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Metallic Means
The metallic means (also ratios or constants) of the successive natural numbers are the continued fractions: n + \cfrac = ;n,n,n,n,\dots= \frac. The golden ratio (1.618...) is the metallic mean between 1 and 2, while the silver ratio (2.414...) is the metallic mean between 2 and 3. The term "bronze ratio" (3.303...), or terms using other names of metals (such as copper or nickel), are occasionally used to name subsequent metallic means. The values of the first ten metallic means are shown at right. Notice that each metallic mean is a root of the simple quadratic equation: x^2-nx=1, where n is any positive natural number. As the golden ratio is connected to the pentagon (first diagonal/side), the silver ratio is connected to the octagon (second diagonal/side). As the golden ratio is connected to the Fibonacci numbers, the silver ratio is connected to the Pell numbers, and the bronze ratio is connected to . Each Fibonacci number is the sum of the previous number times one plus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( or \phi) denotes the golden ratio. The constant \varphi satisfies the quadratic equation \varphi^2 = \varphi + 1 and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of \varphi—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tridecagon
In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon. Regular tridecagon A '' regular tridecagon'' is represented by Schläfli symbol . The measure of each internal angle of a regular tridecagon is approximately 152.308 degrees, and the area with side length ''a'' is given by :A = \fraca^2 \cot \frac \simeq 13.1858\,a^2. Construction As 13 is a Pierpont prime but not a Fermat prime, the regular tridecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisector. The following is an animation from a ''neusis construction'' of a regular tridecagon with radius of circumcircle \overline = 12, according to Andrew M. Gleason, based on the angle trisection by means of the Tomahawk (light blue). An approximate construction of a regular tridecagon using straightedge and compass is shown here. Another possible animation of an approximate construction, also possible with using straightedg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plastic Number
In mathematics, the plastic number (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, ) is a mathematical constant which is the unique real solution of the cubic equation : x^3 = x + 1. It has the exact value : \rho = \sqrt \sqrt Its decimal expansion begins with . Properties Recurrences The powers of the plastic number satisfy the third-order linear recurrence relation for . Hence it is the limiting ratio of successive terms of any (non-zero) integer sequence satisfying this recurrence such as the Padovan sequence (also known as the Cordonnier numbers), the Perrin numbers and the Van der Laan numbers, and bears relationships to these sequences akin to the relationships of the golden ratio to the second-order Fibonacci and Lucas numbers, akin to the relationships between the silver ratio and the Pell numbers. The plastic number satisfies the nested radical recurrence : \rho = \sqrt Numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7). The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be Positive integer, positive. A ratio may be specified either by giving both constituting numbers, written as "''a'' to ''b''" or "''a'':''b''", or by giving just the value of their quotient Equal quotients correspond to equal ratios. Consequently, a ratio may be considered as an ordered pair of numbers, a Fraction (mathematics), fraction with the first number in the numerator and the second in the denom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithmetic mean'', also known as "arithmetic average", is a measure of central tendency of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''1, ''x''2, ..., x''n'' is typically denoted using an overhead bar, \bar. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the ''sample mean'' (\bar) to distinguish it from the mean, or expected value, of the underlying distribution, the ''population mean'' (denoted \mu or \mu_x).Underhill, L.G.; Bradfield d. (1998) ''Introstat'', Juta and Company Ltd.p. 181/ref> Outside probability and statistics, a wide range of other notions of mean are o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant (mathematics)
In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings: * A fixed and well-defined number or other non-changing mathematical object. The terms '' mathematical constant'' or '' physical constant'' are sometimes used to distinguish this meaning. * A function whose value remains unchanged (i.e., a constant function). Such a constant is commonly represented by a variable which does not depend on the main variable(s) in question. For example, a general quadratic function is commonly written as: :a x^2 + b x + c\, , where , and are constants (or parameters), and a variable—a placeholder for the argument of the function being studied. A more explicit way to denote this function is :x\mapsto a x^2 + b x + c \, , which makes the function-argument status of (and by extension the constancy of , and ) clear. In this example , and are co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Triangle
