not. Steiner tried to help him get the work published in Crelle's Journal. But for some unknown reason this did not happen either. Parts of the work were published in English by Cayley in 1860. The first publication of the entire publication took place in 1901 after Schläfli's death. The first review of the book appeared in the Dutch mathematics journal Nieuw Archief voor de Wiskunde in 1904 and was written by the Dutch mathematician Pieter Hendrik Schoute . ${\displaystyle n}$ ${\displaystyle n}$

An excerpt from the introduction to the "Theory of Multiple Continuity":

“ Advertisement for a treatise on the theory of multiple continuity

The treatise which I have the honor of presenting to the Imperial Academy of Sciences contains an attempt to establish and work on a new branch of analysis which, as it were, an analytical geometry of dimensions, those of plane and space as special cases for contained in itself. I call the same theory of multiple continuity in general in the same sense as, for example, the geometry of space can be called a theory of threefold continuity. As in this one group of values ​​of the three coordinates determines a point, so in that one group of given values ​​of the variables should determine a solution. I use this expression because in one or more equations with many variables, every sufficient group of values ​​is also called that; The only unusual thing about the naming is that I keep it even when there is no equation between the variables. In this case I call the totality of all solutions the -fold totality; on the other hand , if equations are given, the total of their solutions is called -fold, -fold, -fold, ... continuum. From the idea of ​​the all-round continuity of the solutions contained in a totality develops that of the independence of their mutual position from the system of used variables, insofar as new variables can take their place through transformation. This independence is expressed in the immutability of what I call the distance between two given solutions ( ), ( ) and in the simplest case through ${\displaystyle n}$ ${\displaystyle n=2,3}$ ${\displaystyle n}$ ${\displaystyle x,y,\ldots }$ ${\displaystyle n}$ ${\displaystyle 1,2,3,\ldots }$ ${\displaystyle n-1}$ ${\displaystyle n-2}$ ${\displaystyle n-3}$ ${\displaystyle x,y,\ldots }$ ${\displaystyle x',y',\ldots }$

${\displaystyle {\sqrt {(x'-x)^{2}+(y'-y)^{2}+\ldots }}}$

define by simultaneously calling the system of variables an orthogonal one, [...] "

Schläfli first understood points in -dimensional space as solutions of linear equations, in order to then carry out the brilliant train of thought, to consider a system without equations in order to obtain all possible points of (as we would call it today). He spread this concept in the articles he published in the 1850s and 1860s, and it developed quickly. In 1867 he started an article with the words We consider the space of the tuples of points. [...] . This not only suggests that he had got the theory under control, but also that his audience no longer needed lengthy explanations. ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n}$

Polytopes

In the theory of multiple continuity , Schläfli defines so-called polyschemas , which are now called polytopes . They are the multidimensional analogues of polygons and polyhedra . He developed their theories and found, among other things, the multidimensional variant of Euler's polyhedron substitute . He also determined the regular polytopes, i.e. H. the -dimensional relatives of the regular polygons and the Platonic solids . It turned out that there are six of them in four-dimensional space and three in all higher-dimensional spaces. ${\displaystyle n}$

Although Schläfli was well known by his colleagues in the second half of the 19th century, especially for his contributions to complex analysis, his early geometric work did not receive much attention for a long time. At the beginning of the 20th century, Pieter Hendrik Schoute worked with Alicia Boole Stott on polytopes. She proved Schläfli's result on regular polytopes once again, but only for four-dimensional space, and then discovered Schläfli's book. Willem Abraham Wijthoff later studied semi-regular polytopes. His work was continued by HSM Coxeter , John Horton Conway, and others. There are still many unsolved problems in this area, which is based on the work of Ludwig Schläfli.

Trivia

• The Schläfli symbol is named after Ludwig Schläfli. ${\displaystyle \left\{p,q,r,\dots \right\}}$
• Ludwig Schläfli was to become a businessman like his father. But he did the worst possible business because he could not understand that an item was sold more expensive than it was bought.
• Schläfli took a state examination in theology and was (after a few complications with the trial sermon) to be found in the Bern directory of persons entitled to the pastoral office. But he probably never had one.
• In the Exact Sciences Library at the University of Bern, the words Three quarks for Muster Mark, Einstein and Schläfli are reminiscent of Schläfli's work in Bern.

literature

• Ludwig Schläfli: Theory of Multiple Continuity . Ed .: HJ Graf, on behalf of the Swiss memorandum commission. Natural Research Society. Zürcher und Furrer, Zurich 1901 ( cornell.edu [accessed December 22, 2018] First print of Schläfli's main work, around 50 years after it was written).
• [Sch] Ludwig Schläfli, Collected Treatises, 3 volumes, Birkhäuser 1950, 1953, 1956
• [DSB] Johann Jakob Burckhardt : Schläfli, Ludwig . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 12 : Ibn Rushd - Jean-Servais Stas . Charles Scribner's Sons, New York 1975, p. 170-173 .
• Johann Jakob Burckhardt Ludwig Schläfli , Elements of Mathematics, Supplement 4, 1948
• [ADB] Moritz Cantor :  Schläfli, Ludwig . In: Allgemeine Deutsche Biographie (ADB). Volume 54, Duncker & Humblot, Leipzig 1908, pp. 29-31.
• [Kas] Abraham Gotthelf Kästner , Mathematical Beginnings of the Analysis of Infinite , Göttingen, 1761
  • Comment: This is the third volume of Kästner's Mathematical Beginnings . It can be viewed online at Göttingen Digitization Center .
• Ruth Kellerhals The mathematician Ludwig Schläfli , DMV Mitteilungen 1996, No. 4, p. 35.
• Jürg Hüsler:  Schläfli, Ludwig. In: New German Biography (NDB). Volume 23, Duncker & Humblot, Berlin 2007, ISBN 978-3-428-11204-3 , p. 20 f. ( Digitized version ).

Web links

• John J. O'Connor, Edmund F. Robertson :  Ludwig Schläfli. In: MacTutor History of Mathematics archive .
• Information on four-dimensional polytopes
• Explanations of the Schläfli symbol.
• Literature by and about Ludwig Schläfli in the catalog of the German National Library
• Erwin Neuenschwander: Schläfli, Ludwig. In: Historical Lexicon of Switzerland .
• Urs Wüthrich: The genius that peddlers should become . In: Berner Zeitung, July 21, 2014

Individual evidence

1. ^ ^{A b} Moritz Cantor:  Schläfli, Ludwig . In: Allgemeine Deutsche Biographie (ADB). Volume 54, Duncker & Humblot, Leipzig 1908, pp. 29-31.