Arthur Cayley
Arthur Cayley was a British mathematician. He helped; as a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, where he excelled in Greek, French and Italian, as well as mathematics, he worked as a lawyer for 14 years. He postulated the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, verified it for matrices of order 2 and 3, he was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws. When mathematicians spoke of "groups", they had meant permutation groups. Cayley tables and Cayley graphs as well. Arthur Cayley was born in Richmond, England, on 16 August 1821, his father, Henry Cayley, was a distant cousin of Sir George Cayley, the aeronautics engineer innovator, descended from an ancient Yorkshire family. He settled in Russia, as a merchant, his mother was daughter of William Doughty. According to some writers she was Russian, his brother was the linguist Charles Bagot Cayley.
Arthur spent his first eight years in Saint Petersburg. In 1829 his parents were settled permanently near London. Arthur was sent to a private school. At age 14 he was sent to King's College School; the school's master observed indications of mathematical genius and advised the father to educate his son not for his own business, as he had intended, but to enter the University of Cambridge. At the unusually early age of 17 Cayley began residence at Cambridge; the cause of the Analytical Society had now triumphed, the Cambridge Mathematical Journal had been instituted by Gregory and Robert Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects, suggested by reading the Mécanique analytique of Lagrange and some of the works of Laplace. Cayley's tutor at Cambridge was George Peacock and his private coach was William Hopkins, he finished his undergraduate course by winning the place of Senior Wrangler, the first Smith's prize. His next step was to take the M.
A. degree, win a Fellowship by competitive examination. He continued to reside at Cambridge University for four years; because of the limited tenure of his fellowship it was necessary to choose a profession. He made a specialty of conveyancing, it was while he was a pupil at the bar examination that he went to Dublin to hear Hamilton's lectures on quaternions. His friend J. J. Sylvester, his senior by five years at Cambridge, was an actuary, resident in London. During this period of his life, extending over fourteen years, Cayley produced between two and three hundred papers. At Cambridge University the ancient professorship of pure mathematics is denominated by the Lucasian, is the chair, occupied by Isaac Newton. Around 1860, certain funds bequeathed by Lady Sadleir to the University, having become useless for their original purpose, were employed to establish another professorship of pure mathematics, called the Sadleirian; the duties of the new professor were defined to be "to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science."
To this chair Cayley was elected. He gave up a lucrative practice for a modest salary, he at once settled down in Cambridge. More fortunate than Hamilton in his choice, his home life was one of great happiness, his friend and fellow investigator, once remarked that Cayley had been much more fortunate than himself. At first the teaching duty of the Sadleirian professorship was limited to a course of lectures extending over one of the terms of the academic year. For many years the attendance was small, came entirely from those who had finished their career of preparation for competitive examinations; the subject lectured on was that of the memoir on which the professor was for the time engaged. The other duty of the chair — the advancement of mathematical science — was discharged in a handsome manner by the long series of memoirs that he published, ranging over every department of pure mathematics, but it was discharged in a much less obtrusive way. In 1872 he was made an honorary fellow of Trinity College, three years an ordinary fellow, which meant stipend as well as honour.
About this time his friends subscribed for a presentation portrait. Maxwell wrote an address to the committee of subscribers; the verses refer to the subjects investigated in several of Cay
Regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used. A regular polyhedron is identified by its Schläfli symbol of the form, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra, four regular star polyhedra, making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. There are five convex regular polyhedra, known as the Platonic solids, four regular star polyhedra, the Kepler–Poinsot polyhedra, five regular compounds of regular polyhedra: The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition: The vertices of the polyhedron all lie on a sphere. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons.
All the solid angles of the polyhedron are congruent. A regular polyhedron has all of three related spheres which share its centre: An insphere, tangent to all faces. An intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices; the regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after them: Tetrahedral Octahedral Icosahedral Any shapes with icosahedral or octahedral symmetry will contain tetrahedral symmetry; the five Platonic solids have an Euler characteristic of 2. This reflects that the surface is a topological 2-sphere, so is true, for example,of any polyhedron, star-shaped with respect to some interior point; the sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not for tetrahedra. In a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, vice versa.
The regular polyhedra show this duality as follows: The tetrahedron is self-dual, i.e. it pairs with itself. The cube and octahedron are dual to each other; the icosahedron and dodecahedron are dual to each other. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron and great icosahedron are dual to each other. The Schläfli symbol of the dual is just the original written backwards. Stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old; some of these stones show not only the symmetries of the five Platonic solids, but some of the relations of duality amongst them. Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery. There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron.
