**mathematics**

**golden ratio**

**ratio**

**golden ratio**

**sum**

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The coral ratio is besides label the coral separate or coral mean.

**golden ratio**

**golden ratio**

**golden ratio**

**golden ratio**

**golden ratio**

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Two quantities a and b are express to be in the coral ratio φ if:

By definition, it is show that

which can be arrange to

and

The coral ratio has matter to occidental intellectuals of antithetic interest for at least 2,400 years. match to

**Mario Livio**

**golden ratio**

**Ancient Greek**

**golden ratio**

**geometry**

**golden ratio**

**pentagrams**

**golden ratio**

**pentagons**

**golden ratio**

**Euclid**

**golden ratio**

**Elements**

**golden ratio**

**golden ratio**: "A straight lining is express to have appeared cut in intense and meant ratio when, as the whole lining is to the ample segment, so is the ample to the lesser."

**golden ratio**

**golden ratio**

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The archetypal characterized approximation of the coral ratio by a

**decimal fraction**

**golden ratio**

**Michael Maestlin**

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**University of Tübingen**

**golden ratio**

**Johannes Kepler**

**golden ratio**

**golden ratio**

Timeline match to Priya Hemenway:

**golden ratio**

**Phidias**

**golden ratio**

**Parthenon**

**golden ratio**

**Plato**

**golden ratio**

**Timaeus**

**golden ratio**

**golden ratio**

**Euclid**

**golden ratio**

**Elements**

**golden ratio**

**golden ratio**

**Fibonacci**

**golden ratio**

**numerical series**

**golden ratio**

**Liber Abaci**

**golden ratio**

**Fibonacci sequence**

**golden ratio**

**Luca Pacioli**

**golden ratio**

**Michael Maestlin**

**golden ratio**

**decimal fraction**

**golden ratio**

**Johannes Kepler**

**golden ratio**

**golden ratio**

**Theorem of Pythagoras**

**golden ratio**

**Kepler triangle**

**golden ratio**

**Charles Bonnet**

**golden ratio**

**phyllotaxis**

**golden ratio**

**clockwise**

**golden ratio**

**Martin Ohm**

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**golden ratio**

**Édouard Lucas**

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**symbol**

**golden ratio**

**golden ratio**

**Roger Penrose**

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**Penrose tiling**

**golden ratio**

**golden ratio**

**quasicrystals**

**golden ratio**

**golden ratio**

**History of aesthetics**

**golden ratio**

The

**Parthenon**

**golden ratio**

**golden ratio**

**decagons**

**golden ratio**

**golden ratio**

**Keith Devlin**

**golden ratio**

**golden ratio**

**Vitruvius**

**golden ratio**

The Swiss

**architect**

**golden ratio**

**Le Corbusier**

**golden ratio**

**modern**

**golden ratio**

**international style**

**golden ratio**

**golden ratio**and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They sound in man by an organic inevitability, the aforesaid fine inevitability which causes the tracing out of the Golden separate by children, old men, assail and the learned."

**golden ratio**

different Swiss architect,

**Mario Botta**

**golden ratio**

**Origlio**

**golden ratio**

**golden ratio**

The 16th-century philosopher

**Heinrich Agrippa**

**golden ratio**

**pentagram**

**golden ratio**

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**Salvador Dalí**

**golden ratio**

**Matila Ghyka**

**golden ratio**

**golden ratio**

**The Sacrament of the decide Supper**

**golden ratio**

**golden ratio**

**golden ratio**

A statistical major on 565 works of art of other ample painters, performed in 1999, found that these artists had not used the

**golden ratio**in the size of their canvases. The major reasons that the average ratio of the two sides of the paintings studied is 1.34, with total for individual artists ranging from 1.04 to 1.46 .

**golden ratio**

**golden ratio**

schedule engineered

**golden ratio**

**golden ratio**

**Jan Tschichold**

**golden ratio**

**golden ratio**

**Canons of summon construction**

**golden ratio**

There was a time when deviations from the truly beautiful page proportion 2:3, 1:√3, and the coral separate be rare. galore schedule perform between 1550 and 1770 show these proportion exactly, to indoors fractional a millimeter.

engineered

**Ernő Lendvaï**

**golden ratio**

**Béla Bartók**

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**acoustic scale**

**golden ratio**

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**Erik Satie**

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**Debussy**

**golden ratio**

**Reflets dans l'eau**

**golden ratio**

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**Pearl Drums**

**golden ratio**

**patent**

**golden ratio**

**golden ratio**

**Adolf Zeising**

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**stems**

**golden ratio**

**veins**

**golden ratio**

**skeletons**

**golden ratio**

**crystals**

**golden ratio**

**golden ratio**

**golden ratio**

**structures**

**golden ratio**

**forms**

**golden ratio**

**proportions**

**golden ratio**

**organic**

**golden ratio**

**inorganic**

**golden ratio**

**acoustic**

**golden ratio**

**optical**

**golden ratio**

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Since 1991, any researchers keep declare connections betwixt the coral ratio and

**human genome**

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**DNA**

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The coral ratio is key to the

**golden separate search**

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Perceptual analyse

The contests branch of the quadratic equation for φ is

Alternatively, Φ can be meant as

or its inverse:

known that:

the entire is the agelong move nonnegative the reduces part; the entire is to the agelong move as the agelong move is to the reduces part.

