Architecture

The Golden Mean does apply to architectural composition in the context of scaling hierarchy that organizes complexity. It is used to determine pleasing dimensional relationships between the width of a building and its height,and even the position of the columns supporting the structure.

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Music

In the music of the twentieth century the golden ratio has found fertile ground spreading out of all proportion: Debussy, Stockhausen, Bartók, Stravinsky, are just some of the best known names that have voluntarily chosen the use of these magical proportions.

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Design

The golden ratio has been used throughout history to create design elements that have an ideal visual appeal. Because the shape is rooted in nature and mathematics, it’s the perfect combination of balance and harmony.

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Nature

The golden ratio is sometimes called the "divine proportion," because of its frequency in the natural world. The number of petals on a flower, for instance, will often be a Fibonacci number. The seeds of sunflowers and pine cones twist in opposing spirals of Fibonacci numbers.

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Architecture

The Parthenon

The Parthenon in Athens, built by the ancient Greeks from 447 to 438 BC, illustrates the application of the Golden Ratio. The symbol for the golden ratio, the Greek letter phi-, was named after the sculptor Phidias. The golden ratio appears in several constructions and layouts of the Parthenon. Phidias widely used the golden ratio in his works of sculpture. The space between the columns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece. The Parthenon also displays golden ratios in the height of the cross beams in relation to its height, and also in the rectangular designs below the sculptures.

The Taj Mahal

The Taj Mahal, located in Agra, India, is India’s most well known structure. It was built by Shah Jahan in 1653 and was designed using the Golden Ratio. This is why it looks so perfect. The rectangles that served as the basic outline for the exterior of the building were all in the Golden Proportion. The Taj Mahal also displays golden proportions in the width of its grand central arch to its width, and also in the height of the windows inside the arch to the height of the main section below the domes. It was shown that the ideal of the Taj is rationally and intently based on the Golden section, as so as all non-Golden regularities are hierarchically dependent to the Golden ones.

The Great Pyramid of Giza

It is believed to be 4,600 years old, which was long before the Greeks. The largest of the pyramids in Giza contains the use of phi and the golden ratio. The golden ratio is represented as the ratio of the length/height of the triangular face to half the length of the square base. Analysis of the site of the Great Pyramid of Giza reveals that the positions and relative sizes of the pyramids are based on the golden ratio. Two main theories were used by architects: one stating that Pi was used to set the proportion base – height, the other that this proportion was determined using the Golden Ratio Phi. The interesting point is that both the theories fits the Great Pyramid proportions with two or three decimal precision.

Chartres Cathedral

In a book Ad Quadratum, Frederik Macody Lund, a historian who studied the geometry of several Gothic structures, claims that the Cathedral of Chartres (begun in the 12th century) is designed according to the golden ratio. The Medieval builders of churches and cathedrals approached the design of their buildings in much the same way as the Greeks. They tried to connect geometry and art. Inside and out, their building were intricate construction based on the concept of golden ratio. The architects believed in a connection between geometrical design and artistic beauty when incorporating the Golden Ratio into their construction.

Notre Dame

Notre Dame in Paris, which was built in between 1163 and 1250 appears to have golden ratio proportions in a number of its key proportions of design. It was claimed as the first Christian church in Paris, Notre Dame was built upon the Roman temple of Jupiter. The west façade of the church was completed around the year 1200, and it is here where the presence of the use of the golden ratios is visible. Although it is rather asymmetrical in its design and difficult to measure photographically because of parallax distortions, the golden ratio lines of the green, blue and red rectangles conform closely to the major architectural lines. It reflects these proportions in the heights of each major stage of the structure as well as in the width of the columns at the top. Each application of the golden ratio is different and unique, but each adds to its overall architectural beauty and harmony with the design principles found in nature.

Le Corbussier

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He developed his doctrine for the proportions of construction. Le Corbusier developed the Modulor in the long tradition of Vitruvius, Leonardo da Vinci’s Vitruvian Man, the work of Leone Battista Alberti, and other attempts to discover mathematical proportions in the human body and then to use that knowledge to improve both the appearance and function of architecture. The system is based on human measurements, the double unit, the Fibonacci numbers, and the golden ratio.

