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The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices.
Consider a triangle with sides of lengths a , b , and c in decreasing order. A golden rhombus is a rhombus whose diagonals are in the golden ratio.
The rhombic triacontahedron is a convex polytope that has a very special property: all of its faces are golden rhombi.
The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:. A closed-form expression for the Fibonacci sequence involves the golden ratio:.
The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence or any Fibonacci-like sequence , as shown by Kepler : .
For example:. The golden ratio has the simplest expression and slowest convergence as a continued fraction expansion of any irrational number see Alternate forms above.
It is, for that reason, one of the worst cases of Lagrange's approximation theorem and it is an extremal case of the Hurwitz inequality for Diophantine approximations.
This may be why angles close to the golden ratio often show up in phyllotaxis the growth of plants. The multiple and the constant are always adjacent Fibonacci numbers.
The golden ratio appears in the theory of modular functions as well. This gives an iteration that converges to the golden ratio itself,.
These iterations all converge quadratically ; that is, each step roughly doubles the number of correct digits.
The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute n digits of the golden ratio is proportional to the time needed to divide two n -digit numbers.
An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them.
The ratio of Fibonacci numbers F and F , each over digits, yields over 10, significant digits of the golden ratio. Both Egyptian pyramids and the regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios.
The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle of size semi-base by apothem , joining the medium-length edges to make the apothem.
This Kepler triangle  is the only right triangle proportion with edge lengths in geometric progression ,   just as the 3—4—5 triangle is the only right triangle proportion with edge lengths in arithmetic progression.
The Rhind papyrus has another pyramid problem as well, again with rational slope expressed as run over rise. This triangle has a face angle of Egyptian pyramids very close in proportion to these mathematical pyramids are known.
In the mid-nineteenth century, Friedrich Röber studied various Egyptian pyramids including those of Khafre , Menkaure , and some of the Giza , Saqqara , and Abusir groups.
He did not apply the golden ratio to the Great Pyramid of Giza, but instead agreed with John Shae Perring that its side-to-height ratio is For all the other pyramids he applied measurements related to the Kepler triangle, and claimed that either their whole or half-side lengths are related to their heights by the golden ratio.
In , the pyramidologist John Taylor misinterpreted Herodotus c. Similarly, Howard Vyse reported the great pyramid height Michael Rice  asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the golden ratio and that it is part of the mathematics of the pyramids, citing Giedon For example, Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements.
In fact, the entire story about the Greeks and golden ratio seems to be without foundation. From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries.
The Section d'Or 'Golden Section' was a collective of painters , sculptors, poets and critics associated with Cubism and Orphism.
Livio, for example, claims that they did not,  and Marcel Duchamp said as much in an interview. Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings,  though other experts including critic Yve-Alain Bois have discredited these claims.
From Wikipedia, the free encyclopedia. This article is about the number. For the calendar dates, see Golden number time.
Ratio between two quantities whose sum is at the same ratio to the larger one. List of numbers Irrational numbers. Further information: Mathematics and architecture.
Further information: Mathematics and art and History of aesthetics before the 20th century. Main article: Canons of page construction.
Main article: Patterns in nature. Further information: Mathematics and art. Golden angle List of works designed with the golden ratio Metallic mean Plastic number Sacred geometry Supergolden ratio.
The sum of the two solutions is one, and the product of the two solutions is negative one. In , Roger Herz-Fischler traced the error back to Taylor.
Math Vault. Retrieved Karl Fink's Geschichte der Elementar-Mathematik 2nd ed. Chicago: Open Court Publishing Co. Norton, p. The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.
The New York Times. New York: Sterling. Euclid's Elements of Geometry. Mathematics Teacher. Historia Mathematica. The MacTutor History of Mathematics archive.
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