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67-558: Hamiltonian paths and cycles are named after William Rowan Hamilton , who invented the icosian game , now also known as Hamilton's puzzle , which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron . Hamilton solved this problem using the icosian calculus , an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hamilton). This solution does not generalize to arbitrary graphs. Despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied

134-823: A Latin copy of Euclid ; and at twelve he studied Newton 's Arithmetica Universalis . By age 16, he had covered much of the Principia , as well as some more recent works on analytic geometry and differential calculus . In mid-1822, Hamilton began a systematic study of Laplace 's Mécanique Céleste . During this period, he encountered what he believed to be a logical error in Mécanique Céleste , an observation which led Hamilton to be introduced to John Brinkley , then Royal Astronomer of Ireland. In November and December of 1822, he completed his first three original mathematical papers. On his first visit to Dunsink Observatory, he showed two of them to Brinkley, who requested that

201-426: A Hamiltonian cycle is called a Hamiltonian graph . Similar notions may be defined for directed graphs , where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices are connected with arrows and the edges traced "tail-to-head"). A Hamiltonian decomposition is an edge decomposition of a graph into Hamiltonian circuits. A Hamilton maze is a type of logic puzzle in which

268-409: A Hamiltonian path if, for every non-adjacent vertex pairs the sum of their degrees and their shortest path length is greater than n . The above theorem can only recognize the existence of a Hamiltonian path in a graph and not a Hamiltonian Cycle. Many of these results have analogues for balanced bipartite graphs , in which the vertex degrees are compared to the number of vertices on a single side of

335-461: A basic understanding of them. At the age of seven, he had already made progress in Hebrew , and before he was 13, he had acquired, under the care of his uncle, a dozen languages: classical and modern European languages, Persian , Arabic , Hindustani , Sanskrit , Marathi and Malay . The emphasis of Hamilton's early education on languages is attributed to the wish of his father to see him employed by

402-400: A closed walk passing through each edge of G exactly once. This tour corresponds to a Hamiltonian cycle in the line graph L ( G ) , so the line graph of every Eulerian graph is Hamiltonian. Line graphs may have other Hamiltonian cycles that do not correspond to Euler tours, and in particular the line graph L ( G ) of every Hamiltonian graph G is itself Hamiltonian, regardless of whether

469-642: A corresponding member of the Saint Petersburg Academy of Sciences . Later, in 1864, the newly established United States National Academy of Sciences elected its first Foreign Associates, and decided to put Hamilton's name on top of their list. A plaque under the Broom Bridge, associated with the discovery of quaternions, was unveiled by Éamon de Valera on 13 November 1958. Since 1989, the National University of Ireland, Maynooth , has organised

536-465: A deeper mathematical structure than had been previously understood, in particular a symmetry between momentum and position. The credit for discovering what are now called the Lagrangian and Lagrange's equations also belongs to Hamilton. Both the Lagrangian mechanics and Hamiltonian approaches have proven important in the study of continuous classical systems in physics, and quantum mechanical systems:

603-403: A graph as being ( k , l ) -sparse if every nonempty subgraph with n vertices has at most kn − l edges, and ( k , l ) -tight if it is ( k , l ) -sparse and has exactly kn − l edges. Thus trees are exactly the (1,1) -tight graphs, forests are exactly the (1,1) -sparse graphs, and graphs with arboricity k are exactly the ( k , k ) -sparse graphs. Pseudoforests are exactly

670-405: A graph is Hamiltonian if it has enough edges . The Bondy–Chvátal theorem operates on the closure cl( G ) of a graph G with n vertices, obtained by repeatedly adding a new edge uv connecting a nonadjacent pair of vertices u and v with deg( v ) + deg( u ) ≥ n until no more pairs with this property can be found. Bondy–Chvátal Theorem (1976)  —  A graph

737-687: A more restrictive definition of sparse is used like | E | = O ( | V | log ⁡ | V | ) {\displaystyle |E|=O(|V|\log |V|)} or even | E | = O ( | V | ) {\displaystyle |E|=O(|V|)} . Upper density is an extension of the concept of graph density defined above from finite graphs to infinite graphs. Intuitively, an infinite graph has arbitrarily large finite subgraphs with any density less than its upper density, and does not have arbitrarily large finite subgraphs with density greater than its upper density. Formally,