It is possible that the Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua in the late 19th century of a dodecahedron made of soapstone, dating back more than 2,500 years. The earliest known written records of the regular convex solids originated from Classical Greece; when these solids were all discovered and by whom is not known, but Theaetetus, was the first to give a mathematical description of all five. H. S. M. Coxeter credits Plato with having made models of them, mentions that one of the earlier Pythagoreans, Timaeus of Locri, used all five in a correspondence between the polyhedra and the nature of the universe as it was perceived – this correspondence is recorded in Plato's dialogue Timaeus. Euclid's reference to Plato led to their common description as the Platonic solids. One might characterise the Greek definition as follows: A regular polygon is a planar figure with all edges equal and all corners equal A regular polyhedron is a solid figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.
This definition rules out, for example, the square pyramid, or the shape formed by joining two tetrahedra together. This concept of a regular polyhedron would remain unchallenged for 2000 years. Regular star polygons such as the pentagram were known to the ancient Greeks – the pentagram was used by the Pythagoreans as their secret sign, but they did not use them to construct polyhedra, it was not until the early 17th century that Johannes Kepler realised that pentagrams could be used as the faces of regular star polyhedra. Some of these star polyhedra may have been discovered
St Mark's Basilica
The Patriarchal Cathedral Basilica of Saint Mark known as Saint Mark's Basilica, is the cathedral church of the Roman Catholic Archdiocese of Venice, northern Italy. It is the most famous of the city's churches and one of the best known examples of Italo-Byzantine architecture, it lies at the eastern end of the Piazza San Marco and connected to the Doge's Palace. It was the chapel of the Doge, has been the city's cathedral only since 1807, when it became the seat of the Patriarch of Venice, archbishop of the Roman Catholic Archdiocese of Venice at San Pietro di Castello; the basic structure of the church dates from 1060 to 1100, the large amount of subsequent work has been to embellish this rather than replace elements. The famous main facade has an ornamented roofline, Gothic, the gold ground mosaics that now cover all the upper areas of the interior took centuries to complete. In the 13th century the external height of the domes was increased by hollow drums raised on a wooden framework and covered with metal.
Without this, they would not now be visible from the piazza. For its opulent design, gold ground mosaics, its status as a symbol of Venetian wealth and power, from the 11th century on the building has been known by the nickname Chiesa d'Oro, it achieves an Oriental feeling of exoticism through blending Byzantine and Islamic elements, but remains unique, a product of Italian workers of all sorts. The first St Mark's was a building next to the Doge's Palace, ordered by the doge in 828, when Venetian merchants stole the supposed relics of Mark the Evangelist from Alexandria, completed by 832; the church was burned in a rebellion in 976, when the populace locked Pietro IV Candiano inside to kill him, restored or rebuilt in 978. Nothing certain is known of the form of these early churches. From 1063 the present basilica was constructed; the consecration is variously recorded as being in 1084–85, 1093, 1102 and 1117 reflecting a series of consecrations of different parts. The size of the church was increased in all directions to the north and south, the wooden domes replaced by brick, which required thickening such walls as were retained.
In 1094 the supposed body of Saint Mark was rediscovered in a pillar by Vitale Faliero, doge at the time. The building incorporates a low tower, believed by some to have been part of the original Doge's Palace; the Pala d'Oro ordered from Constantinople was installed on the high altar in 1105. In 1106 the church, its mosaics, were damaged by a serious fire in that part of the city; the main features of the present structure were all in place by except for the narthex or porch, the facade. The basic shape of the church has a mixture of Italian and Byzantine features, notably "the treatment of the eastern arm as the termination of a basilican building with main apse and two side chapels rather than as an equal arm of a centralized structure". In the first half of the 13th century the narthex and the new facade were constructed, most of the mosaics were completed and the domes were covered with second much higher domes of lead-covered wood in order to blend in with the Gothic architecture of the redesigned Doge's Palace.
