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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.
An infinite series can be derived to express phi: 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 the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer 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: 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. If the side lengths of a triangle form a geometric progression and are in the ratio 1: A triangle whose sides are in the ratio 1: 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: 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 originally shown by Kepler: 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 also appears in hyperbolic geometry , as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: 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.
The medial right triangle of this "golden" pyramid see diagram , with sides 1: 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.
A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus the source of a large part of modern knowledge of ancient Egyptian mathematics , based on the 3: 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, Röber studied various Egyptian pyramids including Khafre, Menkaure and some of the Giza, Sakkara, and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors identified as the Kepler triangle; many other mathematical theories of the shape of the pyramids have also been explored.
Whether the relationship to the golden ratio in these pyramids is by design or by accident remains open to speculation.
Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell , mathematician and historian, claimed in that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3: 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 mathematics of the Pyramids, citing Giedon The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements.
Similarly, Howard Vyse , according to Matila Ghyka,  reported the great pyramid height From Wikipedia, the free encyclopedia.
This article is about the number. For the pop music album, see The Golden Ratio album. For the calendar dates, see Golden number time. List of numbers Irrational numbers.
History of aesthetics preth-century and Mathematics and art. Canons of page construction. If the constraint on a and b each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation.
The sum of the two solutions is one, and the product of the two solutions is negative one. The knight on his quest: University of Delaware Press.
The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design. The New York Times.
The College Mathematics Journal. Euclid's Elements of Geometry. The MacTutor History of Mathematics archive. Phi In Art, Nature, and Science.
Translated by Benjamin Jowett. The Internet Classics Archive. Retrieved 30 May Elementary number theory in nine chapters 2nd ed.
Die Macht der Zahl: Was die Numerologie uns weismachen will. The Curves of Life. Classic Puzzles, Paradoxes, and Problems: Devlin The Math Instinct: Science, Philosophy, Architecture , p.
Analysing Architecture pp. Mirrors of the Unseen: Yet both claims, along with various others in a similar vein, live on. Retrieved September 26, Archived from the original on February 1, Retrieved April 9, Teaching Mathematics in the Block pp.
A Study of Composition in Art pp. The Form of the Book , pp. Determined by Jan Tschichold The lower outer corner of the text area is fixed by a diagonal as well.
A most remarkable measure". Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?
Famous problems and their mathematicians. The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
The mathematics of harmony: A credit card has a form of the golden rectangle. Cracking the Da Vinci code: The Golden Ratio also crops up in some very unlikely places: An Analysis of His Music.
The Dynamics of Delight: Architecture and Aesthetics New York: Archived from the original on December 19, Retrieved December 2, An experiment on harmony ", Huygens-Fokker.
Accessed December 1, Neue Lehre van den Proportionen des meschlischen Körpers. GoldenEye was developed over a period of two and a half years by an inexperienced team led by Martin Hollis , who had previously worked on the coin-op version of Killer Instinct.
It was partially conceived as an on-rails shooter inspired by Sega 's Virtua Cop , before being redesigned as a free-roaming shooter.
The game was highly acclaimed by the gaming media and sold over eight million copies worldwide, making it the third-best-selling Nintendo 64 game.
Retrospectively, GoldenEye is considered an important game in the history of first-person shooters for demonstrating the viability of game consoles as platforms for the genre, and for signalling a transition from the then-standard Doom -like approach to a more realistic style.
It pioneered features such as atmospheric single-player missions, stealth elements, and a console multiplayer deathmatch mode. The game is frequently cited as one of the greatest video games of all time.
A spiritual successor , Perfect Dark , was released in , while a reimagining of the game, also titled GoldenEye , was released in GoldenEye is a first-person shooter that features both single and multiplayer modes.
In the single-player mode, the player takes the role of James Bond through a series of free-roaming 3D levels. Each level requires the player to complete a certain set of objectives—such as collecting or destroying specified items, rescuing hostages, or meeting with friendly non-player characters —and then exit the stage.
The arsenal of weapons includes pistols , submachine guns , assault rifles , grenades , and throwing knives , among others.
Most of the game's firearms are modelled on real-life counterparts although their names are altered , while others are based on fictitious devices featured in the James Bond films, such as the Golden Gun and Moonraker laser.
The weapons vary in characteristics such as rate of fire, degree of penetration , and type of ammunition used, and inflict different levels of damage depending on which body part they hit.
