Aleksandr Petrovich Kotelnikov (russian: Алекса́ндр Петро́вич Коте́льников; October 20, 1865 – March 6, 1944) was a Russian and Soviet
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
specializing 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
kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
.
Biography
Aleksandr was the son of , a colleague of
Nikolai Lobachevsky
Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kn ...
. The subject of
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
was
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, a departure from tradition. The early exposure to Lobachevsky's work eventually led to Aleksandr undertaking the job of editing Lobachevsky's works.
Kotelnikov studied at
Kazan University
Kazan (Volga region) Federal University (russian: Казанский (Приволжский) федеральный университет, tt-Cyrl, Казан (Идел буе) федераль университеты) is a public research uni ...
, graduating in 1884. He began teaching at a gymnasium.
Having an interest in
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, he did graduate study. His thesis was ''The Cross-Product Calculus and Certain of its Applications in Geometry and Mechanics''. His work contributed to the development of
screw theory
Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawe ...
and kinematics. Kotelnikov began instructing at the university in 1893. His
habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...
thesis was ''The Projective Theory of Vectors'' (1899).
In
Kiev
Kyiv, also spelled Kiev, is the capital and most populous city of Ukraine. It is in north-central Ukraine along the Dnieper, Dnieper River. As of 1 January 2021, its population was 2,962,180, making Kyiv the List of European cities by populat ...
, Kotelnikov was professor and head of the department of pure mathematics until 1904. Returning to
Kazan
Kazan ( ; rus, Казань, p=kɐˈzanʲ; tt-Cyrl, Казан, ''Qazan'', IPA: ɑzan is the capital and largest city of the Republic of Tatarstan in Russia. The city lies at the confluence of the Volga and the Kazanka rivers, covering a ...
, he headed the mathematics department until 1914. He was at the
Kyiv Polytechnic Institute
)
, image = NTUU KPI logo.png
, image_size = 220px
, caption = Seal of the Kyiv Polytechnic Institute
, established = 1898
, students = 36,000 (approximately)
, admini ...
directing the department of Theoretical Mechanics until 1924, when he moved to
Moscow
Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 million ...
and took up teaching at
Bauman Technical University
The Bauman Moscow State Technical University, BMSTU (russian: link=no, Московский государственный технический университет им. Н. Э. Баумана (МГТУ им. Н. Э. Баумана)), some ...
.
In addition to the Works of Lobachevsky, Kotelnikov was also the editor of the collected works of
Nikolay Zhukovsky Nikolay Zhukovsky may refer to:
*Nikolay Zhukovsky (revolutionary) (1833–1895), Russian revolutionary
*Nikolay Zhukovsky (scientist)
Nikolay Yegorovich Zhukovsky ( rus, Никола́й Его́рович Жуко́вский, p=ʐʊˈkofskʲ ...
, the father of Russian
aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
.
One reviewer put Kotelnikov at the head of a chain of investigations of ''Spaces over Algebras''. Successive researchers included D.N. Zeiliger, A.P. Norden, and B. A. Rosenfel'd.
Dual quaternions
Kotelnikov advanced an algebraic method of representing Euclidean motions that had been introduced by
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) 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 ...
. Though developed to render motions in three-dimensional space, an eight-dimensional algebra of doubled
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
was used. Clifford had shown that a space of rotations entailed
elliptic space
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
described by
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s in his four-dimensional quaternions. According to
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry.
Education and career
Blaschke was the son of mathematician Josef Blaschke, who taugh ...
, it was Kotelnikov who initiated a "conversion principle" to take a dual rotation acting on elliptic space to a motion of
, three-dimensional Euclidean space:
If ''r'' is one of the square roots of minus one in
, then an underline (
) represents the elliptic line in the plane perpendicular to ''r'' (Blaschke: the united elliptic line). Using the inner product on
formed by taking the product of a quaternion with its conjugate, the condition
:
is equivalent to
:
and implies that elliptic lines
are perpendicular. Under these conditions, the Kotelnikov conversion to Euclidean motion is represented as
:
and where
is the
screw axis
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a scre ...
.
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry.
Education and career
Blaschke was the son of mathematician Josef Blaschke, who taugh ...
(1958) "Anwendung dualer Quaternionen auf Kinematik", Annales Academiae Scientiarum Fennicae
''Annales Fennici Mathematici'' (formerly ''Annales Academiæ Scientiarum Fennicæ Mathematica'' and ''Annales Academiæ Scientiarum Fennicæ'') is a peer-reviewed scientific journal published by the Finnish Academy of Science and Letters since 19 ...
(1958), 1-13; ''Gesammelte Werke'', volume 2
Applications of dual quaternions to kinematics
translation by D. H. Delphenich
Other works
* 1925: ''Introduction to Theoretical Mechanics'', Moscow-Leningrad
* 1927: ''The Principle of Relativity and Lobachevsky's Geometry'', Kazan
* 1950: ''The Theory of Vectors and Complex Numbers'', Moscow-Leningrad
References
Literature
* A.T. Grigorian (1976) "Aleksandr Petrovich Kotelnikov",
Dictionary of Scientific Biography
The ''Dictionary of Scientific Biography'' is a scholarly reference work that was published from 1970 through 1980 by publisher Charles Scribner's Sons, with main editor the science historian Charles Gillispie, from Princeton University. It consi ...
.
* B.L. Laptev & B.A. Rozenfel'd (1996) ''Mathematics of the 19th Century: Geometry'', page 87,
Birkhäuser Verlag
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields:
* Springer continues to publish science (particu ...
.
*
{{DEFAULTSORT:Kotelnikov, Aleksandr
1865 births
1944 deaths
19th-century mathematicians from the Russian Empire
20th-century Russian mathematicians
Mathematicians from Kazan
Academic staff of Bauman Moscow State Technical University
Academic staff of the D. Mendeleev University of Chemical Technology of Russia
Academic staff of Kazan Federal University
Academic staff of Kyiv Polytechnic Institute
Stalin Prize winners
Recipients of the Order of the Red Banner of Labour
Russian mathematicians
Soviet mathematicians