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Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
(at the time called function theory) who was one of the key figures in developing the notion of higher-
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al spaces. The concept of multidimensionality is pervasive in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, has come to play a pivotal role in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, and is a common element in science fiction.


Life and career


Youth and education

Ludwig spent most of his life in
Switzerland ). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are installed in Bern, while other federal institutions, such as the federal courts, are in other cities (Bellinzona, Lausanne, Luzern, Neuchâtel ...
. He was born in Grasswil (now part of
Seeberg Seeberg is a municipality in the Oberaargau administrative district in the canton of Bern in Switzerland. The lake Burgäschisee is located on the border with Aeschi. On 1 January 2016 the former municipality of Hermiswil merged into Seebe ...
), his mother's hometown. The family then moved to the nearby Burgdorf, where his father worked as a
tradesman A tradesman, tradeswoman, or tradesperson is a skilled worker that specializes in a particular trade (occupation or field of work). Tradesmen usually have work experience, on-the-job training, and often formal vocational education in contrast ...
. His father wanted Ludwig to follow in his footsteps, but Ludwig was not cut out for practical work. In contrast, because of his mathematical gifts, he was allowed to attend the Gymnasium in
Bern german: Berner(in)french: Bernois(e) it, bernese , neighboring_municipalities = Bremgarten bei Bern, Frauenkappelen, Ittigen, Kirchlindach, Köniz, Mühleberg, Muri bei Bern, Neuenegg, Ostermundigen, Wohlen bei Bern, Zollikofen , website ...
in 1829. By that time he was already learning differential calculus from
Abraham Gotthelf Kästner Abraham Gotthelf Kästner (27 September 1719 – 20 June 1800) was a German mathematician and epigrammatist. He was known in his professional life for writing textbooks and compiling encyclopedias rather than for original research. Georg Chr ...
's ''Mathematische Anfangsgründe der Analysis des Unendlichen'' (1761). In 1831 he transferred to the Akademie in Bern for further studies. By 1834 the Akademie had become the new
Universität Bern The University of Bern (german: Universität Bern, french: Université de Berne, la, Universitas Bernensis) is a university in the Swiss capital of Bern and was founded in 1834. It is regulated and financed by the Canton of Bern. It is a compreh ...
, where he started studying theology.


Teaching

After graduating in 1836, he was appointed a secondary school teacher in
Thun , neighboring_municipalities= Amsoldingen, Heiligenschwendi, Heimberg, Hilterfingen, Homberg, Schwendibach, Spiez, Steffisburg, Thierachern, Uetendorf, Zwieselberg , twintown = , website = www.thun.ch Thun (french: Thou ...
. He stayed there until 1847, spending his free time studying mathematics and
botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or phytologist is a scientist who specialises in this field. The term "botany" comes from the Ancient Greek w ...
while attending the university in Bern once a week. A turning point in his life came in 1843. Schläfli had planned to visit Berlin and become acquainted with its mathematical community, especially
Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...
, a well known Swiss mathematician. But unexpectedly Steiner showed up in Bern and they met. Not only was Steiner impressed by Schläfli's mathematical knowledge, he was also very interested in Schläfli's fluency in Italian and French. Steiner proposed Schläfli to assist his Berlin colleagues
Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
,
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
,
Carl Wilhelm Borchardt Carl Wilhelm Borchardt (22 February 1817 – 27 June 1880) was a German mathematician. Borchardt was born to a Jewish family in Berlin. His father, Moritz, was a respected merchant, and his mother was Emma Heilborn. Borchardt studied under ...
and himself as an interpreter on a forthcoming trip to Italy. Steiner sold this idea to his friends in the following way, which indicates Schläfli must have been somewhat clumsy at daily affairs: :... während er den Berliner Freunden den neugeworbenen Reisegefaehrten durch die Worte anpries, der sei ein ländlicher Mathematiker bei Bern, für die Welt ein Esel, aber Sprachen lerne er wie ein Kinderspiel, den wollten sie als Dolmetscher mit sich nehmen. DB English translation: :... while he (Steiner) praised/recommended the new travel companion to his Berlin friends with the words that he (Schläfli) was a provincial mathematician working near Bern, an 'ass for the world' (i.e., not very practical), but that he learned languages like child's play, and that they should take him with them as a translator. Schläfli accompanied them to Italy, and benefited much from the trip. They stayed for more than six months, during which time Schläfli even translated some of the others' mathematical works into Italian.