Stealth is a significant element of the gameplay; frequent gunfire can alert distant guards,  and activated alarms can trigger infinitely- respawning enemies.
Therefore, to avoid gunfights with numerous opponents, it is advantageous to eliminate soldiers and security cameras before they spot or hear the player.
Certain weapons incorporate suppressor or telescopic sight attachments to aid the player in killing enemies discreetly. Four save files are available to track the player's progress through the game's twenty missions, each of which may be played on "Agent", "Secret Agent", or "Agent" difficulty settings.
Completing certain missions within particular target times enables the player to unlock bonus cheat options which make various changes to the gameplay.
The multiplayer mode allows two, three, or four players to compete against each other in five different types of split screen deathmatch games: It can be played as a free-for-all game or in teams.
The player with the Golden Gun is unable to pick up body armour while opponents can. The flag carrier cannot use weapons but can still collect them to keep opponents from stocking ammunition.
Two years after the Severnaya mission, Bond investigates an unscheduled test firing of a missile in Kyrgyzstan , believed to be a cover for the launch of a satellite known as GoldenEye.
As Bond leaves the silo, he is ambushed by Ourumov and a squad of Russian troops. Ourumov manages to escape during the encounter. The remainder of the game takes place in Bond visits Monte Carlo to investigate the frigate La Fayette , where he rescues several hostages and plants a tracker bug on the Pirate helicopter before it is stolen by the Janus crime syndicate.
The two escape the complex seconds before it is destroyed, on the orders of Ourumov, by the GoldenEye satellite's EMP. Bond and Natalya escape from Trevelyan, but are arrested by the Russian police and taken to the military archives for interrogation.
Bond eventually manages to escape the interrogation room, rescue Natalya, and communicate with Defence Minister Dimitri Mishkin , who has verified Bond's claim of Ourumov's treachery.
Petersburg,  eventually reaching an arms depot used by Janus. There, Bond destroys its weaponry stores and then hitches a ride on Trevelyan's ex-Soviet missile train ,  where he kills Ourumov and rescues Natalya.
Natalya accompanies Bond to the Caribbean. Surveying the Cuban jungle aerially, their light aircraft is shot down.
Unscathed, Bond and Natalya perform a ground search of the area's heavily guarded jungle terrain, but are ambushed by Xenia,  who is quickly killed by Bond.
Bond sneaks Natalya into the control centre to disrupt transmissions to the GoldenEye satellite and force it to burn up in the Earth's atmosphere.
Trevelyan attempts to re-align it in a final attempt to restore contact with the GoldenEye,  but Bond ultimately destroys machinery vital to controlling the antenna and defeats Trevelyan in a gunfight on a platform above the dish.
GoldenEye was developed by Rare and directed by Martin Hollis , who had previously worked as a second programmer on the coin-op version of Killer Instinct.
Sega 's light gun shooter Virtua Cop was the primary influence on the game, although id Software 's seminal first person shooter Doom and the Nintendo 64 launch title Super Mario 64 were also credited.
The team in charge of the project visited the studios of the GoldenEye film several times to collect photographs and blueprints of the sets that were used.
Details such as bullet marks on walls, cartridge cases being ejected from guns, and objects exploding were part of the design. Work on GoldenEye began in January  with a team of three people that was hired by Hollis: The first year was spent producing art assets and developing the engine, which originally only allowed the player and enemies to move around a virtual environment.
Originally, the development team considered the possibility for the game to have both on-rails and free-roaming modes because they did not know how the Nintendo 64 controller would be;  GoldenEye ' s gas plant location was modelled by Hilton with a predetermined path in mind.
There are rooms with no direct relevance to the level. There are multiple routes across the level. After the first year of development, Rare decided to add more staff to the project.
The first addition was designer David Doak , who helped with the level designs and worked on the game's AI scripting.
If you fired the same gun again within a certain amount of time, it did a larger radius test and I think there was a third even larger radius after that.
It meant if you found one guy and shot him in the head and then didn't fire again, the timer would reset". Though decidedly unrealistic, this was an intentional feature made to encourage the player to use windows to covertly spy on enemies.
A second programmer, Steve Ellies, was hired by Hollis six months later. He mentioned the film Heat as an influence.
The final Nintendo 64 hardware could render polygons faster than the SGI Onyx workstation the development team had been using.