The UN building

The current headquarters for the United Nations was constructed on an 18-acre piece of land in the East side of Manhattan. The organization was under the supervision of lead architect, Wallace K. Harrison from the United States. Although Harrison is not typically known to use the golden ratio in his designs, a French architect Charles E. Jeanneret was on a team of assisting architects. Charles E. Jeanneret was known to frequently use the golden ratio in his architectural designs. When constructing the United Nations headquarters, the team of architects decided to use this ratio in a couple of different ways. When looking closely at the building we observe that many of the windows in fact have the golden ratio when comparing their width and height.

Music

Mozart's works

Mozart’s based many of his works on the Golden Ratio – especially his piano sonatas. Mozart arranged his piano sonatas so that the number of bars in the development and recapitulation divided by the number of bars in the exposition would equal approximately 1.618, The Golden Ratio.

Musical composition

The Golden Ratio has found a great application in the timings of musical compositions. As an example, the climax of songs is often found at roughly the phi point (61.8%) of the song, as opposed to the middle or end of the song. In a 32 bar song, this would occur in the 20th bar.

Haylock's works

Haylock discovered a number of occurrences of Golden Ratio in the first movement of Beethoven’s first Symphony. That’s the one that starts with the famous motto theme: “da, da, da, daah!”

Generating a scale

The golden ratio in music has some interesting properties that the Western equal-tempered scale could scarcely hope to reproduce. If you’ve some highly accurate micro-tunable instruments and perhaps some scale-designing software such as Scala, LMSO, then your next step is to design a scale that features the golden ratio.

Design

Instrument Design

Jody Espina, a highly regarded jazz musician and designer of Jody Jazz Saxophone and Clarinet Mouthpieces has applied Phi in the design of his premier saxophone mouthpiece, the JodyJazz DV. “ Every measurement was analyzed with Phi in mind and used when applicable. This included the length of the bore, the width of the shank walls, the beak of the mcp, the depth of the bore at the facing and others.” – Jody Espina

Typography

The easiest way to start using the golden ratio is to implement it within your typographical graphic design elements. The Golden ratio when incorporated in the design of anything will have a significantly higher appeal than the elements without this ratio.

Speaker Wire

The high-end cable manufacturer Cardas Audio applies the Golden Ratio to the design of their speaker wire which they call “Golden Section Stranding.” The individual conductor strands of their cables are arranged in a Phi proportion to the others allowing them to be uniquely musical and pure.

Logo Design

The Golden Ratio is used to add aesthetic appeal directly to a company’s branding. Even if the logo itself isn’t shaped like a golden rectangle or triangle, it can still employ elements that use golden proportions.

Nature

Shells

The unique properties of the Golden Rectangle provides another example. This shape, a rectangle in which the ratio of the sides a/b is equal to the golden mean (phi), can result in a nesting process that can be repeated into infinity — and which takes on the form of a spiral. It's call the logarithmic spiral, and it abounds in nature.

Flower petals

The number of petals in a flower consistently follows the Fibonacci sequence. Phi appears in petals on account of the ideal packing arrangement as selected by Darwinian processes; each petal is placed at 0.618034 per turn (out of a 360° circle) allowing for the best possible exposure to sunlight and other factors.

Spiral Galaxies

Not surprisingly, spiral galaxies also follow the familiar Fibonacci pattern. The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees. As an interesting aside, spiral galaxies appear to defy Newtonian physics. As early as 1925, astronomers realized that, since the angular speed of rotation of the galactic disk varies with distance from the center, the radial arms should become curved as galaxies rotate.

Tree Branches

The Fibonacci sequence can also be seen in the way tree branches form or split. A main trunk will grow until it produces a branch, which creates two growth points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems. A good example is the sneezewort. Root systems and even algae exhibit this pattern.

Hurricanes

Seen in the eye of the storm, the Golden Ratio is very prevailent in Hurricanes. The changes in pressure create the central circle, this formation is referred to as the Coriolis Effect. The changes in pressure and speed of the winds both below the formation and above are responsible for the spawning of the original spiral shape.

Pine Cones

Nature is replete with spirals, so perhaps it’s no surprise that they are found in pine cones. The more interesting thing is that the number of spirals found on pine cones are almost always Fibonacci numbers. The ratio of two neighboring Fibonacci numbers is an approximation of the golden ratio (e.g. 8/5 = 1.6). This is commonly represented by drawing a series of squares on graph paper and then drawing a spiral across the squares.