Wikipedysta:Januszkaja/brudnopis

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Biblioteka matematyczna

Geometria

• Hilbert D., Cohn-Vossen S.: Geometria poglądowa. Warszawa: PWN, 1956. Brak numerów stron w książce
• Coxeter H. S. M.: Wstęp do geometrii dawnej i nowej. Warszawa: PWN, 1967. Brak numerów stron w książce
• Kapusta J.: K-dron. Opatentowana nieskończoność. Warszawa: WSiP, 1995. Brak numerów stron w książce
• Forder H. G.: The Foundations of Euclidean Geometry. New York: 1958. Brak numerów stron w książce
• Sommerville D. M. Y.: The Elements of Non-Euclidean Geometry. London: 1929. Brak numerów stron w książce
• Gauss C. F.: Werke. T. 8. Göttingen: 1900. Brak numerów stron w książce
• Иовлев Н. Н.: Введение в элементарную геометрию и тригонометрию Лобачевского. Москва-Ленинград: 1930. Brak numerów stron w książce
• Kostin W.: Podstawy geometrii. Warszawa: 1961. Brak numerów stron w książce
• И. М. Виноградов (red.): Математическая энциклопедия. T. 3. Москва: Советская энциклопедия, 1982, s. 404.

Algebra

• Bourbaki N.: Algèbre commutative. Paris: Hermann, 1961-1965, seria: Éléments de matématique. Brak numerów stron w książce
• Боревич З. И., Шафаревич И. Р.: Теория чисел. Москва: Наука, 1985. (ros.). Brak numerów stron w książce
• Atiyah M. F., Macdonald I. G.: Introduction to commutative algebra. Addison-Wesley, 1969. Brak numerów stron w książce
• van der Waerden B. L.: Algebra. Springer-Verlag, 1967. Brak numerów stron w książce

Ośmiościan szóstkowy

Ośmiościan szóstkowy jest Wielościanem Catalana dualny do Wielościanu Archimedesa sześcio-ośmiościanu ściętego. As such it is face-transitive but with irregular face polygons. It looks a bit like an inflated rhombic dodecahedron—if one replaces each face of the rhombic dodecahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis dodecahedron. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron.

Symmetry

It has O_h octahedral symmetry. Its collective edges represent the reflection planes of the symmetry.

Dimensions

If its smallest edges have length 1, its surface area is ${\tfrac {6}{7}}\scriptstyle {\sqrt {783 436{\sqrt {2}}}}$ and its volume is ${\tfrac {1}{7}}\scriptstyle {\sqrt {3(2194 1513{\sqrt {2}})}}$.

See also

• Disdyakis triacontahedron
• Bisected hexagonal tiling
• Great rhombihexacron—A uniform dual polyhedron with the same surface topology

References

• Szablon:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic dodecahedron)

External links

• Szablon:Mathworld2
• Disdyakis Dodecahedron (Hexakis Octahedron) Interactive Polyhedron Model

W geometrii i kombinatoryce wielościennej, Kleeścianem wielościanu jest wielościan powstałym z niego przez doklejenie do każdej ze ścian ostrosłupa o podstawie przystającej do tej ściany. Nazwa Kleeścian została stworzona na cześć matematyka Victora Klee.

Examples

Czworościan poczwórny jest Kleeścianem czworościanu, ośmiościan potrójny jest Kleeścianem ośmiościanu, a dwudziestościan potrójny jest Kleeścianem dwudziestościanu. W każdym z tych przypadków Kleeścian jest utworzony przez doklejenie do każdej ściany odpowiedniego wielościanu ostrosłupa prawidłowego czworokątnego.

Kleetopes of the Platonic solids

Czworościan potrójny, Kleeścian czworościanu.

Sześciościan poczwórny, Kleeścian sześcianu.

Ośmiościan potrójny, Kleeścian ośmiościanu.

Dwunastościan piątkowy, Kleeścian dwunastościanu.

Dwudziestościan potrójny, Kleeścian dwudziestościanu.

