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The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section.
Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.
Though Heinz Bohlen proposed the non-octave-repeating cents scale based on combination tones , the tuning features relations based on the golden ratio.
As a musical interval the ratio 1. Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio".
The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law.
In , the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.
However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.
The golden ratio is key to the golden-section search. The golden ratio is an irrational number. Below are two short proofs of irrationality:.
If we call the whole n and the longer part m , then the second statement above becomes. Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication.
The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial. Having degree 2, this polynomial actually has two roots, the other being the golden ratio conjugate.
The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution see, for example, Thomson problem.
However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.
This method was used to arrange the mirrors of the student-participatory satellite Starshine Application examples you can see in the articles Pentagon with a given side length , Decagon with given circumcircle and Decagon with a given side length.
Both of the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio.
The golden triangle can be characterized as an isosceles triangle ABC with the property that bisecting the angle C produces a new triangle CXB which is a similar triangle to the original.
In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio.
This result is a straightforward consequence of the intersecting chords theorem and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer H.
Coxeter who published it in Odom's name as a diagram in the American Mathematical Monthly accompanied by the single word "Behold! The golden ratio plays an important role in the geometry of pentagrams.
Each intersection of edges sections other edges in the golden ratio. 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 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? Goofs In the first battle scene as the Seven Brothers are rushing forward to meet their foes, one of them stumbles over a sword that is lying on the ground, apparently from a previous take.
Leyland Van Helsing : Well who's he sir? British Consul : A sort of Tong leader. Awful blackguard. Leyland Van Helsing : How very jolly.
Alternate Versions The UK cinema version was cut by the BBFC to remove shots of a man spitting blood during a fight scene and a frog being cut up in a street market.
The video release was cut by 12 secs by the BBFC to edit a throat being punctured in a fight scene, an ear clap and to remove a shot of one of the village girls being stripped during the village attack.
The cuts were fully restored in the DVD release. Was this review helpful to you? Yes No Report this. Add the first question. Language: English Mandarin.
Filming Locations: Hong Kong, China. Runtime: 89 min 75 min re-edited American. Sound Mix: Mono. Color: Color Eastmancolor. Edit page. Louis, who had eventually subdued his rival's claim on the battlefield,  made a first attempt to clarify the process in the Declaration of Rhense of , which renounced any papal involvement and had restricted the right to choose a new king to the prince-electors.
Firstly, the Bull explicitly named the seven Prince-electors Kurfürsten who were to choose the King and also defined the Reichserzämter , their largely ceremonial offices at court: .
Secondly, the principle of majority voting was explicitly stated for the first time in the Empire. The Bull prescribed that four out of seven votes would always suffice to elect a new King; as a result, three Electors could no longer block the election.
Thirdly, the Electoral principalities were declared indivisible , and succession to them was regulated to ensure that the votes would never be divided.
Finally, the Bull cemented a number of privileges for the Electors, confirming their elevated role in the Empire. It is therefore also a milestone in the establishment of largely independent states in the Empire, a process to be concluded only centuries later, notably with the Peace of Westphalia of 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.
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Trailers and Videos. Crazy Credits. Alternate Versions. Rate This. Director: Fatih Akin. Writers: Fatih Akin screenplay , Heinz Strunk novel.
Added to Watchlist. From metacritic. New Holiday Movies to Stream Now. Lingua Estrangeira. Streaming Movies Marked as Favourite. Disturbing films.2/5/ · GOLDENE 7, viele Gewinne win 1/2 - Duration: Rubbel Freak 43, views. Diamond 7's Doubler 💎🎰 - New Scratchcards - Duration: SPayne Scratchies Recommended for oratelnetworks.com: Sepher Saadat. Der Goldene Schnitt 7; ©; Bridal Gown. The slim sleeves are in one with the shoulder yoke. NOTE: The trademark "Der Goldene Schnitt" pre-dates the Lutterloh name, which was not yet used in Die „Goldene 7“ und ihre goldigen Aussichten. Die „Goldene 7“ ist ein Rubbellos mit 10 Gewinnchancen. Der Höchstgewinn beträgt € (Chance 1: 1,2 Mio.), ein Los kostet 5 €. Die „Goldene 7“ lässt sich als kleine Aufmerksamkeit mit großen Gewinnchancen gut verschenken.