Thomas Kirkman
Encyclopedia
Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician
. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, and was listed by Alexander Macfarlane
as one of ten leading 19th-century British mathematicians. Kirkman's schoolgirl problem
, an existence theorem for Steiner triple systems that founded the field of combinatorial design
theory, is named after him.
, in the north west of England, the son of a local cotton dealer. In his schooling at the Bolton Grammar School, he studied classics, but no mathematics was taught in the school. He was recognized as the best scholar at the school, and the local vicar guaranteed him a scholarship at Cambridge, but his father would not allow him to go. Instead, he left school at age 14 to work in his father's office.
Nine years later, defying his father, he went to Trinity College Dublin, working as a private tutor to support himself during his studies. There, among other subjects, he first began learning mathematics. He earned a B.A. in 1833 and returned to England in 1835.
and became the curate
in Bury
and then in Lymm
. In 1839 he was invited to become rector
of Croft
and Southworth, a newly founded parish in Lancashire
, where he would stay for 52 years until his retirement in 1892. Theologically, Kirkman supported the anti-literalist
position of John William Colenso
, and was also strongly opposed to materialism
. He published many tracts and pamphlets on theology, as well as a book Philosophy Without Assumptions (1876).
Kirkman married Eliza Wright in 1841; they had seven children. To support them, Kirkman supplemented his income with tutoring, until Eliza inherited enough property to secure their living. The rectorship itself did not demand much from Kirkman, so from this point forward he had time to devote to mathematics.
Kirkman died 4 February 1895 in Bowdon
, not far from where he was born. His wife died ten days later.
. Despite Kirkman's and Woolhouse's contributions to the problem, Steiner triple systems were named after Jakob Steiner
who wrote a later paper in 1853. Kirkman's second research paper paper, in 1848, concerned hypercomplex number
s.
In 1848, Kirkman published First Mnemonical Lessons, a book on mathematical mnemonic
s for schoolchildren. It was not successful, and Augustus de Morgan
criticized it as "the most curious crochet I ever saw".
Next, in 1849, Kirkman studied the Pascal lines
determined by the intersection points of opposite sides of a hexagon inscribed within a conic section
. Any six points on a conic may be joined into a hexagon in 60 different ways, forming 60 different Pascal lines. Extending previous work of Steiner, Kirkman showed that these lines intersect in triples to form 60 points (now known as the Kirkman points), so that each line contains three of the points and each point lies on three of the lines. That is, these lines and points form a projective configuration of type 603603.
In 1850, Kirkman observed that his 1846 solution to Woolhouse's problem had an additional property, which he set out as a puzzle in the Lady's and Gentleman's Diary:
This problem became known as Kirkman's schoolgirl problem
, subsequently to become Kirkman's most famous result. He published several additional works on combinatorial design theory in later years.
Beginning in 1853, Kirkman began working on combinatorial enumeration problems concerning polyhedra
, beginning with a proof of Euler's formula
and concentrating on simple polyhedra (the polyhedra in which each vertex has three incident edges). He also studied Hamiltonian cycles in polyhedra, and provided an example of a polyhedron with no Hamiltonian cycle, prior to the work of William Rowan Hamilton
on the Icosian game
. He showed that every polyhedron can be generated from a pyramid by face-splitting and vertex-splitting operations, and he studied self-dual polyhedra
.
Kirkman was inspired to work in group theory
by a prize offered beginning in 1858 (but in the end never awarded) by the French Academy of Sciences
. His contributions in this area include an enumeration of the transitive group action
s on sets of up to ten elements. However, as with much of his work on polyhedra, Kirkman's work in this area was weighed down by newly invented terminology and, perhaps because of this, did not significantly influence later researchers.
In the early 1860s, Kirkman fell out with the mathematical establishment and in particular with Arthur Cayley
and James Joseph Sylvester
, over the poor reception of his works on polyhedra and groups and over issues of priority. Much of his later mathematical work was published (often in doggerel
) in the problem section of the Educational Times and in the obscure Proceedings of the Literary and Philosophical Society of Liverpool. However, in 1884 he began serious work on knot theory
, and with Peter Guthrie Tait
published an enumeration of the knots with up to ten crossings. He remained active in mathematics even after retirement, until his death in 1895.
for his research on quaternion
s and partitions. He was also an honorary member of the
Literary and Philosophical Society of Manchester and the Literary and Philosophical Society of Liverpool, and a foreign member of the Dutch Society of Science
.
Since 1994, the Institute of Combinatorics and its Applications
has handed out an annual Kirkman medal, named after Kirkman, to recognize outstanding combinatorial research by a mathematician within four years of receiving a doctorate.
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....
. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, and was listed by Alexander Macfarlane
Alexander Macfarlane
Alexander Macfarlane was a Nova Scotia lawyer and political figure. He was a member of the Canadian Senate from 1870 to 1898. His surname also appears as McFarlane in some sources....
as one of ten leading 19th-century British mathematicians. Kirkman's schoolgirl problem
Kirkman's schoolgirl problem
Kirkman's schoolgirl problem is a problem in combinatorics proposed by Thomas Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary...
, an existence theorem for Steiner triple systems that founded the field of combinatorial design
Combinatorial design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have specified numerical properties....
theory, is named after him.
Early life and education
Kirkman was born 31 March 1806 in BoltonBolton
Bolton is a town in Greater Manchester, in the North West of England. Close to the West Pennine Moors, it is north west of the city of Manchester. Bolton is surrounded by several smaller towns and villages which together form the Metropolitan Borough of Bolton, of which Bolton is the...
, in the north west of England, the son of a local cotton dealer. In his schooling at the Bolton Grammar School, he studied classics, but no mathematics was taught in the school. He was recognized as the best scholar at the school, and the local vicar guaranteed him a scholarship at Cambridge, but his father would not allow him to go. Instead, he left school at age 14 to work in his father's office.
Nine years later, defying his father, he went to Trinity College Dublin, working as a private tutor to support himself during his studies. There, among other subjects, he first began learning mathematics. He earned a B.A. in 1833 and returned to England in 1835.
Ordination and ministry
On his return to England, Kirkman was ordained into the ministry of the Church of EnglandChurch of England
The Church of England is the officially established Christian church in England and the Mother Church of the worldwide Anglican Communion. The church considers itself within the tradition of Western Christianity and dates its formal establishment principally to the mission to England by St...
and became the curate
Curate
A curate is a person who is invested with the care or cure of souls of a parish. In this sense "curate" correctly means a parish priest but in English-speaking countries a curate is an assistant to the parish priest...
in Bury
Bury
Bury is a town in Greater Manchester, England. It lies on the River Irwell, east of Bolton, west-southwest of Rochdale, and north-northwest of the city of Manchester...
and then in Lymm
Lymm
Lymm is a large village and civil parish within the Warrington borough of Cheshire, in North West England. Lymm was an urban district of Cheshire from 1894 to 1974. The civil parish of Lymm incorporates the hamlets of Booths Hill, Broomedge, Church Green, Deansgreen, Heatley, Heatley Heath, Little...
. In 1839 he was invited to become rector
Rector
The word rector has a number of different meanings; it is widely used to refer to an academic, religious or political administrator...
of Croft
Croft, Cheshire
Croft is a village and civil parish in Warrington, Cheshire, England, about five miles north of Warrington town centre. According to the 2001 Census it had a population of 2,920. It is a rural, residential district....
and Southworth, a newly founded parish in Lancashire
Lancashire
Lancashire is a non-metropolitan county of historic origin in the North West of England. It takes its name from the city of Lancaster, and is sometimes known as the County of Lancaster. Although Lancaster is still considered to be the county town, Lancashire County Council is based in Preston...
, where he would stay for 52 years until his retirement in 1892. Theologically, Kirkman supported the anti-literalist
Biblical literalism
Biblical literalism is the interpretation or translation of the explicit and primary sense of words in the Bible. A literal Biblical interpretation is associated with the fundamentalist and evangelical hermeneutical approach to Scripture, and is used almost exclusively by conservative Christians...
position of John William Colenso
John William Colenso
John William Colenso , first Anglican bishop of Natal, mathematician, theologian, Biblical scholar and social activist.-Biography:Colenso was born at St Austell, Cornwall, on 24 January 1814...
, and was also strongly opposed to materialism
Materialism
In philosophy, the theory of materialism holds that the only thing that exists is matter; that all things are composed of material and all phenomena are the result of material interactions. In other words, matter is the only substance...
. He published many tracts and pamphlets on theology, as well as a book Philosophy Without Assumptions (1876).
Kirkman married Eliza Wright in 1841; they had seven children. To support them, Kirkman supplemented his income with tutoring, until Eliza inherited enough property to secure their living. The rectorship itself did not demand much from Kirkman, so from this point forward he had time to devote to mathematics.
Kirkman died 4 February 1895 in Bowdon
Bowdon, Greater Manchester
Bowdon is a suburban village and electoral ward in the Altrincham area of the Metropolitan Borough of Trafford, in Greater Manchester, England.-History:...
, not far from where he was born. His wife died ten days later.
Mathematics
Kirkman's first mathematical publication was in the Cambridge and Dublin Mathematical Journal in 1846, on a problem involving Steiner triple systems that had been published two years earlier in the Lady's and Gentleman's Diary by Wesley S. B. WoolhouseWesley S. B. Woolhouse
Wesley Stoker Barker Woolhouse was an actuary with diverse interests in music theory, the design of steam locomotives, measurements, and many other fields, publishing books in all these fields....