Sześciościan poczwórny jest Kleeścianem sześcianu, utworzonym przez doklejenie do każdej jego ściany ostrosłupa prawidłowego czworokątnego, a dwunastościan piątkowy jest Kleeścianem dwunastościanu, utworzonym przez doklejenie do każdej jego ściany ostrosłupa prawidłowego pięciokątnego.

Some other convex Kleetopes

The disdyakis dodecahedron, the Kleetope of the rhombic dodecahedron.

The disdyakis triacontahedron, the Kleetope of the rhombic triacontahedron.

The tripentakis icosidodecahedron, the Kleetope of the icosidodecahedron.

The base polyhedron of a Kleetope does not need to be a Platonic solid. For instance, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron, formed by replacing each rhombus face of the dodecahedron by a rhombic pyramid, and the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. In fact, the base polyhedron of a Kleetope does not need to be Face-transitive, as can be seen from the tripentakis icosidodecahedron above.

The Goldner–Harary graph may be represented as the graph of vertices and edges of the Kleetope of the triangular bipyramid.

The base polyhedron of a Kleetope does not even need to be convex:

Some nonconvex Kleetopes, based on the Kepler-Poinsot solids

The small stellapentakis dodecahedron, the Kleetope of the small stellated dodecahedron.

The great stellapentakis dodecahedron, the Kleetope of the great stellated dodecahedron.

The great pentakis dodecahedron, the Kleetope of the great dodecahedron.

The great triakis icosahedron, the Kleetope of the great icosahedron.

Definitions

One method of forming the Kleetope of a polytope Szablon:Math is to place a new vertex outside Szablon:Math, near the centroid of each facet. If all of these new vertices are placed close enough to the corresponding centroids, then the only other vertices visible to them will be the vertices of the facets from which they are defined. In this case, the Kleetope of Szablon:Math is the convex hull of the union of the vertices of Szablon:Math and the set of new vertices.

Alternatively, the Kleetope may be defined by duality and its dual operation, truncation: the Kleetope of Szablon:Math is the dual polyhedron of the truncation of the dual of Szablon:Math.

Properties and applications

If Szablon:Math has enough vertices relative to its dimension, then the Kleetope of Szablon:Math is dimensionally unambiguous: the graph formed by its edges and vertices is not the graph of a different polyhedron or polytope with a different dimension. More specifically, if the number of vertices of a Szablon:Math-dimensional polytope Szablon:Math is at least Szablon:Math, then Szablon:Math is dimensionally unambiguous.

If every Szablon:Math-dimensional face of a Szablon:Math-dimensional polytope Szablon:Math is a simplex, and if Szablon:Math, then every Szablon:Math-dimensional face of Szablon:Math is also a simplex. In particular, the Kleetope of any three-dimensional polyhedron is a simplicial polyhedron, a polyhedron in which all facets are triangles.

Kleetopes may be used to generate polyhedra that do not have any Hamiltonian cycles: any path through one of the vertices added in the Kleetope construction must go into and out of the vertex through its neighbors in the original polyhedron, and if there are more new vertices than original vertices then there are not enough neighbors to go around. In particular, the Goldner–Harary graph, the Kleetope of the triangular bipyramid, has six vertices added in the Kleetope construction and only five in the bipyramid from which it was formed, so it is non-Hamiltonian; it is the simplest possible non-Hamiltonian simplicial polyhedron. If a polyhedron with Szablon:Math vertices is formed by repeating the Kleetope construction some number of times, starting from a tetrahedron, then its longest path has length Szablon:Math; that is, the shortness exponent of these graphs is Szablon:Math, approximately 0.630930. The same technique shows that in any higher dimension Szablon:Math, there exist simplicial polytopes with shortness exponent Szablon:Math. Similarly, Szablon:Harvtxt used the Kleetope construction to provide an infinite family of examples of simplicial polyhedra with an even number of vertices that have no perfect matching.

Kleetopes also have some extreme properties related to their vertex degrees: if each edge in a planar graph is incident to at least seven other edges, then there must exist a vertex of degree at most five all but one of whose neighbors have degree 20 or more, and the Kleetope of the Kleetope of the icosahedron provides an example in which the high-degree vertices have degree exactly 20.