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804-775: A party with John Marshall and his family, he stayed at Dunsink with Hamilton. On a second tour in England with Nimmo in 1831, Hamilton parted from him at Birmingham , to visit the Lawrence sisters and family on his mother's side in the Liverpool area. They met up again in the Lake District , where they climbed Helvellyn and had tea with Wordsworth. Hamilton returned to Dublin, via Edinburgh and Glasgow. Hamilton visited Samuel Taylor Coleridge at Highgate , in 1832, helped by an unexpected letter of introduction given to him by Sarah Lawrence on

871-598: A pilgrimage called the Hamilton Walk , in which mathematicians take a walk from Dunsink Observatory to the bridge, where no trace of the carving remains, though a stone plaque does commemorate the discovery. The Hamilton Institute is an applied mathematics research institute at Maynooth University and the Royal Irish Academy holds an annual public Hamilton lecture at which Murray Gell-Mann , Frank Wilczek , Andrew Wiles and Timothy Gowers have all spoken. 2005

938-435: A poet. In 1825, Hamilton met Arabella Lawrence , younger sister of Sarah Lawrence , a significant correspondent and frank critic of his poetry. It was a contact he made through Maria Edgeworth's circle. Hamilton, now Royal Astronomer of Ireland, took up residence at Dunsink Observatory where he spent the rest of his life. He was there from 1827 until his death in 1865. In his early years at Dunsink, Hamilton observed

1005-484: A quaternion as an ordered four-element multiple of real numbers, and described the first element as the "scalar" part, and the remaining three as the "vector" part. He coined the neologisms "tensor" and "scalar", and was the first to use the word "vector" in the modern sense. Hamilton looked into the solution of the quintic in the theory of equations , examining the results arrived at by Niels Henrik Abel , George Jerrard and others in their respective research. There

1072-442: A theory of a single function, now known as Hamilton's principal function , that brings together mechanics and optical theory. It helped to establish the foundations of the wave theory of light in mathematical physics . He proposed it when he first predicted its existence in the third supplement to his Systems of Rays , read in 1832. The Royal Irish Academy paper was finally entitled Theory of Systems of Rays (23 April 1827), and

1139-586: A visit to Liverpool in March of that year. He also paid a call, with Arabella, on the family of William Roscoe , who had died in 1831. He was a Christian, described as "a lover of the Bible, an orthodox and attached member of the Established Church", and as having a "profound conviction of the truth of revealed religion". While attending Trinity College, Hamilton proposed to his friend's sister, whose refusal drove

1206-608: A year earlier by Thomas Kirkman , who, in particular, gave an example of a polyhedron without Hamiltonian cycles. Even earlier, Hamiltonian cycles and paths in the knight's graph of the chessboard , the knight's tour , had been studied in the 9th century in Indian mathematics by Rudrata , and around the same time in Islamic mathematics by al-Adli ar-Rumi . In 18th century Europe, knight's tours were published by Abraham de Moivre and Leonhard Euler . A Hamiltonian path or traceable path

1273-517: Is Hamilton's paper on fluctuating functions in Fourier analysis , and the invention of the hodograph . Of his investigations into the solutions, especially by numerical approximation , of certain classes of physically-important differential equations, only parts were published, at intervals, in the Philosophical Magazine . Hamilton also introduced the icosian game or Hamilton's puzzle . It

1340-665: Is Hamiltonian if and only if its closure is Hamiltonian. As complete graphs are Hamiltonian, all graphs whose closure is complete are Hamiltonian, which is the content of the following earlier theorems by Dirac and Ore. Dirac's Theorem (1952)  —  A simple graph with n vertices ( n ≥ 3 {\displaystyle n\geq 3} ) is Hamiltonian if every vertex has degree n 2 {\displaystyle {\tfrac {n}{2}}} or greater. Ore's Theorem (1960)  —  A simple graph with n vertices ( n ≥ 3 {\displaystyle n\geq 3} )

1407-481: Is Hamiltonian if the sum of full degrees of every pair of distinct non-adjacent vertices is greater than or equal to 2 n − 1 {\displaystyle 2n-1} The number of vertices must be doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph. Rahman– Kaykobad (2005)  —  A simple graph with n vertices has

