Eduard Study
Encyclopedia
Eduard Study was a German
mathematician
known for work on invariant theory
of ternary form
s (1889) and for the study of spherical trigonometry
. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.
Study was born in Coburg
in the Duchy of Saxe-Coburg-Gotha. He died in Bonn
.
was at Leipzig, and Study returned there as Privatdozent. In 1888 he moved to Marburg and in 1893 embarked on a speaking tour in the U.S.A. He appeared at the primordial International Congress of Mathematicians in Chicago as part of the World's Columbian Exposition
and took part in mathematics at Johns Hopkins University
. Back in Germany, in 1894, he was appointed extraordinary professor at Göttingen. Then he gained the rank of full professor in 1897 at Greifswald. In 1904 he was called to the University of Bonn
as the position held by Rudolf Lipschitz
was vacant. There he settled until retirement in 1927.
through the space group
. The second part of his article constructs a seven-dimensional space out of “dual biquaternions”, that is numbers
where a, b, c, and d are dual numbers and {1, i, j, k} multiply as in the quaternion group
. He uses these conventions:
The multiplication table is found on page 520 of volume 39 (1891) in Mathematische Annalen
under the title “Von Bewegungen und Umlegungen, I. und II. Abhandlungen”.
Eduard Study cites William Kingdon Clifford
as an earlier source on these biquaternions. In 1901 Study published Geometrie der Dynamen to highlight the applications of this algebra. Due to Eduard Study’s profound and early exploitation of this eight-dimensional associative algebra
, it is frequently referred to as Study Biquaternions. Study’s achievement is celebrated, for example, in A History of Algebra (1985) by B. L. van der Waerden, who also cites Clifford’s earlier note.
Since the space group is important in robotics
, the Study biquaternions are a technical tool, now sometimes referred to as dual quaternion
s. For example, Joe Rooney has profiled the use of this algebra by several modelers of mechanics (see external link).
. This 34 page article was expanded to 138 pages in 1908 by Élie Cartan
, who surveyed the hypercomplex systems in Encyclopédie des sciences mathématiques pures et appliqueés. Cartan acknowledged Eduard Study’s priority in his title with the words "after Eduard Study".
In the 1993 biography of Cartan by Akivis and Rosenfeld, one reads:
In 1985 Helmut Karzel and Günter Kist developed "Study's quaternions" as the kinematic algebra corresponding to the group of motions of the Euclidean plane. These quaternions arise in "Kinematic algebras and their geometries" alongside ordinary quaternions and the ring of 2 × 2 real matrices which Karzel and Kist cast as the kinematic algebras of the elliptic plane and hyperbolic plane respectively. See the "Motivation and Historical Review" at page 437 of Rings and Geometry, R. Kaya editor.
Thus in the study of classical associative algebra
s over R there are two special ones: Study’s quaternions (4D) and Study’s biquaternions (8D).
s and line coordinates
was noted by Heinrich Guggenheimer
in 1963 in his book Differential Geometry (see pages 162–5). He cites and proves the following theorem of Study: The oriented lines in R3 are in one-to-one correspondence with the points of the dual unit sphere in D3. Later he says "A differentiable curve A(u) on the dual unit sphere, depending on a real parameter u, represents a differentiable family of straight lines in R3: a ruled surface
. The lines A(u) are the generators or rulings of the surface." Guggenheimer also shows the representation of the Euclidean motions in R3 by orthogonal dual matrices.
(60:321–378). Some of its contents were anticipated by Guido Fubini
a year before. The distance Study refers to is a Hermitian form on complex projective space
. Since then this metric
has been called the Fubini–Study metric. Study was careful in 1905 to distinguish the hyperbolic and elliptic cases in Hermitian geometry.
. Like James Joseph Sylvester
, Paul Gordan believed that invariant theory could contribute to the understanding of chemical valence
. In 1900 Gordan and his student G. Alexejeff contributed an article on an analogy between the coupling problem for angular momenta
and their work on invariant theory to the Zeitschrift für Physikalische Chemie (v. 35, p. 610). In 2006 Wormer and Paldus summarized Study's role as follows:
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
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....
known for work on invariant theory
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
of ternary form
Ternary form
Ternary form, sometimes called song form, is a three-part musical form, usually schematicized as A-B-A. The first and third parts are musically identical, or very nearly so, while the second part in some way provides a contrast with them...
s (1889) and for the study of spherical trigonometry
Spherical trigonometry
Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.