References

• StanislavS. Jendro'l StanislavS., TomášT. Madaras TomášT., Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs, „Tatra Mountains Mathematical Publications”, 2005, s. 149–153 ..
• A.A. Goldner A.A., F.F. Harary F.F., Note on a smallest nonhamiltonian maximal planar graph, „Bull. Malaysian Math. Soc.”, 1 (6), 1975, s. 41–42 .. See also the same journal 6(2):33 (1975) and 8:104-106 (1977). Reference from listing of Harary's publications.
• BrankoB. Grünbaum BrankoB., Unambiguous polyhedral graphs, „Israel Journal of Mathematics”, 4 (1), 1963, s. 235–238, DOI: 10.1007/BF02759726 ..
• BrankoB. Grünbaum BrankoB., Convex Polytopes, Wiley Interscience, 1967 . Brak numerów stron w książce.
• J.W.J.W. Moon J.W.J.W., L.L. Moser L.L., Simple paths on polyhedra, „Pacific Journal of Mathematics”, 1963, s. 629–631 ..
• Michael D.M.D. Plummer Michael D.M.D., Extending matchings in planar graphs IV, „Discrete Mathematics”, 1–3 (109), 1992, s. 207–219, DOI: 10.1016/0012-365X(92)90292-N ..

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Move from Nottingham to Sneinton

In 1773 George's father moved to Nottingham, which at the time had a reputation for being a pleasant town with open spaces and wide roads. By 1831, however, the population had increased nearly five times, in part due to the budding industrial revolution, and the city became known as one of the worst slums in England. There were frequent riots by starving workers, often associated with special hostility towards bakers and millers on the suspicion that they were hiding grain to drive up food prices.

For these reasons, in 1807, George Green senior bought a plot of land in Sneinton, a small town about a mile away from Nottingham. On this plot of land he built a "brick wind corn mill", now famously referred to as Green's Windmill. It was technologically impressive for its time, but required nearly twenty-four hour maintenance, which was to become George Green's burden for the next twenty years.

Adult life

Miller

Just as with baking, Green found the responsibilities of operating the mill annoying and tedious. Grain from the fields was arriving continuously at the mill's doorstep, and the sails of the windmill had to be constantly adjusted to the windspeed, both to prevent damage in high winds, and to maximise rotational speed in low winds. The millstones that would continuously grind against each other, could wear down or cause a fire if they ran out of grain to grind. Every month the stones, which weighed over a ton, would have to be replaced or repaired.

Family life

In 1823 Green formed a relationship with Jane Smith, the daughter of William Smith, hired by Green Senior as mill manager. Although Green and Jane Smith never married, Jane eventually became known as Jane Green and the couple had seven children together; all but the first had Green as a baptismal name. The youngest child was born 13 months before Green's death. Green provided for his common-law wife and children in his will.

Nottingham Subscription Library

When Green was thirty, he became a member of the Nottingham Subscription Library. This library exists today, and was likely one of the only sources of Green's advanced mathematical knowledge. Unlike more conventional libraries, the subscription library was exclusive to a hundred or so subscribers, and the first on the list of subscribers was the Duke of Newcastle. This library catered to requests for specialised books and journals that satisfied the particular interests of their subscribers.

1828 essay

The title page to Green's original essay on what is now known as Green's theorem.

In 1828, Green published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, which is the essay he is most famous for today. It was published privately at the author's expense, because he thought it would be presumptuous for a person like himself, with no formal education in mathematics, to submit the paper to an established journal. When Green published his Essay, it was sold on a subscription basis to 51 people, most of whom were friends and probably could not understand it.

The wealthy landowner and mathematician Edward Bromhead bought a copy and encouraged Green to do further work in mathematics. Not believing the offer was sincere, Green did not contact Bromhead for two years.

Mathematician

By 1829, the time when Green's father died, the senior Green had become one of the gentry due to his considerable accumulated wealth and land owned, roughly half of which he left to his son and the other half to his daughter. The young Green, now thirty-six years old, consequently was able to use this wealth to abandon his miller duties and pursue mathematical studies.

