In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. It is an irrational number, a solution to the quadratic equation x 2 − x − 1 = 0, with a value of: φ = 1 + 5 2 = 1.6180339887 …. The golden ratio is called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number. Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio; the golden ratio has been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.
The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ. One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a = a a + b a = 1 + b a = 1 + 1 φ. Therefore, 1 + 1 φ = φ. Multiplying by φ gives φ + 1 = φ 2 which can be rearranged to φ 2 − φ − 1 = 0. Using the quadratic formula, two solutions are obtained: 1 + 5 2 = 1.618 033 988 7 … and 1 − 5 2 = − 0.618 033 988 7 … Because φ is the ratio between positive quantities, φ is positive: φ = 1 + 5 2 = 1.61803 39887 … According to Mario Livio: Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties....
Biologists, musicians, architects and mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction, surprising Pythagoreans. Euclid's Elements provides several propositions and their proofs employing the golden ratio and contains the first known definition: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser; the golden ratio was studied peripherally over the next millennium. Abu Kamil employed it in his geometric calculations of decagons.
Iwao Takamoto was an American animator, television producer, film director. He began his career as a production and character designer for Walt Disney Animation Studios films such as Cinderella and the Tramp, Sleeping Beauty, he moved to Hanna-Barbera Productions, where he designed a great majority of the characters, including Scooby-Doo and Astro, became a director and producer. Takamoto was born in California, his father emigrated from Hiroshima to the United States for his health, returned to Japan only once, to marry his wife. At 15 years of age, Takamoto graduated from Thomas Jefferson High School in Los Angeles. After the bombing of Pearl Harbor, Takamoto's family, like many Japanese Americans, was forced to move to the Manzanar internment camp, they spent the rest of World War II there, it was at the camp that Takamoto received basic illustration training from two co-internees. Takamoto first entered the cartoon world after the end of the war. Without the benefit of a formal portfolio of his work, he created a sketchbook of, by his own admission, "everything I saw".
It was based on this sketchbook. He was hired as an assistant animator by Walt Disney Animation Studios in 1945. Takamoto became an assistant to Milt Kahl, he worked as an animator and character designer on such titles as Cinderella, Peter Pan and the Tramp, Sleeping Beauty, One Hundred and One Dalmatians. Takamoto joined Hanna-Barbera Productions, he is arguably best known as a character designer. He was responsible for the original character design of such characters as Scooby-Doo, The Jetsons' dog Astro, Penelope Pitstop, he worked as a producer at Hanna-Barbera, supervising shows such as The Addams Family, Hong Kong Phooey, Jabberjaw. He directed several feature length animated films, including Charlotte's Jetsons: The Movie; the inspiration for Scooby-Doo's creation as a Great Dane came from an employee of the Hanna-Barbera company, who bred this dog. Takamoto was Vice-President of Creative Design at Hanna-Barbera and was responsible for overseeing Hanna-Barbera's many merchandising lines as well as design work for their Animation Art Dept.
In 1996, he received the Winsor McCay Award for lifetime achievement and contributions in the animation field. In 2005 he received the Golden Award from the Animation Guild, to honor his more than 50 years of service in the animation field. After Time-Warner merged with Turner Broadcasting in 1996, Takamoto became Vice President of Special Projects for Warner Brothers Animation. Iwao Takamoto married Jane M. Baer in 1957, they met at Disney. They had Michael. In 1963, he met Barbara Farber, the assistant to the public relations director at Hanna-Barbera, Arnie Carr. Part of her job was studio tours, how she and Takamoto met. Takamoto married Barbara in 1964 and remained married to her for 44 years, until his death in 2007. Barbara had a daughter from Leslie. Takamoto died on January 8, 2007, at Cedars-Sinai Medical Center in Los Angeles from a heart attack at the age of 81. Throughout the week following his death, Adult Swim put up a bumper reading "Iwao Takamoto ", he is buried at Mount Sinai Memorial Park Cemetery in Los Angeles in Gardens of Blessing, Section 3, Lot 1390, Space 3.
