THE GOLDEN RATIO

Certain pictures, objects, and animals appeal to the human mind more than others.  Proportions and images of symmetry often contribute to our fascination with them.  Often, when examined carefully, you may find a common “coincidence” between man made objects and those found naturally in nature.  This fluke, however, may be used to ascertain various mathematical relationships between these objects.  This paper will introduce the golden ratio and weigh its significance on math, art, and nature.

1.6180339887…. has been given many names varying from the “golden ratio” first coined by the Greeks, to the “golden rectangle” and  “golden section”, “phi” named after Phidias a renowned Greek sculptor, as well as the “divine proportion” conceived by Leonardo da Vinci.  (Blacker, The Golden Ratio)  Simply put, the golden ratio is the length to width of rectangles used in art and nature.  This ratio is considered to be the most agreeable arrangement, mathematically and artistically, to the eye.

Perhaps the first to use the golden ratio were the Egyptians.  Many (if not all) of the pyramids were made with the golden ratio kept solely in mind… as if they were made only using the ratio.  Later, the Greeks began using it in their architecture as well as their sculptures.  Phidias and others popularized the golden ratio by basing their achievements on it.  The Parthenon, specifically, as well as several other buildings and sculptures were the subject of which the ratio has left its mark.

The Greeks and the Egyptians were by far not the only people to have been affected by the number.  Famous painters and mathematicians have also recognized the ratio’s significance.

Perhaps the most famous and blatant use of the golden ratio has come in the works presented by Leonardo da Vinci.  Born in 1452 Leonardo da Vinci was born quite insignificant but rose to fame through his incredible array of skills.

The definitive polymath, he had almost too many gifts, including superlative male beauty, a splendid singing voice, magnificent physique, mathematical excellence, scientific daring… (Beckett, 117)

He studied at various places including Milan and Florence and the Vatican.  It is in these cities that he became famous.  He masterfully uses the golden ratio in the Mona Lisa framing her head as well as the rest of her body parts in exact proportion to the golden rectangle.  Furthermore, he goes on in such works as the Vitruvian Man and Virgin and Child with St. Anne to incorporate the golden rectangle into everything he possibly can.  He was by no means enthralled in art.

Instead, his great passions were mathematics and the natural world, and he compiled volumes of detailed drawings and notes on anatomy, botany, geology, meteorology, architectural design, and mechanics. (Stokstad, 693)

Toward the end of his life math, particularly the golden ratio, began to dominate everything he created.  Leonardo da Vinci died in 1519.

Another painter that used, primarily, golden rectangles was Piet Mondrian.  In, Composition with Gray and Light Brown virtually every rectangle has the “pleasing” dimensions.

Still, more possibilities abound.  It has been proven that famous composers such as Bach, Beethoven and Bartok have used the golden ratio between intervals of their masterpieces.  (Fibonacci & The Golden Ratio, Internet)

The greatest mathematical concept associated with the golden ratio has, no doubt, been in connection with the Fibonacci sequence.  The Fibonacci sequence is a list of numbers revealed by adding together the two numbers before it.  For example:

As the list grows, when you divide any number by the previous number, your result approaches the golden ratio.  This, without a doubt, links the mathematical world with the world of nature.  For, examples of the Fibonacci sequence abound in nature.  Pine cones, nautilus shells, daisy petals, galaxies and even the clouds above our heads not only hint about the golden ratio but reveal it.

For example, in the pine cone, if you count the number of rows going in one direction and compare it to the amount going in the opposite direction the result will almost always be two numbers in the Fibonnaci sequence.  A nautilus’ shell as well as the clouds and galaxies also appear in the same fashion.  Daisies, as well as other flowers, also have the peculiar habit of bearing seeds that spiral in corresponding Fibonnaci numbers.

Among other naturally occurring golden ratio examples in nature are the ratio of a persons shoulder to the tip of his fingers and from his elbow to his fingers.  Still, you can find the ratio in freaks of nature such as lightning.

Still there are even more types of golden ratios.  For instance take, for example, the golden pentagon that relates the Perrin sequence to successive triangles through the formulas

where s[n] equals the height of the nth term.  The ratio, however, comes from the expansiveness of one triangle to the next resulting in a proportion that approaches 1.324717957.  (Brown, Golden Rectangle)

There is somewhat of an uncanny relationship between the natural world and the mathematical world.  It is examples such as those above that have coerced the human mind into thinking that a natural world is a mathematical world.  We may come closer and closer to the truth… but, we may never know.

BIBLIOGRAPHY

Beckett, Sister Wendy.  The Story of Painting:  The Essential Guide to the History of Western Art.  New York:  Dorling Kindersley, 1994.

Blacker, Steve and Jeanette Polanski and Marc Schwach.  “Golden Ratio:  Fibonacci in Nature.”  Dec. 8, 1999.  http://www.geom.umn.edu/~demo5337/s97b/spiral.html

Blacker, Steve and Jeanette Polanski and Marc Schwach.  “Golden Ratio.”  Dec. 8, 1999.  http://www.geom.umn.edu/~demo5337/s97b/art.htm

Brown, Kevin.  “Math Pages:  The Golden Pentagon.” Dec. 11, 1999.  http://www.seanet.com/~ksbrown/kmath153.htm

“Fibonacci & The Golden Ratio.”  Dec. 8, 1999. http://www.aegsp.br/hs/fib/

Knott, Dr. Ron.  “Fibonacci Numbers and Nature.”  Dec. 8, 1999.  http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits

Snyder, Brian Joseph.  “The Golden Mean – The Golden Section.”  Dec. 7, 1999.  http://www.netreach.net/people/waterboy/phiratio/

Stokstad, Marilyn.  Art History.  New York:  Prentice Hall, 1999.