Cambridge

Members of the Nottingham Subscription Library who knew Green repeatedly insisted that he obtain a proper University education. In particular, one of the library's most prestigious subscribers was Sir Edward Bromhead, with whom Green shared many correspondences; he insisted that Green go to Cambridge.

In 1832, aged nearly forty, Green was admitted as an undergraduate at Gonville and Caius College, Cambridge. He was particularly insecure about his lack of knowledge of Greek and Latin, which were prerequisites, but it turned out not to be as hard for him to learn as he believed, as the expected mastery was not as high as he had expected. In the mathematics examinations, he won the first-year mathematical prize. He graduated BA in 1838 as a 4th Wrangler (the 4th highest scoring student in his graduating class, coming after James Joseph Sylvester who scored 2nd).

College fellow

Following his graduation, Green was elected a fellow of the Cambridge Philosophical Society. Even without his stellar academic standing, the Society had already read and made note of his Essay and three other publications, and so Green was warmly welcomed.

The next two years provided an unparalleled opportunity for Green to read, write and discuss his scientific ideas. In this short time he published an additional six publications with applications to hydrodynamics, sound and optics.

Final years and posthumous fame

The Grave of Green, in the church not far from his mill

In his final years at Cambridge, Green became rather ill, and in 1840 he returned to Sneinton, only to die a year later. There are rumours that at Cambridge, Green had "succumbed to alcohol", and some of his earlier supporters, such as Sir Edward Bromhead, tried to distance themselves from him.

Green's work was not well known in the mathematical community during his lifetime. Besides Green himself, the first mathematician to quote his 1828 work was the Briton Robert Murphy (1806–1843) in his 1833 work. In 1845, four years after Green's death, Green's work was rediscovered by the young William Thomson (then aged 21), later known as Lord Kelvin, who popularised it for future mathematicians. According to the book "George Green" by D.M. Cannell, William Thomson noticed Murphy's citation of Green's 1828 essay but found it difficult to locate Green's 1828 work; he finally got some copies of Green's 1828 work from William Hopkins in 1845.

Green's work on the motion of waves in a canal anticipates the WKB approximation of quantum mechanics, while his research on light-waves and the properties of the ether produced what is now known as the Cauchy-Green tensor. Green's theorem and functions were important tools in classical mechanics, and were revised by Schwinger's 1948 work on electrodynamics that led to his 1965 Nobel prize (shared with Feynman and Tomonaga). Green's functions later also proved useful in analysing superconductivity. On a visit to Nottingham in 1930, Albert Einstein commented that Green had been 20 years ahead of his time. The theoretical physicist, Julian Schwinger, who used Green's functions in his ground-breaking works, published a tribute, entitled "The Greening of Quantum Field Theory: George and I," in 1993.

The George Green Library at the University of Nottingham is named after him, and houses the majority of the University's Science and Engineering Collection. In 1986, Green's Windmill was restored to working order. It now serves both as a working example of a 19th-century windmill and as a museum and science centre dedicated to Green.

Westminster Abbey has a memorial stone for Green in the nave adjoining the graves of Sir Isaac Newton and Lord Kelvin.

Source of knowledge

Szablon:Unreferenced section It is unclear to historians exactly where Green obtained information on current developments in mathematics, as Nottingham had little in the way of intellectual resources. What is even more mysterious is that Green had used "the Mathematical Analysis", a form of calculus derived from Leibniz that was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton who had his own methods that were thus championed in England). This form of calculus, and the developments of mathematicians such as Laplace, Lacroix and Poisson were not taught even at Cambridge, let alone Nottingham, and yet Green had not only heard of these developments, but also improved upon them.

It is speculated that only one person educated in mathematics, John Toplis, headmaster of Nottingham High School 1806–1819, graduate from Cambridge and an enthusiast of French mathematics, is known to have lived in Nottingham at the time.

