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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] |
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 cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arc Length
ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * Airport Regions Conference, a European organization of major airports * Amalgamated Roadstone Corporation, a British stone quarrying company * American Record Company (1904–1908, re-activated 1979), one of two United States record labels by this name * American Record Corporation (1929–1938), a United States record label also known as American Record Company * ARC (American Recording Company) (1978-present), a vanity label for Earth, Wind & Fire * ARC Document Solutions, a company based in California, formerly American Reprographics Company * Amey Roadstone Construction, a former British construction company * Aqaba Railway Corporation, a freight railway in Jordan * ARC/Architectural Resources Cambridge, Inc., Cambridge, Massachuse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spain
, image_flag = Bandera de España.svg , image_coat = Escudo de España (mazonado).svg , national_motto = ''Plus ultra'' (Latin)(English: "Further Beyond") , national_anthem = (English: "Royal March") , image_map = , map_caption = , image_map2 = , capital = Madrid , coordinates = , largest_city = Madrid , languages_type = Official language , languages = Spanish , ethnic_groups = , ethnic_groups_year = , ethnic_groups_ref = , religion = , religion_ref = , religion_year = 2020 , demonym = , government_type = Unitary parliamentary constitutional monarchy , leader_title1 = Monarch , leader_name1 = Felipe VI , leader_title2 = Prime Minister , leader_name2 = Pedro Sánchez , legislature = C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Portugal
Portugal, officially the Portuguese Republic ( pt, República Portuguesa, links=yes ), is a country whose mainland is located on the Iberian Peninsula of Southwestern Europe, and whose territory also includes the Atlantic archipelagos of the Azores and Madeira. It features the westernmost point in continental Europe, and its Iberian portion is bordered to the west and south by the Atlantic Ocean and to the north and east by Spain, the sole country to have a land border with Portugal. Its two archipelagos form two autonomous regions with their own regional governments. Lisbon is the capital and largest city by population. Portugal is the oldest continuously existing nation state on the Iberian Peninsula and one of the oldest in Europe, its territory having been continuously settled, invaded and fought over since prehistoric times. It was inhabited by pre-Celtic and Celtic peoples who had contact with Phoenicians and Ancient Greek traders, it was ruled by the Ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistically Random
A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll or the digits of π exhibit statistical randomness. Statistical randomness does not necessarily imply "true" randomness, i.e., objective unpredictability. Pseudorandomness is sufficient for many uses, such as statistics, hence the name ''statistical'' randomness. ''Global randomness'' and ''local randomness'' are different. Most philosophical conceptions of randomness are global—because they are based on the idea that "in the long run" a sequence looks truly random, even if certain sub-sequences would ''not'' look random. In a "truly" random sequence of numbers of sufficient length, for example, it is probable there would be long sequences of nothing but repeating numbers, though on the whole the sequence might be random. ''Local'' randomness refers to the idea that there can be minimum sequence lengths in which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandelbrot Set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications; mathematically, one would say that the boundary of the Mandelbrot set is a ''fractal curve''. The "style" of this recursive detail depends on the region of the set boundary being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nanometer
330px, Different lengths as in respect to the molecular scale. The nanometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: nm) or nanometer (American and British English spelling differences#-re, -er, American spelling) is a units of measurement, unit of length in the International System of Units (SI), equal to one billionth (short scale) of a metre () and to 1000 picometres. One nanometre can be expressed in scientific notation as , and as metres. History The nanometre was formerly known as the millimicrometre – or, more commonly, the millimicron for short – since it is of a micron (micrometre), and was often denoted by the symbol mμ or (more rarely and confusingly, since it logically should refer to a ''millionth'' of a micron) as μμ. Etymology The name combines the SI prefix ''nano-'' (from the Ancient Greek , ', "dwarf") with the parent unit name ''metre'' (from Greek , ', "unit of measurement"). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space-filling Curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called ''Peano curves'', but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano. Definition Intuitively, a curve in two or three (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a ''curve'': In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sierpiński Curve
Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n \to \infty completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve. Because the Sierpiński curve is space-filling, its Hausdorff dimension (in the limit n \to \infty ) is 2 . The Euclidean length of the nth iteration curve S_n is : l_n = (1+\sqrt 2) 2^n - (2-\sqrt 2) , i.e., it grows ''exponentially'' with n beyond any limit, whereas the limit for n \to \infty of the area enclosed by S_n is 5/12 \, that of the square (in Euclidean metric). Uses of the curve The Sierpiński curve is useful in several practical applications because it is more symmetrical than other commonly studied space-filling curves. For example, it has been used as a basis for the rapid construction of an approximate solution to the Travelling Salesman Problem (w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are continuous on U. If f is k-dif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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