As with most Venetian buildings, the main structure is built in brick, with the arches given moulded terracotta or brick decoration, with stone columns, horizontal mouldings, some other details. The brick remains in place, but covered over except in a few places; the basic structure of the building has not been much altered. Its decoration has changed over time, though the overall impression of the interior with a dazzling display of gold ground mosaics on all ceilings and upper walls remains the same; the original unadorned structure would have looked different, but it is that gradual decoration was always intended. The succeeding centuries the period after the Venetian-led conquest of Constantinople in the Fourth Crusade of 1204 and the fourteenth century, all contributed to its adornment, with many elements being spolia brought in from ancient or Byzantine buildings, such as mosaics, capitals, or friezes; the Venetian sculptors of other capitals and friezes copied the Byzantine style so that some of their work can only be distinguished with difficulty.
The exterior brickwork became covered with marble cladding and carvings, some much older than the building itself, such as the statue of the Four Tetrarchs. The latest structural additions include the closing-off of the Baptistery and St Isidor's Chapel, the carvings on the upper facade and the Sacristy, the closing-off of the Zen Chapel. During the 13th century the emphasis of the church's function seems to have changed from being the private chapel of the Doge to that of a "state church", with increased power for the procurators, it was the location for the great public ceremonies of the state, such as the installation and burials of Doges, though as space ran out and the demand for grander tombs increased, from the 15th century Santi Giovanni e Paolo became the usual burial place. The function of the basilica remained the same until 1807, after the
Gravitation (M. C. Escher)
Gravitation is a mixed media work by the Dutch artist M. C. Escher completed in June 1952, it was first printed as a black-and-white lithograph and coloured by hand in watercolour. It depicts a nonconvex regular polyhedron known as the small stellated dodecahedron; each facet of the figure has a trapezoidal doorway. Out of these doorways protrude the heads and legs of twelve turtles without shells, who are using the object as a common shell; the turtles are in six coloured pairs with each turtle directly opposite its counterpart. Printmaking Locher, J. L.. The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0
M. C. Escher
Maurits Cornelis Escher was a Dutch graphic artist who made mathematically-inspired woodcuts and mezzotints. Despite wide popular interest, Escher was for long somewhat neglected in the art world in his native Netherlands, he was 70. In the twenty-first century, he became more appreciated, with exhibitions across the world, his work features mathematical objects and operations including impossible objects, explorations of infinity, symmetry, perspective and stellated polyhedra, hyperbolic geometry, tessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, Harold Coxeter and crystallographer Friedrich Haag, conducted his own research into tessellation. Early in his career, he drew inspiration from nature, making studies of insects and plants such as lichens, all of which he used as details in his artworks, he traveled in Italy and Spain, sketching buildings, townscapes and the tilings of the Alhambra and the Mezquita of Cordoba, became more interested in their mathematical structure.
Escher's art became well known among scientists and mathematicians, in popular culture after it was featured by Martin Gardner in his April 1966 Mathematical Games column in Scientific American. Apart from being used in a variety of technical papers, his work has appeared on the covers of many books and albums, he was one of the major inspirations of Douglas Hofstadter's Pulitzer Prize-winning 1979 book Gödel, Bach. Maurits Cornelis Escher was born on 17 June 1898 in Leeuwarden, the Netherlands, in a house that forms part of the Princessehof Ceramics Museum today, he was the youngest son of the civil engineer George Arnold Escher and his second wife, Sara Gleichman. In 1903, the family moved to Arnhem, where he attended primary and secondary school until 1918. Known to his friends and family as "Mauk", he was a sickly child and was placed in a special school at the age of seven. Although he excelled at drawing, his grades were poor, he took piano lessons until he was thirteen years old. In 1918, he went to the Technical College of Delft.
From 1919 to 1922, Escher attended the Haarlem School of Architecture and Decorative Arts, learning drawing and the art of making woodcuts. He studied architecture, but he failed a number of subjects and switched to decorative arts, studying under the graphic artist Samuel Jessurun de Mesquita. In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra and Ravello. In the same year, he traveled through Spain, visiting Madrid and Granada, he was impressed by the Italian countryside and, in Granada, by the Moorish architecture of the fourteenth-century Alhambra. The intricate decorative designs of the Alhambra, based on geometrical symmetries featuring interlocking repetitive patterns in the coloured tiles or sculpted into the walls and ceilings, triggered his interest in the mathematics of tessellation and became a powerful influence on his work. Escher returned to Italy and lived in Rome from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, like himself attracted to Italy – whom he married in 1924.