List of publications

• An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. By George Green, Nottingham. Printed for the Author by T. Wheelhouse, Nottingham. 1828. (Quarto, vii 72 pages.)
• Mathematical Investigations concerning the Laws of the Equilibrium of Fluids analogous to the Electric Fluid, with other similar Researches. By George Green, Esq., Communicated by Sir Edward Ffrench Bromhead, Bart., M.A., F.R.S.L. and E. (Cambridge Philosophical Society, read 12 November 1832, printed in the Transactions 1833. Quatro, 64 pages.) Vol. III, Part I.
• On the Determination of the Exterior and Interior Attractions of Ellipsoids of Variable Densities. By George Green, Esq., Caius College. (Cambridge Philosophical Society, read 6 May 1833, printed in the Transactions 1835. Quarto, 35 pages.) Vol. III, Part III.
• Researches on the Vibration of Pendulums in Fluid Media. By George Green, Esq., Communicated by Sir Edward Ffrench Bromhead, Bart., M.A., F.R.S.S. Lond. and Ed. (Royal Society of Edinburgh, read 16 December 1833, printed in the Transactions 1836, Quarto, 9 pages.) Vol. III, Part I.
• On the Motion of Waves in a Variable Canal of Small Width and Depth. By George Green, Esq., BA, of Caius College. (Cambridge Philosophical Society, read 15 May 1837, printed in the Transactions 1838. Quarto, 6 pages.) Vol. VI, Part IV.
• On the Reflexion and Refraction of Sound. By George Green, Esq., BA, of Caius College, Cambridge. (Cambridge Philosophical Society, read 11 December 1837, printed in the Transactions 1838. Quarto, 11 pages.) Vol. VI, Part III.
• On the Laws of Relexion and Refraction of Light at the common Surface of two non-crystallized Media. By George Green, Esq., BA, of Caius College. (Cambridge Philosophical Society, read 11 December 1837, printed in the Transactions 1838. Quarto, 24 pages.) Vol. VII, Part I.
• Note on the Motion of Waves in Canals. By George Green, Esq., BA, of Caius College. (Cambridge Philosophical Society, read 18 February 1839, printed in the Transactions 1839. Quarto, 9 pages.) Vol. VII, Part I.
• Supplement to a Memoir on the Reflexion and Refraction of Light. By George Green, Esq., BA, of Caius College. (Cambridge Philosophical Society, read 6 May 1839, printed in the Transactions 1839. Quarto, 8 pages.) Vol. VII, Part I.
• On the Propagation of Light in Crystallized Media. By George Green, BA, Fellow of Caius College. (Cambridge Philosophical Society, read 20 May 1839, printed in the Transactions 1839. Quarto, 20 pages.) Vol. VII, Part II.

Przypisy

A zoospore is a motile asexual spore that uses a flagellum for locomotion. Also called a swarm spore, these spores are created by some algae, bacteria and fungi to propagate themselves.

Flagella

There are two types of flagellated zoospores, tinsel or "decorated", and whiplash.

Tinsellated flagella have lateral filaments perpendicular to the main axis which allow for more surface area, and disturbance of the medium, giving it the property of a rudder, that is, the purpose of being used for steering.

• Whiplash flagella are straight, to power the zoospore through its medium. There is also the 'default' zoospore, which only has the propelling, 'whiplash' flagella.

• Both tinsel and whiplash flagella beat in a sinusoidal wave pattern, but when both are present, the tinsel will beat in the opposite direction of the whiplash, to give 2 axes of control of motility.

• There can be many combinations for location of the flagella, such as posterior tinsel; posterior whiplash, anterior tinsel; and anterior whiplash.

Oomycetes and heterokont algae produce distinct bi-flagellated zoospores:

The phyla Chytridiomycota (Kingdom Fungi), Oomycota (Kingdom Chromista), and Hyphochytridiomycota within (Kingdom Chromista), produce zoospores with flagella in the same order as described above (e.g. Hyphochytridiomycota produces anterior whiplash and none else). These phyla number 1000 , 580 and 16 species respectively.

Zoosporangium

A zoosporangium is the sexual structure (sporangium) in which the zoospores develop in a plant, fungi, or protists (such as the Oomycota)

See also

• Angiosperm
• Chytridiomycota
• Fern
• Flagellum
• Gametangium

References

• C.J. Alexopolous, Charles W. Mims, M. Blackwell et al., Introductory Mycology, 4th ed. (John Wiley and Sons, Hoboken NJ, 2004) ISBN 0-471-52229-5