Later life

Schläfli kept up a correspondence with Steiner till 1856. The vistas that had been opened up to him encouraged him to apply for a position at the university in Bern in 1847, where he was appointed(?) in 1848. He stayed until his retirement in 1891, and spent his remaining time studying
Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
and translating the
Hindu Hindus (; ) are people who religiously adhere to Hinduism.Jeffery D. Long (2007), A Vision for Hinduism, IB Tauris, , pages 35–37 Historically, the term has also been used as a geographical, cultural, and later religious identifier for ...
scripture Religious texts, including scripture, are texts which various religions consider to be of central importance to their religious tradition. They differ from literature by being a compilation or discussion of beliefs, mythologies, ritual prac ...
''
Rig Veda The ''Rigveda'' or ''Rig Veda'' ( ', from ' "praise" and ' "knowledge") is an ancient Indian collection of Vedic Sanskrit hymns (''sūktas''). It is one of the four sacred canonical Hindu texts (''śruti'') known as the Vedas. Only one Sh ...
'' into German, until his death in 1895.


Higher dimensions

Schläfli is one of the three architects of multidimensional geometry, together with Arthur Cayley and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
. Around 1850 the general concept of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
had not been developed – but linear equations in n variables were well understood. In the 1840s
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irela ...
had developed his
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s and John T. Graves and Arthur Cayley the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s. The latter two systems worked with bases of four and, respectively, eight elements, and suggested an interpretation analogous to the
cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
s in three-dimensional space. From 1850 to 1852 Schläfli worked on his magnum opus, ''Theorie der vielfachen Kontinuität'', in which he initiated the study of the linear geometry of n-dimensional space. He also defined the n-dimensional sphere and calculated its volume. He then wanted to have this work published. It was sent to the Akademie in Vienna, but was refused because of its size. Afterwards it was sent to Berlin, with the same result. After a long bureaucratic pause, Schläfli was asked in 1854 to write a shorter version, but he did not do so. Steiner then tried to help him getting the work published in '' Crelle's Journal'', but somehow things didn't work out. The exact reasons remain unknown. Portions of the work were published by Cayley in English in 1860. The first publication of the entire manuscript was only in 1901, after Schläfli's death. The first review of the book then appeared in the Dutch mathematical journal ''Nieuw Archief voor de Wiskunde'' in 1904, written by the Dutch mathematician
Pieter Hendrik Schoute Pieter Hendrik Schoute (21 January 1846, Wormerveer – 18 April 1913, Groningen) was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry. He started his career as a civil engineer, but became a professor of ...
. During this period, Riemann held his famous Habilitationsvortrag ''Über die Hypothesen welche der Geometrie zu Grunde liegen'' in 1854, and introduced the concept of an n-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. The concept of higher-dimensional spaces was starting to flourish. Below is an excerpt from the preface to ''Theorie der vielfachen Kontinuität'': :Die Abhandlung, die ich hier der Kaiserlichen Akademie der Wissenschaften vorzulegen die Ehre habe, enthält einen Versuch, einen neuen Zweig der Analysis zu begründen und zu bearbeiten, welcher, gleichsam eine analytische Geometrie von n Dimensionen, diejenigen der Ebene und des Raumes als spezielle Fälle fuer n=2,3 in sich enthielte. Ich nenne denselben Theorie der vielfachen Kontinuität überhaupt in demselben Sinne, wie man zum Beispiel die Geometrie des Raumes eine Theorie der dreifachen Kontinuität nennen kann. Wie in dieser eine Gruppe von Werten der drei Koordinaten einen Punkt bestimmt, so soll in jener eine Gruppe gegebener Werte der n Variabeln x,y,\ldots eine Lösung bestimmen. Ich gebrauche diesen Ausdruck, weil man bei einer oder mehreren Gleichungen mit vielen Variabeln jede genügende Gruppe von Werten auch so nennt; das Ungewöhnliche der Benennung liegt nur darin, daß ich sie auch noch beibehalte, wenn gar keine Gleichung zwischen den Variabeln gegeben ist. In diesem Falle nenne ich die Gesamtheit aller Lösungen die n-fache Totalität; sind hingegen 1,2,3, \ldots Gleichungen gegeben, so heißt bzw. die Gesamtheit ihrer Lösungen n-1-faches, n-2-faches, n-3-faches, ... Kontinuum. Aus der Vorstellung der allseitigen Kontinuität der in einer Totalität enthaltenen Lösungen entwickelt sich diejenige der Unabhängigkeit ihrer gegenseitigen Lage von dem System der gebrauchten Variabeln, insofern durch Transformation neue Variabeln an ihre Stelle treten können. Diese Unabhängigkeit spricht sich aus in der Unveränderlichkeit dessen, was ich den Abstand zweier gegebener Lösungen (x,y,\ldots), (x', y',\ldots) nenne und im einfachsten Fall durch :definiere, indem ich gleichzeitig das System der Variabeln ein orthogonales heiße, .. English translation: :The treatise I have the honour of presenting to the Imperial Academy of Science here, is an attempt to found and develop a new branch of analysis that would, as it were, be a geometry of n dimensions, containing the geometry of the plane and space as special cases for n=2,3. I call this the theory of multiple continuity in generally the same sense, in which one can call the geometry of space that of triple continuity. Like in that theory the 'group' of values of its coordinates determines a point, so in this one a 'group' of given values of the n variables x,y,\ldots will determine a solution. I use this expression, because one also calls every sufficient 'group' of values thus in the case of one or more equations with many variables; the only thing unusual about this naming is, that I keep it when no equations between the variables is given whatsoever. In this case I call the total (set) of solutions the n-fold totality; whereas when 1,2,3,\ldots equations are given, the total of their solutions is called respectively (an) n-1-fold, n-2-fold, n-3-fold, ... Continuum. From the notion of the solutions contained in a totality comes forth that of the independence of their relative positions (of the variables) in the system of variables used, insofar as new variables could take their place by transformation. This independence is expressed in the inalterability of that, which I call the distance between two given solutions (x,y,\ldots), (x', y',\ldots) and define in the easiest case by: :while at the same time I call a system of variables orthogonal .. We can see how he is still thinking of points in n-dimensional space as solutions to linear equations, and how he is considering a system ''without any equations'', thus obtaining all possible points of the \mathbf^n, as we would put it now. He disseminated the concept in the articles he published in the 1850s and 1860s, and it matured rapidly. By 1867 he starts an article by saying "We consider the space of n-tuples of points. ... This indicates not only that he had a firm grip on things, but also that his audience did not need a long explanation of it.