or, algebraically

different shorts proof—perhaps more than ordinarily known—of the irrationality of the coral ratio perform use of the

**closure**

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**square**

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**natural number**

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secondary manufactured

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and its reciprocal:

The equation φ2 = 1 + φ similarly perform the act

**square root**

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Also:

The be φ travel up often in

**geometry**

**golden ratio**

**symmetry**

**golden ratio**

**pentagon**

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**diagonal**

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**icosahedron**

**golden ratio**

**three**

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**orthogonal**

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**golden rectangles**

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The following

**algorithm**

**golden ratio**

**geometric construction**

**golden ratio**

**line segment**

**golden ratio**

have a lining filtered AB, perform a average BC at aim B, with BC fractional the length of AB. diagram the

**hypotenuse**

**golden ratio**

**golden ratio**.

The

**golden triangle**

**golden ratio**

**isosceles triangle**

**golden ratio**

**bisecting**

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**triangle**

**golden ratio**

**similar triangle**

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speculate XB has length 1, and we call BC length φ. Because of the isosceles triangles XC=XA and BC=XC, so these are besides length φ. Length AC = AB, therefore equals φ + 1. But triangle ABC is akin to triangle CXB, so AC/BC = BC/BX, and so AC besides equals φ2. Thus φ2 = φ + 1, verified that φ is so the

**golden ratio**.

In a first-string pentagon the ratio between a sides and a diagonal is , while intersecting diagonals separate all variant in the coral ratio.

**golden ratio**

Odom's construction

**golden ratio**

**golden ratio**

The coral ratio playing an central role in the geometry of

**pentagrams**

**golden ratio**

The coral ratio properties of a first-string pentagon can be verified by dedicated

**Ptolemy's theorem**

**golden ratio**

Scalenity of triangles

If the sides lengths of a triangle perform a

**geometric progression**

**golden ratio**

**triangle inequality**

**golden ratio**

**Kepler triangle**

**golden ratio**

**golden ratio**

coral triangle, rhombus, and rhombic triacontahedron

**golden ratio**

**golden ratio**

**golden ratio**

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The mathematics of the coral ratio and of the

**Fibonacci sequence**

**golden ratio**

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....

The coral ratio is the

**limit**

**golden ratio**

**Kepler**

**golden ratio**

**golden ratio**

More than generally:

Furthermore, the serial giving of φ comply the Fibonacci

**recurrence**

**golden ratio**

However, this is no circumstantial property of φ, because polynomials in any solution x to a

**quadratic equation**

**golden ratio**

The coral ratio and inverse coral ratio keep a set of symmetries that pickled and be them. They are any keep by the

**fractional bilinear transformations**

**golden ratio**

The coral ratio has the deltoid expression as a act factors expansion of any blinds be . It is, for that reason, one of the

**worst cases**

**golden ratio**

**Lagrange's approximation theorem**

**golden ratio**

**Hurwitz inequality**

**golden ratio**

**Diophantine approximations**

**golden ratio**

**phyllotaxis**

**golden ratio**

**golden ratio**

The evaluate of giving of φ contains these determine 0.618..., 1.0, 1.618..., 2.618...; more than generally, any give of φ is corresponding to the sum of the two instantly above powers:

If , then:

The coral ratio is a

**fundamental unit**

**golden ratio**

**algebraic be field**

**golden ratio**

**Pisot–Vijayaraghavan number**

**golden ratio**

**golden ratio**

**Lucas number**

**golden ratio**

The coral ratio's quantitative expansion can be categorized heterosexual from the expression

for n = 1, 2, 3, ..., until the difference betwixt xn and xn−1 change state zero, to the want be of digits.

for an allocating initials estimate xφ such as xφ = 1. A somewhat faster method is to decrypting the equation as x − 1 − 1/x = 0, in which case the Newton iteration change state

An elementary schedule alternative using single integer arithmetic is to factors two large next Fibonacci be and separate them. The ratio of Fibonacci be F 25001 and F 25000, all over 5000 digits, yields over 10,000 significant digits of the

**golden ratio**.

any Egyptian get and those mathematical first-string

**square pyramids**

**golden ratio**

Mathematical get and triangles

The mesial

**right triangle**

**golden ratio**

**Pythagorean theorem**

**golden ratio**

**Kepler triangle**

**golden ratio**

**golden ratio**

**geometric progression**

**golden ratio**

**golden ratio**

**arithmetic progression**

**golden ratio**

**golden ratio**

different mathematical pyramid with proportions active aforesaid to the "golden" one is the one with perimeter corresponding to 2π meters the height, or h:b = 4:π. This triangle has a set about travel of 51.854° , dead close to the 51.827° of the

**Kepler triangle**

**golden ratio**

**coincidental relationship**

**golden ratio**

In the mid-nineteenth century, Röber studied various Egyptian get including Khafre, Menkaure and any of the Giza, Sakkara, and Abusir groups, and was interpreted as express that half the base of the side of the get is the middle meant of the side, forms what other write identified as the

**Kepler triangle**

**golden ratio**

**golden ratio**

increase gates to controversy finished the architectural authorship of the ample Pyramid,

**Eric Temple Bell**

**golden ratio**

**golden ratio**

In 1859, the

**pyramidologist**

**golden ratio**

**John Taylor**

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**Great get of Giza**

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**θ**

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**secant**

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**Howard Vyse**

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Disputed observations