Ludwig Schläfli
Encyclopedia
Ludwig Schläfli was a Swiss
geometer
and complex analyst
(at the time called function theory) who was one of the key figures in developing the notion of higher dimension
al spaces. The concept of multidimensionality has since come to play a pivotal role in physics
, and is a common element in science fiction
. Although his ideas have become so widely accepted, he is poorly remembered, even among mathematician
s.
. He was born in Graßwil, his mother's hometown. The family then moved to the nearby Burgdorf, where his father worked as a tradesman
. 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 in 1829. By that time he was already learning differential calculus
from Abraham Gotthelf Kästner
'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, where he started studying theology
.
teacher
in Thun
. He stayed there until 1847, spending his free time studying mathematics and botany
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
, 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, Dirichlet, Carl Wilhelm Borchardt
and himself as an interpreter
on a forthcoming trip to Italy
. Steiner sold this idea to his friends on the following way, which indicates Schläfli must have been somewhat clumsy at daily affairs:
English translation:
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.
and translating the Hindu
scripture Rig Veda into German, until his death in 1895.
and Bernhard Riemann
. Around 1850 the general concept of Euclidean space
hadn't been developed — but linear equation
s in
variables were well-understood. In the 1840s William Rowan Hamilton
had developed his quaternion
s and John T. Graves
and Arthur Cayley
the octonion
s. The latter two systems worked with bases of four (respectively eight) elements, and suggested an interpretation analogous to the cartesian coordinates 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
-dimensional space. He also defined the 
-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 this he understandably did not. 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
.
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
-dimensional manifold
. The concept of higher dimensional spaces was starting to flourish.
Below is an excerpt from the preface to Theorie der vielfachen Kontinuität:
Anzeige einer Abhandlung über die Theorie der vielfachen Kontinuität
English translation:
We can see how he is still thinking of points in
-dimensional space as solutions to linear equations, and how he is considering a system without any equations, thus obtaining all possible points of the 
, 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 
-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.
s, which are the higher dimensional analogues to polygon
s and polyhedra
. 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
-dimensional cousins of regular polygons and platonic solids. 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 century, especially for his contributions to complex analysis, his early geometrical work didn't get proper attention for a long time. At the beginning of the twentieth century Pieter Hendrik Schoute
started to work on polytopes together with Alicia Boole Stott
. 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
and others. There are still many problems to be solved in this area of investigation opened up by Ludwig Schläfli.
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
geometer
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and complex analyst
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
(at the time called function theory) who was one of the key figures in developing the notion of higher dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al spaces. The concept of multidimensionality has since come to play a pivotal role in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, and is a common element in science fiction
Science fiction
Science fiction is a genre of fiction dealing with imaginary but more or less plausible content such as future settings, futuristic science and technology, space travel, aliens, and paranormal abilities...
. Although his ideas have become so widely accepted, he is poorly remembered, even among mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s.
Youth and education
Ludwig Schläfli spent most of his life in SwitzerlandSwitzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
. He was born in Graßwil, his mother's hometown. The family then moved to the nearby Burgdorf, where his father worked as a tradesman
Tradesman
This article is about the skilled manual worker meaning of the term; for other uses see Tradesperson .A tradesman is a skilled manual worker in a particular trade or craft. Economically and socially, a tradesman's status is considered between a laborer and a professional, with a high degree of both...
. 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
Gymnasium (school)
A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...
in Bern in 1829. By that time he was already learning differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
from Abraham Gotthelf Kästner
Abraham Gotthelf Kästner
Abraham Gotthelf Kästner 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 Christoph Lichtenberg was one of his doctoral students, and admired the man greatly. He became...
'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, where he started studying theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...
.
Teaching
After his graduation in 1836, he was appointed a secondary schoolSecondary school
Secondary school is a term used to describe an educational institution where the final stage of schooling, known as secondary education and usually compulsory up to a specified age, takes place...
teacher
Teacher
A teacher or schoolteacher is a person who provides education for pupils and students . The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional...
in Thun
Thun
Thun is a municipality in the administrative district of Thun in the canton of Bern in Switzerland with about 42,136 inhabitants , as of 1 January 2006....
. He stayed there until 1847, spending his free time studying mathematics and botany
Botany
Botany, plant science, or plant biology is a branch of biology that involves the scientific study of plant life. Traditionally, botany also included the study of fungi, algae and viruses...
while attending the university in Bern once a week.