The couple settled in Rome where their first son, Giorgio Arnaldo Escher, named after his grandfather, was born. Escher and Jetta had two more sons – Arthur and Jan, he travelled visiting Viterbo in 1926, the Abruzzi in 1927 and 1929, Corsica in 1928 and 1933, Calabria in 1930, the Amalfi coast in 1931 and 1934, Gargano and Sicily in 1932 and 1935. The townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra and spending days at a time making detailed drawings of its mosaic patterns, it was here that he became fascinated, to the point of obsession, with tessellation, explaining: It remains an absorbing activity, a real mania to which I have become addicted, from which I sometimes find it hard to tear myself away. The sketches he made in the Alhambra formed a major source for his work from that time on, he studied the architecture of the Mezquita, the Moorish mosque of Cordoba. This turned out to be the last of his long study journeys.
His art correspondingly changed from being observational, with a strong emphasis on the realistic details of things seen in nature and architecture, to being the product of his geometric analysis and his visual imagination. All the same his early work shows his interest in the nature of space, the unusual and multiple points of view. In 1935, the political climate in Italy became unacceptable to Escher, he had no interest in politics, finding it impossible to involve himself with any ideals other than the expressions of his own concepts through his own particular medium, but he was averse to fanaticism and hypocrisy. When his eldest son, was forced at the age of nine to wear a Ballila uniform in school, the family left Italy and moved to Château-d'Œx, where they remained for two years; the Netherlands post office had Escher design a semi-postal stamp for the "Air Fund" in 1935, again in 1949 he designed Netherlands stamps. These were for the 75th anniversary of the Universal Postal Union. Escher, who had
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century, he was born in London, received his BA and PhD from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald, he was most noted for his work on higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10, he felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal. Coxeter went up to Cambridge in 1926 to read mathematics. There he earned his BA in 1928, his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, Solomon Lefschetz.
Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics. In 1934 he spent a further year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and P. Du Val, H. T. Flather, John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays published by W. W. Rouse Ball in 1892, he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and a Fellow of the Royal Society in 1950, he met M. C. Escher in 1954 and the two became lifelong friends, he inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, he published twelve books. Since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor, he was made a Fellow of the Royal Society in 1950 and in 1997 he was awarded their Sylvester Medal.
In 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he received the Jeffery–Williams Prize. 1940: Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46: 380-407, MR 2,10 doi:10.1007/BF01181449 1942: Non-Euclidean Geometry, University of Toronto Press, MAA. 1954: "Uniform Polyhedra", Philosophical Transactions of the Royal Society A 246: 401–50 doi:10.1098/rsta.1954.0003 1949: The Real Projective Plane 1957: Generators and Relations for Discrete Groups 1980: Second edition, Springer-Verlag ISBN 0-387-09212-9 1961: Introduction to Geometry 1963: Regular Polytopes, Macmillan Company 1967: Geometry Revisited 1970: Twisted honeycombs 1973: Regular Polytopes, Dover edition, ISBN 0-486-61480-8 1974: Projective Geometry 1974: Regular Complex Polytopes, Cambridge University Press 1981:, Zero-Symmetric Graphs, Academic Press. 1985: Regular and Semi-Regular Polytopes II, Mathematische Zeitschrift 188: 559–591 1987 Projective Geometry ISBN 978-0-387-40623-7 1988: Regular and Semi-Regular Polytopes III, Mathematische Zeitschrift 200: 3–45 1995: F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: Kaleidoscopes — Selected Writings of H.
S. M. Coxeter. John Wiley and Sons ISBN 0-471-01003-0 1999: The Beauty of Geometry: Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler; the Coxeter Legacy: Reflections and Projections. Providence, R. I.: American Mathematical Society. ISBN 978-0-8218-3722-1. OCLC 62282754. Roberts, Siobhan. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. New York: Walker & Company. ISBN 978-0-8027-1499-2. OCLC 71436884. Archival papers held at University of Toronto Archives and Records Management Services Harold Scott MacDonald Coxeter at the Mathematics Genealogy Project H. S. M. Coxeter, Erich W. Ellers, Branko Grünbaum, Peter McMullen, Asia Ivic Weiss Notices of the AMS: Volume 50, Number 10. Www.donaldcoxeter.com www.math.yorku.ca/dcoxeter webpages dedicated to him Jaron's World: Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C. Escher video of a lecture by H. S. M. Coxeter, April 28, 2000
Venice
Venice is a city in northeastern Italy and the capital of the Veneto region. It is situated on a group of 118 small islands that are separated by canals and linked by over 400 bridges; the islands are located in the shallow Venetian Lagoon, an enclosed bay that lies between the mouths of the Po and the Piave rivers. In 2018, 260,897 people resided in the Comune di Venezia, of whom around 55,000 live in the historical city of Venice. Together with Padua and Treviso, the city is included in the Padua-Treviso-Venice Metropolitan Area, considered a statistical metropolitan area, with a total population of 2.6 million. The name is derived from the ancient Veneti people who inhabited the region by the 10th century BC; the city was the capital of the Republic of Venice. The 697–1797 Republic of Venice was a major financial and maritime power during the Middle Ages and Renaissance, a staging area for the Crusades and the Battle of Lepanto, as well as an important center of commerce and art in the 13th century up to the end of the 17th century.