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1474-468: Is Hamiltonian if, for every pair of non-adjacent vertices, the sum of their degrees is n or greater. The following theorems can be regarded as directed versions: Ghouila–Houiri (1960)  —  A strongly connected simple directed graph with n vertices is Hamiltonian if every vertex has a full degree greater than or equal to n . Meyniel (1973)  —  A strongly connected simple directed graph with n vertices

1541-405: Is a path that visits each vertex of the graph exactly once. A graph that contains a Hamiltonian path is called a traceable graph . A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle , Hamiltonian circuit , vertex tour or graph cycle is a cycle that visits each vertex exactly once. A graph that contains

1608-429: Is a prediction for transparent biaxial crystals (i.e. monoclinic , orthorhombic or triclinic crystals). A ray of light entering such a crystal at a certain angle would emerge as a hollow cone of rays. This discovery was known as conical refraction . Hamilton found it from the geometry of the wave surface introduced by Augustin-Jean Fresnel , which has singular point . There is a basic mathematical explanation of

1675-415: Is based on the concept of a Hamiltonian path in graph theory . Hamilton introduced, as a method of analysis, both quaternions and biquaternions , the extension to eight dimensions by the introduction of complex number coefficients . When his work was assembled in 1853, the book Lectures on Quaternions had "formed the subject of successive courses of lectures, delivered in 1848 and subsequent years, in

1742-442: Is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph . The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem. The graph density of simple graphs is defined to be the ratio of the number of edges | E | with respect to

1809-485: Is fundamental to modern theoretical physics , particularly his reformulation of Newtonian mechanics . Hamiltonian mechanics including its Hamilitonian function are now central both to electromagnetism and quantum mechanics . Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819), who lived in Dublin at 29 Dominick Street , later renumbered to 36. Hamilton's father, who

1876-470: The (1,0) -sparse graphs, and the Laman graphs arising in rigidity theory are exactly the (2,3) -tight graphs. Other graph families not characterized by their sparsity can also be described in this way. For instance the facts that any planar graph with n vertices has at most 3 n – 6 edges (except for graphs with fewer than 3 vertices), and that any subgraph of a planar graph is planar, together imply that

1943-666: The British East India Company . An expert mental calculator , the young Hamilton was capable of working out some calculations to many decimal places. In September 1813, the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, a year older than Hamilton. The two were pitted against each other in a mental arithmetic contest, with Colburn emerging as the clear victor. In reaction to his defeat, Hamilton spent less time studying languages, and more on mathematics. At age ten, he stumbled across

2010-606: The Central Bank of Ireland in 2005 to commemorate 200 years since his birth. It is believed by some modern mathematicians that Hamilton's work on quaternions was satirized by Charles Lutwidge Dodgson in Alice in Wonderland . In particular, the Mad Hatter's tea party was meant to represent the folly of quaternions and the need to revert to Euclidean geometry . In September 2022 evidence

2077-560: The Hamiltonian cycle polynomial of its weighted adjacency matrix defined as the sum of the products of the arc weights of the digraph's Hamiltonian cycles. This polynomial is not identically zero as a function in the arc weights if and only if the digraph is Hamiltonian. The relationship between the computational complexities of computing it and computing the permanent was shown by Grigoriy Kogan. William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865)

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2144-578: The Royal Medal of the Royal Society the following year. He was to win it again in 1848. In 1835, being secretary to the meeting of the British Association which was held that year in Dublin, Hamilton was knighted by the lord-lieutenant . Other honours rapidly succeeded, among which his election in 1837 to the president's chair in the Royal Irish Academy , and the rare distinction of being made

2211-557: The Halls of Trinity College, Dublin". Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research. When he died, Hamilton was working on a definitive statement of quaternion science. His son William Edwin Hamilton brought the Elements of Quaternions , a hefty volume of 762 pages, to publication in 1866. As copies ran short, a second edition

2278-453: The bipartition rather than the number of vertices in the whole graph. Theorem  —  A 4-connected planar triangulation has a Hamiltonian cycle. Theorem  —  A 4-connected planar graph has a Hamiltonian cycle. An algebraic representation of the Hamiltonian cycles of a given weighted digraph (whose arcs are assigned weights from a certain ground field) is