Study was born in Coburg
Coburg
Coburg is a town located on the Itz River in Bavaria, Germany. Its 2005 population was 42,015. Long one of the Thuringian states of the Wettin line, it joined with Bavaria by popular vote in 1920...
in the Duchy of Saxe-Coburg-Gotha. He died in Bonn
Bonn
Bonn is the 19th largest city in Germany. Located in the Cologne/Bonn Region, about 25 kilometres south of Cologne on the river Rhine in the State of North Rhine-Westphalia, it was the capital of West Germany from 1949 to 1990 and the official seat of government of united Germany from 1990 to 1999....
.
Career
Eduard Study began his university career in Jena, Strasbourg, Leipzig, and Munich. He loved to study biology, especially entomology. He was awarded the doctorate in mathematics at the University of Munich in 1884. Paul Gordan, an expert in invariant theoryInvariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
was at Leipzig, and Study returned there as Privatdozent. In 1888 he moved to Marburg and in 1893 embarked on a speaking tour in the U.S.A. He appeared at the primordial International Congress of Mathematicians in Chicago as part of the World's Columbian Exposition
World's Columbian Exposition
The World's Columbian Exposition was a World's Fair held in Chicago in 1893 to celebrate the 400th anniversary of Christopher Columbus's arrival in the New World in 1492. Chicago bested New York City; Washington, D.C.; and St...
and took part in mathematics at Johns Hopkins University
Johns Hopkins University
The Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...
. Back in Germany, in 1894, he was appointed extraordinary professor at Göttingen. Then he gained the rank of full professor in 1897 at Greifswald. In 1904 he was called to the University of Bonn
University of Bonn
The University of Bonn is a public research university located in Bonn, Germany. Founded in its present form in 1818, as the linear successor of earlier academic institutions, the University of Bonn is today one of the leading universities in Germany. The University of Bonn offers a large number...
as the position held by Rudolf Lipschitz
Rudolf Lipschitz
Rudolf Otto Sigismund Lipschitz was a German mathematician and professor at the University of Bonn from 1864. Peter Gustav Dirichlet was his teacher. He supervised the early work of Felix Klein....
was vacant. There he settled until retirement in 1927.
Euclidean space group and biquaternions
In 1891 Eduard Study published “Of Motions and Translations, in two parts”. It treats Euclidean spaceEuclidean 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...
through the space group
Space group
In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
. The second part of his article constructs a seven-dimensional space out of “dual biquaternions”, that is numbers
where a, b, c, and d are dual numbers and {1, i, j, k} multiply as in the quaternion group
Quaternion group
In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
. He uses these conventions:
The multiplication table is found on page 520 of volume 39 (1891) in Mathematische Annalen
Mathematische Annalen
Mathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...
under the title “Von Bewegungen und Umlegungen, I. und II. Abhandlungen”.
Eduard Study cites William Kingdon Clifford
William Kingdon Clifford
William Kingdon Clifford FRS was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour, with interesting applications in contemporary mathematical physics...
as an earlier source on these biquaternions. In 1901 Study published Geometrie der Dynamen to highlight the applications of this algebra. Due to Eduard Study’s profound and early exploitation of this eight-dimensional associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
, it is frequently referred to as Study Biquaternions. Study’s achievement is celebrated, for example, in A History of Algebra (1985) by B. L. van der Waerden, who also cites Clifford’s earlier note.
Since the space group is important in robotics
Robotics
Robotics is the branch of technology that deals with the design, construction, operation, structural disposition, manufacture and application of robots...
, the Study biquaternions are a technical tool, now sometimes referred to as dual quaternion
Dual quaternion
The set of dual quaternions is an algebra that can be used to represent spatial rigid body displacements. A dual quaternion is an ordered pair of quaternions  = and therefore is constructed from eight real parameters. Because rigid body displacements are defined by six parameters, dual...
s. For example, Joe Rooney has profiled the use of this algebra by several modelers of mechanics (see external link).
Hypercomplex numbers
In 1898 Eduard Study was the author of an article on hypercomplex numbers in the Klein's encyclopediaKlein's encyclopedia
In mathematics, Klein’s encyclopedia refers to a German mathematical encyclopedia published in six volumes from 1898 to 1933. Felix Klein and Wilhelm Meyer were organizers of the encyclopedia. Its title in English is "Encyclopedia of mathematical sciences including their applications", which is...
. This 34 page article was expanded to 138 pages in 1908 by Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
, who surveyed the hypercomplex systems in Encyclopédie des sciences mathématiques pures et appliqueés. Cartan acknowledged Eduard Study’s priority in his title with the words "after Eduard Study".