A turning point in his life came in 1843. Schläfli had planned to visit Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
and become acquainted with its mathematical community, especially Jakob Steiner
Jakob Steiner
Jakob Steiner was a Swiss mathematician who worked primarily in geometry.-Personal and professional life:...
, 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
Italian language
Italian is a Romance language spoken mainly in Europe: Italy, Switzerland, San Marino, Vatican City, by minorities in Malta, Monaco, Croatia, Slovenia, France, Libya, Eritrea, and Somalia, and by immigrant communities in the Americas and Australia...
and French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
.
Steiner proposed Schläfli to assist his Berlin colleagues Carl Gustav Jacob Jacobi, Dirichlet, Carl Wilhelm Borchardt
Carl Wilhelm Borchardt
Carl Wilhelm Borchardt 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 a number of tutors, including Julius Plücker and Jakob Steiner...
and himself as an interpreter
Interpreting
Language interpretation is the facilitating of oral or sign-language communication, either simultaneously or consecutively, between users of different languages...
on a forthcoming trip to Italy
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
. Steiner sold this idea to his friends on 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 anpreis, 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. [ADB]
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 SanskritSanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
and translating the Hindu
Hindu
Hindu refers to an identity associated with the philosophical, religious and cultural systems that are indigenous to the Indian subcontinent. As used in the Constitution of India, the word "Hindu" is also attributed to all persons professing any Indian religion...
scripture Rig Veda into German, until his death in 1895.
Higher dimensions
Schläfli is one of the three architects of multidimensional geometry, together with Arthur CayleyArthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
and Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
. Around 1850 the general concept of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
hadn't been developed — but linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
s in
variables were well-understood. In the 1840s William Rowan HamiltonWilliam Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
had developed his quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s and John T. Graves
John T. Graves
John Thomas Graves was an Irish jurist and mathematician. He was a friend of William Rowan Hamilton, and is credited both with inspiring Hamilton to discover the quaternions and with personally discovering the octonions, which he called the octaves...
and Arthur Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
the octonion
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
s. The latter two systems worked with bases of four (respectively eight) elements, and suggested an interpretation analogous to the cartesian coordinates 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
-dimensional space. He also defined the -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 this he understandably did not. 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 SchoutePieter Hendrik Schoute
Pieter Hendrik Schoute was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry.- References :...
.
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
-dimensional manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. 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
Dimensionen, diejenigen der Ebene und des Raumes als spezielle Fälle fuer 
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 
Variabeln 
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 
-fache Totalität; sind hingegen 
Gleichungen gegeben, so heißt bzw. die Gesamtheit ihrer Lösungen 
-faches, 
-faches, 
-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 (
), (
) 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
dimensions, containing the geometry of the plane and space as special cases for 
. 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 
variables 
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 
-fold totality; whereas when 
equations are given, the total of their solutions is called respectively (an) 
-fold, 
-fold, 
-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 (
), (
) 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
-dimensional space as solutions to linear equations, and how he is considering a system without any equations, thus obtaining all possible points of the , 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 -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 polytopePolytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
s, which are the higher dimensional analogues to polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
s and polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
. 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
-dimensional cousins of regular polygons and platonic solids. 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 century, especially for his contributions to complex analysis, his early geometrical work didn't get proper attention for a long time. At the beginning of the twentieth century Pieter Hendrik Schoute
Pieter Hendrik Schoute
Pieter Hendrik Schoute was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry.- References :...
started to work on polytopes together with Alicia Boole Stott
Alicia Boole Stott
Alicia Boole Stott was the third daughter of George Boole and Mary Everest Boole, born in Cork, Ireland. Before marrying Walter Stott, an actuary, in 1890, she was known as Alicia Boole...
. 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
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
, John Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
and others. There are still many problems to be solved in this area of investigation opened up by Ludwig Schläfli.
Literature
- [Sch] Ludwig Schläfli, Gesammelte Abhandlungen
- [DSB] Dictionary of Scientific Biographies
- [ADB] Allgemeine Deutsche BiographieAllgemeine Deutsche BiographieAllgemeine Deutsche Biographie is one of the most important and most comprehensive biographical reference works in the German language....
, Band 54, S.29—31. Biography by Moritz Cantor, 1896 - [Kas] Abraham Gotthelf KästnerAbraham Gotthelf KästnerAbraham Gotthelf Kästner 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 Christoph Lichtenberg was one of his doctoral students, and admired the man greatly. He became...
, 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 the Göttinger Digitalisierungszentrum.