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Length Of Coast
"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoit Mandelbrot, first published in ''Science'' on 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of ''fractals'', although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals. Overview The paper examines the coastline paradox: the property that the measured length of a stretch of coastline depends on the scale of measurement. Empirical evidence suggests that the smaller the increment of measurement, the longer the measured length becomes. If one were to measure a stretch of coastline with a yardstick, one would get a shorter result than if the same stretch were measured with a ruler. This is because one would be laying the ruler along a more curvilinear route th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Curve
A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length. A famous example is the boundary of the Mandelbrot set. Fractal curves in nature Fractal curves and fractal patterns are widespread, in nature, found in such places as broccoli, snowflakes, feet of geckos, frost crystals, and lightning bolts. See also Romanesco broccoli, dendrite crystal, trees, fractals, Hofstadter's butterfly, Lichtenberg figure, and self-organized criticality. Dimensions of a fractal curve Most of us are used to mathematical curves having dimension one, but as a general rule, fractal curves have different dimensions, also see also fractal dimension and list of fractals by Hausdor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Papers
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Countries By Length Of Coastline
This article contains a list of countries by length of coastline, in kilometers. A coastline of zero indicates that the country is landlocked. Overview The coastline paradox states that a coastline does not have a well-defined length. Measurements of the length of a coastline behave like a fractal, being different at different scale intervals (distance between points on the coastline at which measurements are taken). The smaller the scale interval (meaning the more detailed the measurement), the longer the coastline will be.The smaller the scale interval mathematically, the more detailed and the "greater the scale of the map", in common usage. See scale (map). This "magnifying" effect is greater for convoluted coastlines than for relatively smooth ones. * Data marked The World Factbook or TWF covers 198 countries and 55 territories, from the book published by the Central Intelligence Agency. In addition to coastline lengths, this is the source of the land area used to calculate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coastline Paradox
The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve-like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension. The first recorded observation of this phenomenon was by Lewis Fry Richardson and it was expanded upon by Benoit Mandelbrot. The measured length of the coastline depends on the method used to measure it and the degree of cartographic generalization. Since a landmass has features at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there is no obvious size of the smallest feature that should be taken into consideration when measuring, and hence no single well-defined perimeter to the landmass. Various approximations exist when specific assumptions are made about minimum feature size. The problem is fundamentally different from the measurement of other, simpler edges. It is possib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Making A New Science
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Peano Curve
In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve. Construction Peano's curve may be constructed by a sequence of steps, where the ''i''th step constructs a set ''Si'' of squares, and a sequence ''Pi'' of the centers of the squares, from the set and sequence constructed in the previous step. As a base case, ''S''0 consists of the single unit square, and ''P''0 is the one-element sequence consisting of its center point. In step ''i'', each square ''s'' of ''S''''i'' − 1 is partitioned into nine smaller equal squares, and its center point ''c'' is replace ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koch Snowflake
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch. The Koch snowflake can be built up iteratively, in a sequence of stages. The first stage is an equilateral triangle, and each successive stage is formed by adding outward bends to each side of the previous stage, making smaller equilateral triangles. The areas enclosed by the successive stages in the construction of the snowflake converge to \tfrac times the area of the original triangle, while the perimeters of the successive stages increase without bound. Consequently, the snowflake encloses a finite area, but has an infinite perimeter. Construction The Koch snowflake can be constructed by starting with an equilateral triangle, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Britain
Great Britain is an island in the North Atlantic Ocean off the northwest coast of continental Europe. With an area of , it is the largest of the British Isles, the largest European island and the ninth-largest island in the world. It is dominated by a maritime climate with narrow temperature differences between seasons. The 60% smaller island of Ireland is to the west—these islands, along with over 1,000 smaller surrounding islands and named substantial rocks, form the British Isles archipelago. Connected to mainland Europe until 9,000 years ago by a landbridge now known as Doggerland, Great Britain has been inhabited by modern humans for around 30,000 years. In 2011, it had a population of about , making it the world's third-most-populous island after Java in Indonesia and Honshu in Japan. The term "Great Britain" is often used to refer to England, Scotland and Wales, including their component adjoining islands. Great Britain and Northern Ireland now constitute the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
South Africa
South Africa, officially the Republic of South Africa (RSA), is the southernmost country in Africa. It is bounded to the south by of coastline that stretch along the South Atlantic and Indian Oceans; to the north by the neighbouring countries of Namibia, Botswana, and Zimbabwe; and to the east and northeast by Mozambique and Eswatini. It also completely enclaves the country Lesotho. It is the southernmost country on the mainland of the Old World, and the second-most populous country located entirely south of the equator, after Tanzania. South Africa is a biodiversity hotspot, with unique biomes, plant and animal life. With over 60 million people, the country is the world's 24th-most populous nation and covers an area of . South Africa has three capital cities, with the executive, judicial and legislative branches of government based in Pretoria, Bloemfontein, and Cape Town respectively. The largest city is Johannesburg. About 80% of the population are Black South Afri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lewis Fry Richardson
Lewis Fry Richardson, FRS (11 October 1881 – 30 September 1953) was an English mathematician, physicist, meteorologist, psychologist, and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them. He is also noted for his pioneering work concerning fractals and a method for solving a system of linear equations known as modified Richardson iteration. Early life Lewis Fry Richardson was the youngest of seven children born to Catherine Fry (1838–1919) and David Richardson (1835–1913). They were a prosperous Quaker family, David Richardson operating a successful tanning and leather-manufacturing business. At age 12 he was sent to a Quaker boarding school, Bootham School in York, where he received an education in science, which stimulated an active interest in natural history. In 1898 he went on to Durham College of Science (a college of Durham University) whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