This helped the developers significantly, as some backgrounds rendered at 2 frames per second on the Onyx without even drawing enemies, objects, or Bond's gun.
RGB colour textures cost a lot more in terms of processing power. You could do double the resolution if you used greyscale , so a lot was done like that.
If I needed a bit of colour, I'd add it in the vertex. As a result, the team toned down the killing and added an end credits sequence that introduces all the non-player characters, giving the game a filmic sense.
GoldenEye was released on a megabit cartridge on 25 August This function was originally made as an experimental side project by Rare and was deactivated in the final build of the game, but has since been unlocked through fan-made patches.
Despite low expectations among the gaming media and an unsuccessful showing at the Electronic Entertainment Expo in Atlanta in ,   GoldenEye turned out to be both a critical and a commercial success.
Ocarina of Time , which was released in November and sold nearly the same amount. Graphically, GoldenEye was praised for its varied and detailed environments,   realistic animations,   special effects such as glass transparencies and lingering smoke,   and high frame rate in multiplayer games.
The gameplay was highlighted for its depth. GoldenEye is frequently cited as one of the greatest video games of all time. In , GamePro , placing it ninth in "The 52 Most Important Video Games of All Time", called it the console killer-app of the s and the best game ever licensed from a film.
In a poll, readers of Computer and Video Games voted GoldenEye into first place in a list of "the hundred greatest video games",  and it was ranked fifth in a poll the following year.
Retrospective commentary on GoldenEye in the years following its release included an October review by Mark Reece from NintendoLife. Reece gave the game a rating of eight out of ten, commenting that although the game's multiplayer mode stands up well, its graphics, audio and "fiddly" aiming system are dated.
Video game journalists have praised GoldenEye for proving that it is possible to create a "fun" first-person shooter experience on a console in both single-player and multiplayer modes;  when the game was released, the first-person shooter was primarily a genre for PC gamers.
Journalists noted that the game "opened the genre to a completely new market" and that it was "the first big console [first-person shooter] that truly got it right.
GoldenEye introduced stealth elements not seen in previous first-person shooter games. Combat Evolved in After GoldenEye was released, Rare began development of a spiritual successor titled Perfect Dark.
These games contain several references to GoldenEye , including the design of the health- HUD , the nature of the aiming system, and the Russian dam setting of the opening level of TimeSplitters 2.
The reimagining features Daniel Craig as the playable character, contemporary first-person shooter conventions, entirely new level-layouts, and an online component.
Source , a multiplayer only total conversion mod developed using Valve 's Source engine. From Wikipedia, the free encyclopedia.
GoldenEye American box art. The geometry of some of the structures in the film top was replicated in the game bottom. Archived from the original on 6 October Retrieved 28 August Archived from the original on 3 April Retrieved 4 August Archived from the original on 28 January Archived from the original on 7 February Archived from the original on 16 July Retrieved 23 August Archived from the original on 17 June Archived from the original on 2 April Retrieved 26 August Inside the chemical weapons facility you should quickly and efficiently proceed to the bottling area and place demolition charges on the main gas tanks.
You'll be working with on this assignment, the bottling room is the rendezvous point. Now listen here, Bond.
Be sure to place the mines carefully, otherwise you won't take out all the tanks in the bottling room. His sacrifice should give you enough time to find an aircraft.
A spy satellite has detected increased levels of activity at an old observatory complex near Severnaya Penetrate the base and find out what is going on.
MI6 has become concerned that unscheduled test firings from a missile silo in Kirghizstan are being used to cover the launch of GoldenEye weapons satellites.
Infiltrate the silo and ascertain what is being placed into orbit. Monte Carlo, Part i: A demonstration of the Pirate stealth helicopter by the French military has been unexpectedly postponed.
Official channels insist that nothing is wrong but unofficially M16 has been asked to help salvage a very tricky hostage situation on board the frigate La Fayette.
It seems that a crime syndicate called 'Janus' will stop at nothing in its attempt to hijack the helicopter. We need you to place a tracker on the Pirate so that we can trace its movements after Janus steals it.
He claims he can set up a meeting with Janus. Captured by the Russians, Bond must escape their military prison. Your only chance is to speak directly to Defence Minister Mishkin.
She is being taken by car to the Janus operations base in St. Get after them, ! We can't afford to let the trail go cold and the chances are they'll lead you straight to Trevelyan.
The Janus operations base in St.