The city-state of Venice is considered to have been the first real international financial center, emerging in the 9th century and reaching its greatest prominence in the 14th century. This made Venice a wealthy city throughout most of its history. After the Napoleonic Wars and the Congress of Vienna, the Republic was annexed by the Austrian Empire, until it became part of the Kingdom of Italy in 1866, following a referendum held as a result of the Third Italian War of Independence. Venice has been known as "La Dominante", "La Serenissima", "Queen of the Adriatic", "City of Water", "City of Masks", "City of Bridges", "The Floating City", "City of Canals"; the lagoon and a part of the city are listed as a UNESCO World Heritage Site. Parts of Venice are renowned for the beauty of their settings, their architecture, artwork. Venice is known for several important artistic movements—especially during the Renaissance period—has played an important role in the history of symphonic and operatic music, is the birthplace of Antonio Vivaldi.
Although the city is facing some major challenges, Venice remains a popular tourist destination, an iconic Italian city, has been ranked the most beautiful city in the world. The name of the city, deriving from Latin forms Venetia and Venetiae, is most taken from "Venetia et Histria", the Roman name of Regio X of Roman Italy, but applied to the coastal part of the region that remained under Roman Empire outside of Gothic and Frankish control; the name Venetia, derives from the Roman name for the people known as the Veneti, called by the Greeks Enetoi. The meaning of the word is uncertain, although there are other Indo-European tribes with similar-sounding names, such as the Celtic Veneti and the Slavic Vistula Veneti. Linguists suggest that the name is based on an Indo-European root *wen, so that *wenetoi would mean "beloved", "lovable", or "friendly". A connection with the Latin word venetus, meaning the color'sea-blue', is possible. Supposed connections of Venetia with the Latin verb venire, such as Marin Sanudo's veni etiam, the supposed cry of the first refugees to the Venetian lagoon from the mainland, or with venia are fanciful.
The alternative obsolete form is Vinegia. Although no surviving historical records deal directly with the founding of Venice and the available evidence have led several historians to agree that the original population of Venice consisted of refugees—from nearby Roman cities such as Padua, Treviso and Concordia, as well as from the undefended countryside—who were fleeing successive waves of Germanic and Hun invasions; this is further supported by the documentation on the so-called "apostolic families", the twelve founding families of Venice who elected the first doge, who in most cases trace their lineage back to Roman families. Some late Roman sources reveal the existence of fishermen, on the islands in the original marshy lagoons, who were referred to as incolae lacunae; the traditional founding is identified with the dedication of the first church, that of San Giacomo on the islet of Rialto —said to have taken place at the stroke of noon on 25 March 421. Beginning as early as AD 166–168, the Quadi and Marcomanni destroyed the main Roman town in the area, present-day Oderzo.
This part of Roman Italy was again overrun in the early 5th century by the Visigoths and, some 50 years by the Huns led by Attila. The last and most enduring immigration into the north of the Italian peninsula, that of the Lombards in 568, left the Eastern Roman Empire only a small strip of coastline in the current Veneto, including Venice; the Roman/Byzantine territory was organized as the Exarchate of Ravenna, administered from that ancient port and overseen by a viceroy appointed by the Emperor in Constantinople. Ravenna and Venice were connected only by sea routes, with the Venetians' isolated position came increasing autonomy. New ports were built, including those at Torcello in the Venetian lagoon; the tribuni maiores formed the earliest central standing governing committee of the islands in the lagoon, dating from c. 568. The traditional first doge of Venice, Paolo Lucio A