Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter. According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions. ===Music=== Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis.
Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.
Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835.
Kepler said of these: 18th-century mathematicians Abraham de Moivre, Daniel Bernoulli, and Leonhard Euler used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula".
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 1859, the pyramidologist John Taylor misinterpreted Herodotus () as indicating that the Great Pyramid's height squared equals the area of one of its face triangles.
James Sully used the equivalent English term in 1875. By 1910, mathematician Mark Barr began using the Greek letter Phi (φ) as a symbol for the golden ratio.
The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position". The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section.
James Sully used the equivalent English term in 1875. By 1910, mathematician Mark Barr began using the Greek letter Phi (φ) as a symbol for the golden ratio.
Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat.
However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition used the golden ratio in any compositions is more difficult to determine.
Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat.
Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application.
It has also been represented by tau (τ), the first letter of the ancient Greek τομή ('cut' or 'section'). Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.
It has also been represented by tau (τ), the first letter of the ancient Greek τομή ('cut' or 'section'). Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.
This led to Dan Shechtman's early 1980s discovery of quasicrystals, some of which exhibit icosahedral symmetry. ==Applications and observations== ===Architecture=== The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion.
A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition. A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases.
Archived November 2007.
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