2345-718: The first part was printed in 1828 in the Transactions of the Royal Irish Academy . The more important contents of the second and third parts appeared in the three voluminous supplements (to the first part) which were published in the same Transactions, and in the two papers On a General Method in Dynamics , which appeared in the Philosophical Transactions in 1834 and 1835. In these papers, Hamilton developed his central principle of "Varying Action". A result of this work

2412-558: The goal is to find the unique Hamiltonian cycle in a given graph. Any Hamiltonian cycle can be converted to a Hamiltonian path by removing one of its edges, but a Hamiltonian path can be extended to a Hamiltonian cycle only if its endpoints are adjacent. All Hamiltonian graphs are biconnected , but a biconnected graph need not be Hamiltonian (see, for example, the Petersen graph ). An Eulerian graph G (a connected graph in which every vertex has even degree) necessarily has an Euler tour,

2479-506: The graph G is Eulerian. A tournament (with more than two vertices) is Hamiltonian if and only if it is strongly connected . The number of different Hamiltonian cycles in a complete undirected graph on n vertices is ⁠ ( n – 1)! / 2 ⁠ and in a complete directed graph on n vertices is ( n – 1)! . These counts assume that cycles that are the same apart from their starting point are not counted separately. The best vertex degree characterization of Hamiltonian graphs

2546-611: The graphs in the family are all ( k , l ) -sparse is equivalent to the graphs in the family having bounded degeneracy or having bounded arboricity . More precisely, it follows from a result of Nash-Williams (1964) that the graphs of arboricity at most a are exactly the ( a , a ) -sparse graphs. Similarly, the graphs of degeneracy at most d are ( d , ( d + 1 2 ) ) {\displaystyle \left(d,{\binom {d+1}{2}}\right)} -sparse graphs ( Lick & White 1970 ). Nešetřil & Ossona de Mendez (2010) considered that

2613-512: The heavens quite regularly; He left routine observation to his assistant Charles Thompson. Hamilton's sisters also supported the observatory's work. The introductory lectures by Hamilton in astronomy were celebrated; in addition to his students, they attracted scholars, poets, and women. Felicia Hemans wrote her poem The Prayer of the Lonely Student after hearing one of his lectures. Hamilton invited his four sisters to come and live at

2680-502: The last, and continued the task of finishing the Elements of Quaternions which had occupied the last six years of his life. He died on 2 September 1865, following a severe attack of gout . He is buried in Mount Jerome Cemetery in Dublin. Hamilton made outstanding contributions to classical mechanics and optics . His first discovery was in an early paper that he communicated in 1823 to John Brinkley, who presented it under

2747-408: The maximal density is 1 (for complete graphs ) and the minimal density is 0 ( Coleman & Moré 1983 ). For families of graphs of increasing size, one often calls them sparse if D → 0 {\displaystyle D\rightarrow 0} as | V | → ∞ {\displaystyle |V|\rightarrow \infty } . Sometimes, in computer science ,

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2814-619: The maximum possible edges. For undirected simple graphs , the graph density is: For directed , simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is: where E is the number of edges and V is the number of vertices in the graph. The maximum number of edges for an undirected graph is ( | V | 2 ) = | V | ( | V | − 1 ) 2 {\displaystyle {\binom {|V|}{2}}={\frac {|V|(|V|-1)}{2}}} , so

2881-599: The most natural medium for their description and development, and the function that is now universally known as the Hamiltonian, is the starting point for calculation in almost any area of physics. Many scientists, including Liouville , Jacobi , Darboux , Poincaré , Kolmogorov , Prigogine and Arnold , have extended Hamilton's work, in mechanics , differential equations and symplectic geometry . Hamilton's mathematical studies seem to have been undertaken and carried to their full development without collaboration, and his writings do not belong to any particular school. He

2948-537: The observatory in 1827, and they ran the household until his marriage in 1833. They included Eliza Mary Hamilton (1807–1851), the poet. In 1827, Hamilton wrote to his sister Grace about "some of" the Lawrence sisters having met his sister Eliza in Dublin. Newly appointed to the Observatory, Hamilton set off on a tour in Ireland and England with Alexander Nimmo , who was coaching him on latitude and longitude . One call

3015-457: The papers be developed further. Hamilton complied, and early in 1823, Brinkley approved the amended version. In July of 1823, Hamilton earned a place at Trinity College Dublin by examination, at age 17. His tutor there was Charles Boyton , a family friend, who brought to his attention the contemporary mathematics published by the group at the École Polytechnique in Paris. John Brinkley remarked of