In the 1993 biography of Cartan by Akivis and Rosenfeld, one reads:
- [Study] defined the algebra °H of ‘semiquaternions’ with the units 1, i, ε, η having the properties
- Semiquaternions are often called ‘Study’s quaternions’.
In 1985 Helmut Karzel and Günter Kist developed "Study's quaternions" as the kinematic algebra corresponding to the group of motions of the Euclidean plane. These quaternions arise in "Kinematic algebras and their geometries" alongside ordinary quaternions and the ring of 2 × 2 real matrices which Karzel and Kist cast as the kinematic algebras of the elliptic plane and hyperbolic plane respectively. See the "Motivation and Historical Review" at page 437 of Rings and Geometry, R. Kaya editor.
Thus in the study of classical associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
s over R there are two special ones: Study’s quaternions (4D) and Study’s biquaternions (8D).
Ruled surfaces
Study's work with dual numberDual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and...
s and line coordinates
Line coordinates
In geometry, line coordinates are used to specify the position of a line just as point coordinates are used to specify the position of a point.-Lines in the plane:...
was noted by Heinrich Guggenheimer
Heinrich Guggenheimer
Heinrich Walter Guggenheimer is an American mathematician who has contributed to knowledge in differential geometry, topology, algebraic geometry, and convexity. He has also contributed volumes on Jewish sacred literature....
in 1963 in his book Differential Geometry (see pages 162–5). He cites and proves the following theorem of Study: The oriented lines in R3 are in one-to-one correspondence with the points of the dual unit sphere in D3. Later he says "A differentiable curve A(u) on the dual unit sphere, depending on a real parameter u, represents a differentiable family of straight lines in R3: a ruled surface
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
. The lines A(u) are the generators or rulings of the surface." Guggenheimer also shows the representation of the Euclidean motions in R3 by orthogonal dual matrices.
Hermitian form metric
In 1905 Study wrote "Kürzeste Weg im complexen Gebiet" (Shortest path in complex domains) for Mathematische AnnalenMathematische Annalen
Mathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...
(60:321–378). Some of its contents were anticipated by Guido Fubini
Guido Fubini
Guido Fubini was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric.Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, who was himself a teacher of mathematics...
a year before. The distance Study refers to is a Hermitian form on complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
. Since then this metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
has been called the Fubini–Study metric. Study was careful in 1905 to distinguish the hyperbolic and elliptic cases in Hermitian geometry.
Valence theory
Somewhat surprisingly Eduard Study is known by practitioners of quantum chemistryQuantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
. Like James Joseph Sylvester
James Joseph Sylvester
James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...
, Paul Gordan believed that invariant theory could contribute to the understanding of chemical valence
Valence (chemistry)
In chemistry, valence, also known as valency or valence number, is a measure of the number of bonds formed by an atom of a given element. "Valence" can be defined as the number of valence bonds...
. In 1900 Gordan and his student G. Alexejeff contributed an article on an analogy between the coupling problem for angular momenta
Angular momentum coupling
In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact through spin-orbit interaction, in which case the...
and their work on invariant theory to the Zeitschrift für Physikalische Chemie (v. 35, p. 610). In 2006 Wormer and Paldus summarized Study's role as follows:
- The analogy, lacking a physical basis at the time, was criticised heavily by the mathematician E. Study and ignored completely by the chemistry community of the 1890's. After the advent of quantum mechanics it became clear, however, that chemical valences arise from electron-spin couplings ... and that electron spin functions are, in fact, binary forms of the type studied by Gordan and ClebschClebsch-Gordan coefficientsIn physics, the Clebsch–Gordan coefficients are sets of numbers that arise in angular momentum coupling under the laws of quantum mechanics.In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum...
.
Cited publications
- Sphärische Trigonometrie, orthogonale Substitutionen, und elliptische Functionen: Eine Analytisch-Geometrische Untersuchung. Leipzig, Germany: Teubner, 1893
- Aeltere und neuere Untersuchungen uber Systeme complexer Zahlen, Mathematical Papers Chicago Congress
- Einleitung in die Theorie der Invarianten (1933).
- Geometrie der Dynamen (1903), from Historical Math Monographs at Cornell UniversityCornell UniversityCornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
. - Theorie der allgemeinen und höheren komplexen Grossen in Encyklopädie der mathematischen Wissenschaften, weblink to University of Göttingen.
External links
- Joe Rooney William Kingdon Clifford, Department of Design and Innovation, the Open University, London.