Polytopes

In ''Theorie der Vielfachen Kontinuität'' he goes on to define what he calls ''polyschemes'', nowadays called
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
s, which are the higher-dimensional analogues to
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
s and
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...
. He develops their theory and finds, among other things, the higher-dimensional version of Euler's formula. He determines the regular polytopes, i.e. the n-dimensional cousins of regular polygons and
platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
. It turns out there are six in dimension four and three in all higher dimensions. Although Schläfli was familiar to his colleagues in the second half of the 19th century, especially for his contributions to complex analysis, his early geometrical work failed to attract notice for many years. At the beginning of the twentieth century
Pieter Hendrik Schoute Pieter Hendrik Schoute (21 January 1846, Wormerveer – 18 April 1913, Groningen) was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry. He started his career as a civil engineer, but became a professor of ...
started to work on polytopes together with
Alicia Boole Stott Alicia Boole Stott (8 June 1860 – 17 December 1940) was an Irish mathematician. Despite never holding an academic position, she made a number of valuable contributions to the field, receiving an honorary doctorate from the University of Groni ...
. She reproved Schläfli's result on regular polytopes for dimension 4 only and afterwards rediscovered his book. Later Willem Abraham Wijthoff studied semi-regular polytopes and this work was continued by H.S.M. Coxeter,
John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
and others. There are still many problems to be solved in this area of investigation opened up by Ludwig Schläfli.


See also

*
Regular 4-polytope In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star reg ...
*
Schläfli double six In geometry, the Schläfli double six is a configuration of 30 points and 12 lines, introduced by . The lines of the configuration can be partitioned into two subsets of six lines: each line is disjoint from ( skew with) the lines in its own subse ...
*
Schläfli graph In the mathematical field of graph theory, the Schläfli graph, named after Ludwig Schläfli, is a 16- regular undirected graph with 27 vertices and 216 edges. It is a strongly regular graph with parameters srg(27, 16, 10, 8). ...
*
Schläfli orthoscheme In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges (v_0v_1), (v_1v_2), \dots, (v_ ...
* Schläfli symbol


References

* * chLudwig Schläfli, Gesammelte Abhandlungen * SB''Dictionary of Scientific Biographies'' * DB Allgemeine Deutsche Biographie, Band 54, S.29–31. Biography by
Moritz Cantor Moritz Benedikt Cantor (23 August 1829 – 10 April 1920) was a German historian of mathematics. Biography Cantor was born at Mannheim. He came from a Sephardi Jewish family that had emigrated to the Netherlands from Portugal, another branch o ...
, 1896 * as
Abraham Gotthelf Kästner Abraham Gotthelf Kästner (27 September 1719 – 20 June 1800) was a German mathematician and epigrammatist. He was known in his professional life for writing textbooks and compiling encyclopedias rather than for original research. Georg Chr ...
, ''Mathematische Anfangsgründe der Analysis des Unendlichen'', Göttingen, 1761 ** ''Note:'' This is the third volume of Kästner's ''Mathematische Anfangsgründe'', which can be viewed online at th
Göttinger Digitalisierungszentrum


External links

* * {{DEFAULTSORT:Schlafli, Ludwig 19th-century Swiss mathematicians 1814 births 1895 deaths Polytopes University of Bern alumni