3082-402: The phenomenon, namely that the wave surface is not the boundary of a convex body. A fuller understanding awaited the microlocal analysis of the middle of the 20th century, The step from optics to dynamics in the application of the method of "Varying Action" was made in 1827, and communicated to the Royal Society, in whose Philosophical Transactions for 1834 and 1835 there are two papers on

3149-403: The planar graphs are (3,6) -sparse. However, not every (3,6) -sparse graph is planar. Similarly, outerplanar graphs are (2,3) -sparse and planar bipartite graphs are (2,4) -sparse. Streinu and Theran show that testing ( k , l ) -sparsity may be performed in polynomial time when k and l are integers and 0 ≤ l < 2 k . For a graph family, the existence of k and l such that

3216-484: The precocious Hamilton, "This young man, I do not say will be , but is , the first mathematician of his age." The college awarded Hamilton two optime s, or off-the-chart grades, in Greek and in physics. He was first in every subject and at every examination. He aimed to win a Trinity College fellowship by competitive examination, but this did not happen. Instead, after Brinkley in 1826 was made Bishop of Cloyne , Hamilton

3283-405: The side of the nearby Broom Bridge (which Hamilton called Brougham Bridge). The quaternions involved abandoning the commutative law , a radical step for the time. In the context of this prototype geometric algebra , Hamilton also introduced the cross and dot products of vector algebra, the quaternion product being the cross product minus the dot product as scalar . Hamilton also described

3350-430: The sparsity/density dichotomy makes it necessary to consider infinite graph classes instead of single graph instances. They defined somewhere dense graph classes as those classes of graphs for which there exists a threshold t such that every complete graph appears as a t -subdivision in a subgraph of a graph in the class. To the contrary, if such a threshold does not exist, the class is nowhere dense . Properties of

3417-484: The subject. Hamiltonian mechanics was a powerful new technique for working with equations of motion . Hamilton's advances enlarged the class of mechanical problems that could be solved. His principle of "Varying Action" was based on the calculus of variations , in the general class of problems included under the principle of least action which had been studied earlier by Pierre Louis Maupertuis , Euler , Joseph Louis Lagrange and others. Hamilton's analysis uncovered

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3484-462: The techniques find use in electromagnetism , quantum mechanics , relativity theory and quantum field theory . In the Dictionary of Irish Biography David Spearman writes: The formulation that he devised for classical mechanics proved to be equally suited to quantum theory, whose development it facilitated. The Hamiltonian formalism shows no signs of obsolescence; new ideas continue to find this

3551-399: The title of Caustics in 1824 to the Royal Irish Academy . It was referred as usual to a committee, which recommended further development and simplification before publication. Between 1825 and 1828 the paper was expanded, and became a clearer exposition of a novel method. Over this period, Hamilton gained an appreciation for the nature and importance of optics. In 1827, Hamilton presented

3618-609: The upper density of a graph G is the infimum of the values α such that the finite subgraphs of G with density α have a bounded number of vertices. It can be shown using the Erdős–Stone theorem that the upper density can only be 1 or one of the superparticular ratios 0, ⁠ 1 / 2 ⁠ , ⁠ 2 / 3 ⁠ , ⁠ 3 / 4 ⁠ , ⁠ 4 / 5 ⁠ , … ⁠ n / n + 1 ⁠ (see, e.g., Diestel, edition 5, p. 189). Lee & Streinu (2008) and Streinu & Theran (2009) define

3685-577: The young Hamilton to depression and illness, even to the verge of suicide. He proposed again in 1831 to Ellen de Vere, a sister of the poet Aubrey De Vere , who declined as well. Hamilton eventually married Helen Marie Bayly in 1833, a country preacher's daughter, and had three children with her: William Edwin Hamilton (born 1834), Archibald Henry (born 1835), and Helen Elizabeth (born 1840). Hamilton's married life turned out to be difficult and unhappy as Bayly proved to be pious, shy, timid, and chronically ill. Hamilton retained his faculties unimpaired to

