The Golden Ratio

Ashleigh McFall and Claire Leone golden_ratio01.jpg What is the Golden Ratio?

The Golden Ratio has many names. The most common name is of course, the Golden Ratio. This name was created by Martin Ohm in 1835. Other names that Martin Ohm created in 1835 were the Golden Mean and the Golden Number. Although we frequently use the Golden Ratio, the original name for this mystical number was the Divine Proportion, which was created by Luca Paciolli in 1509. phi.gif
The number that has been associated with the Golden Ratio is called Phi. Phi is very similar to pi, and is considered the cousin of pi. Just like pi, phi is a non-repeating and never ending number and therefore extending to infinity. Phi is approximately 1.61803399887... and continuing on forever, just like its "cousin".

This is the Greek symbol for Phi



Euclid
History Euclid.jpg

The history of the Golden Ratio is not exact but we do know some of the most important contributors. Euclid of Alexandria taught in the ancient city of Alexandria around 300 B.C.E. and studied the Golden Ratio. Of course, he was not the only one. Some of the othersplato1.jpg who were involved with the Golden Ratio were Martin Ohm, Luca Paciolli, Ptolemy I, Kepler and Plato. Martin Ohm made his discoveries and created the names the Golden Ratio and the Golden Number around 1835. Luca Paciolli gave the name to the number the Divine Proportion in 1509. Ptolemy I was a successor to Alexander the Great and made some of his
own discoveries about the mystical number. However Plato was the first
to study the Golden Ratio even before Euclid did. Plato was an impressive mathematician and through his discoveries he prophesied about phi's significance. He possessed a unique talent which enabled him to see the world in perfect shapes and mathematical formulas. This talent gave him the ultimate logical point of view. As you can see the history involved many great mathematicians for the discovery of this mystical number.


Plato

Connection with the Fibonacci Sequence

Fibonacci.jpg

fibonacci.gifThe Fibonacci Sequence which was created by Leonardo de Pisa who was also known as Fibonacci. The Fibonacci Sequence is a pattern that starts with two 1's and then creates the succeeding terms by adding the last two numbers to get the next term. This pattern was discovered through an experiment on the breeding of rabbits. The pattern goes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233... The Fibonacci Sequence has been known as a type of the "Golden Ratio in disguise". The Fibonacci Sequence can be seen in microtubules in a cytoskeleton. It is also seen in logarithmic spirals and disk-shaped galaxies. Leonardo de Pisa


Fibonacci Sequence




Where is the Golden Ratio found?



The Golden Ratio is found in many different places.

nature_3.jpgnature.jpg It is very commonly found in nature. Botanists, scientists who study plants, use the term phyllotaxis to describe how leaves are formed on plants. According to botanists "successive leaves tend to spiral around each new leaf...". There is a common angle of 137.5 degrees that scientists have found that many plants share. It is said that if you "take the 360 degrees that fill out an entire circle and divide it into two angles that conform to the golden ratio, the smaller angle, which is the smaller segment of our now angular line) in 137.5 degrees. Another example of the golden ratio in nature is the logarithmic spiral. The logarithmic spiral was discovered in the seventeen-century by a Swiss mathematician named Jakob Bernoulli. The logarithmic spiral appears in astronomy and nature. In astronomy the logarithmic spiral appears in giant spiral galaxies. Also in nature the logarithmic spiral is found in the shells of mollusks. It is said the the logarithmic spiral is nature's "favorite shape". According to many scientists, "as the spiral grows, the distance of the coil from the center increases with the same ratio for every equal increase in the angle. This is best understood geometrically using the golden ratio to trace out increasingly larger squares within the rectangles."

mona_lisa_2.jpg The golden ratio is also very commonly found in art. The golden ratio was found very often in the work of artist Leonardo Da Vinci. Leonardo Da Vinci explored the human body and the ratios relating to the lengths of various body parts. He called this ratio the "divine proportion". He then featured this "divine proportion" in many of his paintings. This is show in such works as the Old Man. Then in his famous Vitruvian Man and the Mona Lisa it is easy to find golden rectangles connecting all of the pivotal points in the paintings. The golden ratio is present in many other artists' work such as Michelangelo's Holy family and Raphael's Crucifixion.

The golden ratio is also commonly found in architecture. Phidias, who lived from 490 B.C.E. to 430 B.C.E. first used the "divine proportion" in sculpture. In 447 B.C.
parthenon_33585425_std.jpgE. he was appointed to oversee the building of the Parthenon and other buildings in Athens. The Parthenon was built between 447 B.C.E. and 438 B.C.E. Ictinos, the builder of the Parthenon, wrote all about the proportion used in the Parthenon, but the book has since been lost. The most prevalent instance of the golden ratio in the Parthenon is that the front elevation uses the golden rectangle.


The golden ratio is also evident in the thermodynamics of black holes through a unique mathematic property of the golden ratio that means that its square can be obtained simply by adding 1 to phi.


golden_rectangle.jpgThe golden ratio is, of course, also found in geometry and geometric figures. The golden rectangle is one of the most famous and well known examples of the golden ratio. The golden rectangle is a "golden rectangle" because: length (AB) to the height (BC) = phi. One side of the rectangle = 1 and the other equals phi. The golden rectangle is considered one of the most interesting quadrilaterals. The golden rectangle has been around since at least the time of Pythagoras of Samos, who lived from 580 B.C.E. to 500 B.C.E. The golden rectangle is the most pleasing shape to the eye and that is why many artists and architects use this rectangle. The oldest use of the special rectangle is displayed on the Parthenon.













The golden ratio also appears in geometry especially when one is examining a line segment. If you take any line segment and divide it into two parts, so that the longer part of the line segment is in the same proportion to the shorter part as the entire line segment is to the longer part then that is the representation of the golden ratio.
golden_segment.jpg



Mario Livio



Mario Livio was a multi-faceted man. He was an art enthusiast and an astrophysicist. He spent a decent amount of his time studying the Golden Ratio. He later went on to write the book; The Golden Ratio: The Story of Phi, the World's Most Astonishing Numbers.

    • Art enthusiast
    • Astrophysicist
    • The Golden Ratio: The Story of Phi, the World's Most Astonishing Numbers









__Bibliography:__

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