Ludwig Schläfli

Ludwig Schläfli (January 15th, 1814 with Grasswyl - March 20th 1895 with Bern) was a Mathématicien Suisse specialist in Géométrie and in Analyze complexes. He played a key function in the development of the concept of unspecified space of dimension. Perhaps because its ideas appear obvious today, he is known rather little, even among the mathematicians.

Life and career

Youth and education

Ludwig Schläfli passed the major part of its life in Suisse. It was born with Grasswyl, birthplace of his mother. The family then moved for the city close to Burgdorf, where his/her father was commercial. His/her father wanted that Ludwig made the same trade that him, but it did not seem not made for practical work.

On the other hand, thanks to its gift for mathematics, it could enter to the Gymnasium of Bern in 1829. He then learned already the differential Calculus in the book Mathematische Anfangsgründe der Analysis from Unendlichen (mathematical Foundations of the analysis of infinite, 1761) of Abraham Gotthelf Kästner. In 1831, it enters to the Academy of Bern to continue its studies. In 1834, the Academy becomes the news Université of Bern, where it begins studies of Théologie.

Teaching

After its diploma in 1836, it is engaged as teacher in a secondary school with Thun. It remains there until in 1847, spending its spare time to study mathematics and the Botanique while going once per week at the university of Bern.

A turning in its life takes place in 1843. Schläfli intended to visit Berlin and to become acquainted with the mathematical community of the city, in particular of Jakob Steiner, a famous Swiss mathematician. But, in an unexpected way, Steiner goes to Bern and they meet there. Not only Steiner is impressed by mathematical knowledge of Schläfli, but it is also interested by its good knowledge of the French and the Italian .

Steiner proposes in Schläfli to become the assistant of its Berliner colleagues Carl Gustav Jakob Jacobi, Dirichlet, Carl Wilhelm Borchardt and of him even as interprets for a voyage envisaged in Italy.

Schläfli accompanied them in Italy and the voyage was very beneficial for him. They remained there more than six months, during which Schläfli translated even some their work into Italian.

Continuation of its life

Schläfli maintained a correspondence with Steiner until in 1856. The prospects which opened with him encouraged with candidater for a station at the university of Bern in 1847. It was recruited there in 1848 and there remained until its retirement in 1891. He studied then the Sanscrit and translated the Rig Veda into German, until his death in 1895.

Higher dimensions

Schläfli is one of the three architects of the multidimensional geometry with Arthur Cayley and Bernhard Riemann. Around 1850, the general concept of Euclidean Espace did not exist yet, but the linear equations out of N variable were well included/understood. In the years 1840, William Rowan Hamilton had developed the Quaternion S, and John Thomas Graves and Arthur Cayley the Octonion S. These two systems of numbers are described by four or eight real numbers and suggested an interpretation similar to the Cartesian coordinated in the space of dimension three.

Of 1850 with 1852, Schläfli worked with its major work Theorie DER vielfachen Kontinuität (Theory of multiple continuities) in which it initiates the study of the linear geometry in the space of dimension N. It defines also the sphere of dimension N and calculates its volume. He then wanted to publish this work. He was sent to the Academy of Vienna but was refused for his too big length. He was sent then to Berlin with the same result. After a long bureaucratic pause, one asked for Schläfli in 1854 of write a shortened version, but it did not do it. Steiner then tried to help it to make appear this work in the Journal of Crelle, but it was still a failure, for unknown reasons. Parts were translated into English and were published by Cayley in 1860. The first integral publication took place in 1901 only, on a purely posthumous basis.

At the same time, Riemann supported its famous enabling Über die Hypothesen welche der Geometrie zu Grunde liegen (On the assumptions which underlies the geometry) in 1854 and presented the concept of variety of dimension N. The concept of space of high size started to open out.

Polytopes

In Theory der Vielfachen Kontinuität , it defines also what it names the polyschèmes , today called the Polytope S, which are analogues in higher dimensions of the Polygone S and Polyèdre S. It makes the theory and finds of it in particular the analog of the Formule of Euler. It determines the regular polytopes, analogues of the regular polygons and the Platonic solid . There are six polytopes regular in dimension four and three in each dimension larger than four.

Even if Schläfli is rather well-known his/her colleagues, especially for its contributions in Analyze complexes, its work geometrical immediately did not hold the attention. At the beginning of the 20th century, Pieter Hendrik Schoute started to work on the polytopes with Alicia Boole Stott. It D-obtained the results of Schläfli on the regular polytopes in dimension four before taking note of the book of Schläfli. Later, Abraham Willem Wijthoff studied the regular semi polytopes and this work was continued in particular by Coxeter and John Conway. It remains much of problems open in the theory of the polytopes.

Schläfli is also famous for the discovery of the 27 straight lines on the generic cubic surface, which was worth to him to receive the Steiner price of the Académie of Berlin.

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