3752-477: Was an Irish mathematician , physicist and astronomer . He was Andrews Professor of Astronomy at Trinity College Dublin . Hamilton was Dunsink's third director, having worked there from 1827 to 1865. His career included the study of geometrical optics , Fourier analysis , and quaternions , the last of which made him one of the founders of modern linear algebra . He has made major contributions in optics, classical mechanics , and abstract algebra . His work

3819-510: Was appointed in 1827 to the vacant posts left by Brinkley's departure, Andrews Professor of Astronomy and Royal Astronomer of Ireland. Despite having his undergraduate career cut short in this way, he earned degrees in both classics and mathematics (BA in 1827, MA in 1837). In 1824, Hamilton was introduced at Edgeworthstown to the novelist Maria Edgeworth , by the Rev. Richard Butler , the vicar of Trim, County Meath to whom his uncle James Hamilton

3886-448: Was away from Dunsink, staying with sisters, for much of the time from 1840 to 1842. Hamilton's married life was reportedly difficult. In the troubled period of the early 1840s, his sister Sydney ran his household; when Helen returned, he was happier after some depression. Dense graph In mathematics , a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices

3953-773: Was curate. During the same period, his uncle introduced him to the Disney family at Summerhill House , County Meath. The Disney sons attended Trinity College, and Hamilton had friends among them. At Summerhill, he met Catherine Disney, their sister. Hamilton was attracted to Catherine Disney, but her family did not approve and Catherine was required to marry the Rev. William Barlow, a brother of her elder sister's husband. The wedding took place in 1825. Hamilton wrote in 1826 about his feelings for her in an extended poem, "The Enthusiast". Over twenty years later, in 1847, he confided in John Herschel that during this period he might have become

4020-558: Was from Dublin, worked as a solicitor. By the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim , County Meath. Hamilton is said to have shown talent at an early age. His uncle observed that Hamilton, from a young age, had displayed an uncanny ability to acquire languages — a claim which has been disputed by some historians, who claim he had only

4087-520: Was intended by the university authorities who elected him to the Professorship of Astronomy to spend his time as he best could for the advancement of science, without restrictions. Hamilton made his discovery of the algebra of quaternions in 1843. Among much prior related work, in 1840 Benjamin Olinde Rodrigues had reached a result that amounted to their discovery in all but name. Hamilton

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4154-515: Was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional Argand diagram ) to higher spatial dimensions. In working with four dimensions, rather than three, he created quaternion algebra. According to Hamilton, on 16 October he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation occurred to him; Hamilton then carved this equation using his penknife into

4221-460: Was prepared by Charles Jasper Joly , when the book was split into two volumes, the first appearing in 1899 and the second in 1901. The subject index and footnotes in this second edition improved the Elements accessibility. Hamilton was twice awarded the Cunningham Medal of the Royal Irish Academy . The first award, in 1834, was for his work on conical refraction, for which he also received

4288-573: Was presented to counter this suggestion, which appears to have been based on an incorrect understanding of both quaternions and their history. Hamilton married Helen Bayly, daughter of Rev Henry Bayly, Rector of Nenagh, County Tipperary, in 1833; she was a sister of neighbours to the observatory. They had three children: the journalist William Edwin Hamilton (born 1834), Archibald Henry (born 1835) and Helen Eliza Amelia (born 1840). Helen stayed with her widowed mother at Bayly Farm, Nenagh for extended periods, until her mother's death in 1837. She also

4355-525: Was provided in 1972 by the Bondy – Chvátal theorem, which generalizes earlier results by G. A. Dirac (1952) and Øystein Ore . Both Dirac's and Ore's theorems can also be derived from Pósa's theorem (1962). Hamiltonicity has been widely studied with relation to various parameters such as graph density , toughness , forbidden subgraphs and distance among other parameters. Dirac and Ore's theorems basically state that

4422-581: Was the 200th anniversary of Hamilton's birth and the Irish government designated that the Hamilton Year, celebrating Irish science . Trinity College Dublin marked the year by launching the Hamilton Mathematics Institute . Two commemorative stamps were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions. A 10- euro commemorative silver proof coin was issued by

4489-464: Was to Sarah Lawrence's school at Gateacre , near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes. They visited William Wordsworth at Rydal Mount in September of that year, where the writer Caesar Otway was also present. After the visit, Hamilton sent numerous poems to Wordsworth, becoming a "poetic disciple". When Wordsworth visited Dublin in the summer of 1829, in

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