There was a memorial added to the end of the Scooby-Doo film Chill Out, Scooby-Doo!. Takamoto received the Winsor McCay Award, the lifetime achievement award from the International Animated Film Association Hollywood, he received an honorary citation from the Japanese American National Museum. In 2005, he was given a golden award from the Animation Guild. Takamoto's memoirs were published posthumously in 2009 by University Press of Mississippi as Iwao Takamoto: My Life with a Thousand Characters by Iwao Takamoto and Michael Mallory. An intimate memoir, Living With A Legend, was published posthumously in 2012 by TotalRecall Press by his stepdaughter, Leslie E. Stern. Scooby-Doo - television series William Hanna Joseph Barbera Takamoto, Iwao. Iwao Takamoto: My life with a Thousand Characters. Jackson, Mississippi: University Press of Mississippi. ISBN 9781604734775. Stern, Leslie. Living With A Legend. TotalRecall Publications. ISBN 1590950968. Iwao Takamoto on IMDb January 1999 interview in Animation Blast #3 Iwao Takamoto at Find a Grave
Nick Crowe is a Yorkshire artist. He live comic situations, he is a Senior Lecturer in Fine Art Practice at University of London. Crowe works in London and Manchester. Crowe was studied English literature and drama at Hull University, he began working in experimental theatre and performance in 1989, establishing Index Theatre Co-operative with David Whitaker, Julian Hammond, Jane Gant, Adele Fowles and Les Hampson. Index changed its membership over time. Crowe went on to develop his practice in a range of media including film, new media and etched glass, his art uses a range of strategies including the exploration of the role of technology and its effects on everyday life. Since the 1990s he was involved in setting up a range of artist-led initiatives including The Annual Programme in Manschester and The Manchester Pavilion at the Venice Biennale. In 2003, he and regular collaborator Ian Rawlinson co-curated Artranpennine03, a trans-regional exhibition of public art. Several of Crowe's pieces are internet-based.
SERVICE 2000, for example, was a ring of spoof websites for notable art galleries. Other work, like "Discrete Packets" and "Police Radio" use the web as a medium for exploring wider cultural concerns. Crowe works a great deal in more traditional art media including film and photography as well as installation and publishing, he produces engraved glass works which deal with subjects as wide as internet memorial web pages and the occupation of Iraq. Some film and video work, uploaded to YouTube has engaged with the pitfalls and possibilities of the online participatory culture with subtle use of displacement and irony. In 2003, Chisenhale Gallery commissioned Crowe to produce "Getting On", a reflection on age and contemporary Britain, that comprised "a group of metallic-finish, scarlet mobility scooters and a soundtrack of Deutsche Grammophon classical pieces. In the same year, Nick Crowe was shortlisted for the Beck's Futures prize. Since 1994 Crowe has a collaborative practice with Manchester based artist Ian Rawlinson.
Together they have worked on many projects using boredom to deal with issues surrounding national identity and faith. Exhibitions include:'Song for Coal', Yorkshire Sculpture Park, Wakefield, 2014'Ride Across Lake Constance', LoBe, Berlin, 2013'Six White Horses', Whitworth Art Gallery, Manchester, 2013'The Opera', Plataforma Revolver, Lisbon, 2012'Nick Crowe and Ian Rawlinson', SALT, Istanbul, 2012'Towards A Free Society', Institute of Jamais Vu, London, 2012'Stolen Artifact', Kotti Shop, Berlin, 2011'The Fireworks', Gallery Kit, St Louis, 2011'No Sign of Helicopters', Ceri Hand Gallery, Liverpool, 2010'Nick Crowe & Ian Rawlinson', Newlyn Art Gallery, Penzance, 2008'At 25 Metres', FACT, Liverpool, 2007'Nick Crowe and Ian Rawlinson', Manchester Art Gallery. Home page www.croweandrawlinson.net