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The pentagram includes ten isosceles triangles : five acute and five obtuse isosceles triangles. The acute triangles are golden triangles.
The obtuse isosceles triangles are golden gnomons. 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. Metacritic Reviews.
Photo Gallery. Trailers and Videos. Crazy Credits. Alternate Versions. Rate This. While lecturing in China, Van Helsing agrees to help seven kung fu trained siblings reclaim their ancestral mountain village, now the domain of seven powerful vampires and their army of undead slaves.
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Edit Cast Complete credited cast: Peter Cushing Hsi Ching Julie Ege Vanessa Buren Han Chen Wang Leyland Van Helsing Szu Shih Dracula Robert Hanna British Consul Shen Chan Edit Storyline Count Dracula journeys to a remote Chinese village in the guise of a warlord to support six vampires who are dispirited after the loss of a seventh member of their cult.
Edit Did You Know? This codification of prince-electors, though largely based on precedence, was not uncontroversial, especially in regard to the two chief rivals of the ruling House of Luxembourg :.
The bull regulated the whole election process in great detail, listing explicitly where, when, and under which circumstances what should be done by whom, not only for the prince-electors but also for example for the population of Frankfurt , where the elections were to be held, and also for the counts of the regions the prince-electors had to travel through to get there.
The decision to hold the elections in Frankfurt reflected a traditional feeling dating from East Frankish days that both election and coronation ought to take place on Frankish soil.
Quod si facere distulerint infra triginta dies, a die prestiti juramenti prefati continuo numerandos, extunc transactis eisdem triginta diebus amodo panem manducent et aquam et nullatenus civitatem exeant antedictam, nisi prius per ipsos vel majorem partem ipsorum rector seu temporale caput fidelium electum fuerit, ut prefertur.
Besides regulating the election process, the chapters of the Golden Bull contained many minor decrees. For instance, it also defined the order of marching when the emperor was present, both with and without his insignia.
A relatively major decision was made in chapter 15, where Charles IV outlawed any conjurationes, confederationes, and conspirationes , meaning in particular the city alliances Städtebünde , but also other communal leagues that had sprung up through the communal movement in mediaeval Europe.
Most Städtebünde were subsequently dissolved, [ citation needed ] sometimes forcibly, and where refounded, their political influence was much reduced.
The pope 's involvement with the Golden Bull of was basically nonexistent, which was significant in the history of relations between the popes and the emperors.
When Charles IV laid down procedure for electing a King of the Romans, he mentioned nothing about receiving papal confirmation of the election.
From Wikipedia, the free encyclopedia. For other uses, see Golden Bull. January 10, chapters 1—23 ; December 25, chapters 24— The Golden Bull of AD.
Photo Gallery. Trailers and Videos. Crazy Credits. Alternate Versions. Rate This. Director: Fatih Akin. Writers: Fatih Akin screenplay , Heinz Strunk novel.
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Disturbing films. Use the HTML below. You must be a registered user to use the IMDb rating plugin. Edit Cast Credited cast: Jonas Dassler Fritz Honka Tristan Göbel Willi Margarete Tiesel Gerda Voss Adam Bousdoukos