Ludwig Schläfli

Ludwig Schläfli ( born January 15, 1814 Grasswil, today Seeberg, † March 20, 1895 in Bern ) was a Swiss mathematician who worked on geometry and function theory. Played a key role in the development of the concept of dimension, which plays an important role, inter alia, in physics. Although his ideas are now treated in each undergraduate studies in mathematics, Schläfli is relatively unknown even among mathematicians.

Life

Youth and Education

Ludwig Schläfli spent most of his life in Switzerland. He came in Grasswil, his mother's hometown, to the world. Shortly thereafter, his family moved to the nearby castle village where his father worked as a businessman. Ludwig should occur in the footsteps of his father, but he was not made ​​for the practical work.

Due to his mathematical talent, he got the opportunity in 1829 to attend high school in Bern. At that time he already learned the Abraham Gotthelf Kästner differential calculus from Mathematical Foundations of Analysis of the Infinite (1761 ). In 1831 he went to the Academy in Bern to get further training on. 1834 from the Academy, the new University of Bern, where he took up the study of theology.

Teaching

After graduating in 1836, he was appointed teacher in Thun. This employment he went to 1847, where he spent studying mathematics and botany his spare time and once a week the University of Bern visited to further study theology.

The year 1843 marked a turning point in Schläfli life. Schläfli had a visit to Berlin planned to do with the local mathematical community acquaintance, especially with Jakob Steiner, a famous Swiss mathematician. But unexpectedly came to Berne and Steiner met Schläfli. Steiner was not only impressed by Schläfli mathematical knowledge, but also of his excellent language skills in Italian and French.

Steiner suggested Schläfli ago to support his Berlin colleague Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Karl Wilhelm Borchardt and Steiner himself as an interpreter at the upcoming trip to Italy. Steiner praised this idea to his friends in the following manner to (which is an indication that Schläfli was a little awkward in everyday affairs ):

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

Later life

Schläfli remained in contact with Steiner to 1856. Prospects, which were opened to him, encouraged him to apply in 1847 for a position at the University of Bern. He was appointed as a lecturer in 1848, in 1853 as Associate Professor and in 1872 full professor. Schläfli teaching lasted until his retirement in 1891. Until his death in 1895 he devoted himself to the study of Sanskrit and the translation of the Hindu scripture Rig Veda into German.

Higher dimensions

Schläfli is one of the three founders of multi-dimensional geometry, together with Arthur Cayley and Bernhard Riemann. In 1850, the general concept of Euclidean spaces had not yet developed - but linear equations in variables were already well understood. In the 1840s, William Rowan Hamilton 's quaternions and John Thomas Graves and Cayley developed the octaves. These two systems were working with a base of four or eight elements and put an interpretation analogous to the Cartesian coordinates of the three-dimensional space nearby.

From 1850 to 1852 Schläfli worked on his masterpiece theory of multiple continuity, in which he explained the study to the linear geometry of the - dimensional space. He also defined the - dimensional sphere, and calculated their volume. He decided to publish his work, and sent it to the Academy in Vienna, but it was rejected because of its scope. A second attempt in Berlin ended with the same result. Finally Schläfli in 1854 asked to write a shorter version of what he did not. Steiner tried to help, to publish the work in Crelle Journal him. But for unknown reasons, this was not made. Parts of the work were published in 1860 by Cayley in English. The first publication of all correspondence took place only in 1901 Schläfli death. The first review of the book appeared in 1904 in the Netherlands Nieuw Archief voor de mathematics journal Maths and was written by the Dutch mathematician Pieter Hendrik Schoute.

An excerpt from the introduction to the " theory of the multiple continuity ":

Schläfli summarized points in the -dimensional space on first as solutions of linear equations, and then execute the brilliant thought to consider a system with no equations, so as to obtain all possible points of ( as we would call it today). He used this concept in the articles that he published in the 1850s and 1860s, and it developed quickly. In 1867 he began an article with the words We consider the space of tuples of points. [ ... ]. This suggests not only that he had got to grips with the theory, but also that his audience no longer had need long explanations.

Polytopes

In the theory of multiples continuity Schläfli defines so-called Polyschemas which are nowadays called polytopes. They are the multidimensional analogues of polygons and polyhedra. He developed their theories and found, among other things, the multi-dimensional variant of the Euler Polyedersatzes. He also determined the regular polytopes, that is, the -dimensional relatives of regular polygons and Platonic solids. It turned out that there are six of those in the four-dimensional space and three in all higher dimensional spaces.

Although Schläfli in the second half of the 19th century pretty well known among his colleagues, in particular for his contributions to complex analysis, was his early geometrical works for a long time did not get much attention. At the beginning of the 20th century, Pieter Hendrik Schoute dealt with Alicia Boole Stott with polytopes. She proved Schläfli result about regular polytopes again, but only for the four-dimensional space and then discovered Schläfli book. Later studied Willem Abraham Wijthoff 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.
• Ludwig Schläfli should be like his father a businessman. But he made the worst possible business because he could not understand that you sold an item more expensive than that one shopped him.
• Schläfli made ​​a theological state examination and was (according to some complications with the sample sermon ) to find in the Bernese directory of the parish authorized persons. But he has such a well never held.
• In the Library of Exact Sciences, University of Berne, the Three quarks for Muster Mark lettering, Einstein and Schläfli reminiscent of Schläfli activity in Bern.