. Despite Kirkman's and Woolhouse's contributions to the problem, Steiner triple systems were named after Jakob Steiner
Jakob Steiner
Jakob Steiner was a Swiss mathematician who worked primarily in geometry.-Personal and professional life:...
who wrote a later paper in 1853. Kirkman's second research paper paper, in 1848, concerned hypercomplex number
Hypercomplex number
In mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...
s.
In 1848, Kirkman published First Mnemonical Lessons, a book on mathematical mnemonic
Mnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...
s for schoolchildren. It was not successful, and Augustus de Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....
criticized it as "the most curious crochet I ever saw".
Next, in 1849, Kirkman studied the Pascal lines
Pascal's theorem
In projective geometry, Pascal's theorem states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration.- Related results :This theorem...
determined by the intersection points of opposite sides of a hexagon inscribed within a conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
. Any six points on a conic may be joined into a hexagon in 60 different ways, forming 60 different Pascal lines. Extending previous work of Steiner, Kirkman showed that these lines intersect in triples to form 60 points (now known as the Kirkman points), so that each line contains three of the points and each point lies on three of the lines. That is, these lines and points form a projective configuration of type 603603.
In 1850, Kirkman observed that his 1846 solution to Woolhouse's problem had an additional property, which he set out as a puzzle in the Lady's and Gentleman's Diary:
This problem became known as Kirkman's schoolgirl problem
Kirkman's schoolgirl problem
Kirkman's schoolgirl problem is a problem in combinatorics proposed by Thomas Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary...
, subsequently to become Kirkman's most famous result. He published several additional works on combinatorial design theory in later years.
Beginning in 1853, Kirkman began working on combinatorial enumeration problems concerning polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
, beginning with a proof of Euler's formula
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
and concentrating on simple polyhedra (the polyhedra in which each vertex has three incident edges). He also studied Hamiltonian cycles in polyhedra, and provided an example of a polyhedron with no Hamiltonian cycle, prior to the work of William Rowan Hamilton
William 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...
on the Icosian game
Icosian game
The icosian game is a mathematical game invented in 1857 by William Rowan Hamilton. The game's object is finding a Hamiltonian cycle along the edges of a dodecahedron such that every vertex is visited a single time, no edge is visited twice, and the ending point is the same as the starting point...
. He showed that every polyhedron can be generated from a pyramid by face-splitting and vertex-splitting operations, and he studied self-dual polyhedra
Dual graph
In mathematics, the dual graph of a given planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge in G joining two neighboring regions, for a certain embedding of G. The term "dual" is used because this property is symmetric, meaning that if H is a dual...
.
Kirkman was inspired to work in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
by a prize offered beginning in 1858 (but in the end never awarded) by the French Academy of Sciences
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...
. His contributions in this area include an enumeration of the transitive group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
s on sets of up to ten elements. However, as with much of his work on polyhedra, Kirkman's work in this area was weighed down by newly invented terminology and, perhaps because of this, did not significantly influence later researchers.
In the early 1860s, Kirkman fell out with the mathematical establishment and in particular with Arthur Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
and 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...
, over the poor reception of his works on polyhedra and groups and over issues of priority. Much of his later mathematical work was published (often in doggerel
Doggerel
Doggerel is a derogatory term for verse considered of little literary value. The word probably derived from dog, suggesting either ugliness, puppyish clumsiness, or unpalatability in the 1630s.-Variants:...
) in the problem section of the Educational Times and in the obscure Proceedings of the Literary and Philosophical Society of Liverpool. However, in 1884 he began serious work on knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, and with Peter Guthrie Tait
Peter Guthrie Tait
Peter Guthrie Tait FRSE was a Scottish mathematical physicist, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with Kelvin, and his early investigations into knot theory, which contributed to the eventual formation of topology as a mathematical...
published an enumeration of the knots with up to ten crossings. He remained active in mathematics even after retirement, until his death in 1895.
Awards and honours
In 1857, Kirkman was elected as a fellow of the Royal SocietyRoyal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
for his research on 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 partitions. He was also an honorary member of the
Literary and Philosophical Society of Manchester and the Literary and Philosophical Society of Liverpool, and a foreign member of the Dutch Society of Science
Koninklijke Hollandsche Maatschappij der Wetenschappen
The Koninklijke Hollandsche Maatschappij der Wetenschappen , located on the east side of the Spaarne in downtown Haarlem, Netherlands, was established in 1752 and is the oldest society for the sciences in the coutnry. The society has been housed in its present location, called Hodshon Huis, since...
.
Since 1994, the Institute of Combinatorics and its Applications
Institute of Combinatorics and its Applications
The Institute of Combinatorics and its Applications is an international scientific organization. It was formed in 1990 and is based in Winnipeg, Canada.-Aim and membership:...
has handed out an annual Kirkman medal, named after Kirkman, to recognize outstanding combinatorial research by a mathematician within four years of receiving a doctorate.