**The Golden Ratio and The Golden Spiral**

The Golden Spiral

In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.

Phi - The Golden Ratio

Phi is a letter of the Greek alphabet.

It is used to represent the

Golden Ratio.

But what is it's value?

Φ = 1.618033988749894848204586834...

Places where the golden spiral and golden ratio is seen...

The golden spiral is seen in-

Architecture

Photography and Drawings

In Humans ( actually by God )

Nature

examples

Many artists who lived after Phidias have used this proportion.Leonardo Da Vinci called it the "divine proportion" and featured it in many of his paintings, for example in the famous "Mona Lisa". Try drawing a rectangle around her face. You will realize that the measurements are in a golden proportion. You can further explore this by subdividing the rectangle formed by using her eyes as a horizontal divider.

Leonardo Da Vinci

Leonardo did an entire exploration of the human body and the ratios of the lengths of various body parts. “Vitruvian Man” illustrates that the human body is proportioned according to the Golden Ratio.

The “Vitruvian Man”

“Phi“ was named for the Greek sculptor Phidias.The exterior dimensions of the Parthenon in Athens, built in about 440BC, form a perfect golden rectangle.

The Parthenon

The base length of Egyptian pyramids divided by the height of them gives the golden ratio.. amazing

Pyramids

The length of different parts in your arm also fits the golden ratio.

Look at your own hand:

You have ...

2 hands each of which has ...

5 fingers, each of which has ...

3 parts separated by ...

2 knuckles

Golden Ratio in Human Hand and Arm

The dividence of every long line to the short line equals the golden ratio.

Lenght of the face / Wideness of the face Lenght between the lips and eyebrows / Lenght of the nose, Lenght of the face / Lenght between the jaw and eyebrows Lenght of the mouth / Wideness of the nose, Wideness of the nose / Distance between the holes of the nose, Length between the pupils / Length between teh eyebrows.

All contain the golden ratio.

Golden Ratio in the Human Face

In Architecture

In Photography and Drawings

Golden ratio in humans

The Golden Spiral can be seen in the arrangement of seeds on flower heads.

The shape of the inner and outer surfaces of the sea shells, and their curves fit the golden ratio..

Golden Ratio In The Sea Shells

The ratio of the braches of a snowflake results in the golden ratio.

Golden Ratio In the Snowflakes

In nature

Some specific proportions in the bodies of many animals (including humans ) and parts of the shells of mollusks and cephalopods are often claimed to be in the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio. The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio. The nautilus shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one; however, measurements of nautilus shells do not support this claim.

Historian John Man states that the pages of the Gutenberg Bible were "based on the golden section shape". However, according to Man's own measurements, the ratio of height to width was 1.45.

The proportions of different plant components (numbers of leaves to branches, diameters of geometrical figures inside flowers) are often claimed to show the golden ratio proportion in several species.[88] In practice, there are significant variations between individuals, seasonal variations, and age variations in these species. While the golden ratio may be found in some proportions in some individuals at particular times in their life cycles, there is no consistent ratio in their proportions.

In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.[89] The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.

Want To Know More About History Of Golden Spiral :

Phidias (490–430 BC) made the Parthenon statues that seem to embody the golden ratio.

Plato (427–347 BC), in his Timaeus, describes five possible regular solids (the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron), some of which are related to the golden ratio.

Euclid (c. 325–c. 265 BC), in his Elements, gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio" (Greek: ἄκρος καὶ μέσος λόγος).

Fibonacci (1170–1250) mentioned the numerical series now named after him in his Liber Abaci; the ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically.

Luca Pacioli (1445–1517) defines the golden ratio as the "divine proportion" in his Divina Proportione.

Michael Maestlin (1550–1631) publishes the first known approximation of the (inverse) golden ratio as a decimal fraction.

Johannes Kepler (1571–1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers, and describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." These two treasures are combined in the Kepler triangle.

Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.

Martin Ohm (1792–1872) is believed to be the first to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.

Édouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.

Mark Barr (20th century) suggests the Greek letter phi (φ), the initial letter of Greek sculptor Phidias's name, as a symbol for the golden ratio.

Roger Penrose (b. 1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.

Perceptual studies

Studies by psychologists, starting with Fechner, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.

Illustration Using Nautilus Shell

In Photography

Some Disputes

There Are Many, Many, Many More Examples To Show The Golden Spiral In Our Surroundings

**In Flora**

**Miscellaneous**

Did You Notice?

Did you notice that the ZOOM-IN PATTERN is also of the golden spiral form?

Made By

Hrishabh - 5 (pictures)

Jenson - 6 (info)

Jomy - 8 (pictures)

Mahil - 10 (making)

Mihir - 11 (making)

Priyesh 12- (info)