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a^2+b^2=c^2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=\sqrt2 is a right triangle, but (1,1,\sqrt2) is not a Pythagorean triple because \sqrt2 is not an integer. Moreover, 1 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twos Triangle
2 is a number, numeral, and glyph. 2, two or II may also refer to: * AD 2, the second year of the AD era * 2 BC, the second year before the AD era * The month of February Music Bands * Two (metal band) Albums * ''2'' (All Girl Summer Fun Band album), 2003 * ''2'' (Black Country Communion album), 2011 * ''2'' (The Black Heart Procession album), 1999 * ''2'' (Ned Collette album), 2012 * ''2'' (Darker My Love album), 2008 * ''2'' (Mac DeMarco album), 2012 * ''2'' (Dover album), 2007 * ''2'' (The Gloaming album), 2016 * ''2'' (Erkin Koray album), 1976 * ''2'' (Amaia Montero album), 2011 * ''2'' (Mudcrutch album), 2016 * ''2'' (Netsky album), 2012 * ''2'' (Olivia Newton-John album), 2002 * ''2'' (Nik & Jay album), 2004 * ''2'' (Florent Pagny album), 2001 * ''2'' (Pole album), 1999 * ''2'' (Retribution Gospel Choir album), 2010 * ''2'' (Rockapella album), 2000 * ''2'' (Saint Lu album), 2013, by Luise Gruber * ''2'' (Smoking Popes EP), 1993 * ''2'' (Sneaky S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bronze Ratio "tinted" Triangle
Bronze is an alloy consisting primarily of copper, commonly with about 12–12.5% tin and often with the addition of other metals (including aluminium, manganese, nickel, or zinc) and sometimes non-metals, such as phosphorus, or metalloids such as arsenic or silicon. These additions produce a range of alloys that may be harder than copper alone, or have other useful properties, such as strength, ductility, or machinability. The archaeological period in which bronze was the hardest metal in widespread use is known as the Bronze Age. The beginning of the Bronze Age in western Eurasia and India is conventionally dated to the mid-4th millennium BCE (~3500 BCE), and to the early 2nd millennium BCE in China; elsewhere it gradually spread across regions. The Bronze Age was followed by the Iron Age starting from about 1300 BCE and reaching most of Eurasia by about 500 BCE, although bronze continued to be much more widely used than it is in modern times. Because historical artworks were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Silver Ratio "tinted" Triangle
Silver is a chemical element with the symbol Ag (from the Latin ', derived from the Proto-Indo-European ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, white, lustrous transition metal, it exhibits the highest electrical conductivity, thermal conductivity, and reflectivity of any metal. The metal is found in the Earth's crust in the pure, free elemental form ("native silver"), as an alloy with gold and other metals, and in minerals such as argentite and chlorargyrite. Most silver is produced as a byproduct of copper, gold, lead, and zinc refining. Silver has long been valued as a precious metal. Silver metal is used in many bullion coins, sometimes alongside gold: while it is more abundant than gold, it is much less abundant as a native metal. Its purity is typically measured on a per-mille basis; a 94%-pure alloy is described as "0.940 fine". As one of the seven metals of antiquity, silver has had an enduring role in most human cultures. Other than in curre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio "tinted" Triangle
Golden means made of, or relating to gold. Golden may also refer to: Places United Kingdom *Golden, in the parish of Probus, Cornwall *Golden Cap, Dorset *Golden Square, Soho, London *Golden Valley, a valley on the River Frome in Gloucestershire *Golden Valley, Herefordshire United States *Golden, Colorado, a town West of Denver, county seat of Jefferson County *Golden, Idaho, an unincorporated community *Golden, Illinois, a village *Golden Township, Michigan *Golden, Mississippi, a village *Golden City, Missouri, a city *Golden, Missouri, an unincorporated community *Golden, Nebraska, ghost town in Burt County *Golden Township, Holt County, Nebraska *Golden, New Mexico, a sparsely populated ghost town *Golden, Oregon, an abandoned mining town *Golden, Texas, an unincorporated community *Golden, Utah, a ghost town *Golden, Marshall County, West Virginia, an unincorporated community Elsewhere *Golden, County Tipperary, Ireland, a village on the River Suir *Golden Vale, Munster, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5. Etymology The word ''hypotenuse'' is derived from Greek (sc. or ), meaning " idesubtending the right angle" (Apollodorus), ''hupoteinousa'' being the feminine present active participle of the verb ''hupo-teinō'' "to stretch below, to subtend", from ''teinō'' "to stretch, extend". The nominalised participle, , was used for the hypotenuse of a triangle in the 4th century BCE (attested in Plato, ''Timaeus'' 54d). The Greek term was loaned into La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
