Lesson 07: The Golden Ratio

Class# 7: The Golden Ratio                    Sept. 29-Oct. 3, 2014

Materials needed:  ?Strathmore Sketchbook Pad (9 x 12)?Drawing pencils?Eraser?Ruler?Pencils

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.  Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean.  The golden ratio is an irrational mathematical constant, approximately defined by the number Phi(f = 1.618033988749895... )

Long before the Renaissance, artistsand architectshave proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aestheticallypleasing.

History of the Golden Ratio

While the proportion known as the Golden Mean has always existed in mathematics and in the physical universe, it is unknown exactly when it was first discovered and applied by mankind. It is reasonable to assume that it has perhaps been discovered and rediscovered throughout history.

It is believed that the Egyptians may have used both pi and phi in the design of the Great Pyramids, which began to be constructed in 2560 BC.  Below is an image of the Great Pyramids:??http://a3.l3images.myspacecdn.com/images02/92/05ca2979d48a4e309a1443ccdd9bc4d1/?l.jpg

Here is a diagram that shows how the golden ratio may have been used on the great pyramids:?http://weekly.ahram.org.eg/2008/890/her2.jpg

The Greeks based the design of the Parthenon that was the climax of over four centuries of Greek temple architecture on this proportion.  Here is a picture of the Parthenon:??http://upload.wikimedia.org/wikipedia/commons/d/da/The_Parthenon_in_Athens.jpg

Here is a diagram that shows how the Golden Ratio/Golden Rectangle were used to design the Parthenon:??http://www2.rgu.ac.uk/subj/ats/teachingweb/teaching/t26DesignPrinciples/TheGoldenSectio?n/Parthenon.jpg

Later, the Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve balance and beauty. Leonardo Da Vinci, for instance, used it to define all the fundamental proportions of his painting of "The Last Supper," from the dimensions of the table at which Christ and the disciples sat to the proportions of the walls and windows in the background.  This important painting was done in Milan Italy in 1490. ?Here is the last supper:??http://www.jaydax.co.uk/lastsupper/lastsuppertongerlocopyz.jpg

Here’s da Vinci’s painting, The Last Supper, with golden sections highlighted:  ??http://emptyeasel.com/wp-content/uploads/2009/01/leonardo-supper.jpg

Some works in the Dutch artistic movement called De Stijl, or neoplasticism, exhibit golden ratio proportions. Piet Mondrian used the golden section extensively in his neoplasticist, geometrical paintings, created circa 1918–38. ??Here is an example of one such work:http://pietmondrian.co.uk/Piet%20Mondrian.jpg

Many artists, designers and architects continue to use the Golden Ratio in their work to the present day and now that you know what it is you can look for it when you see an artwork.

Now it is time for you to learn to draw a Golden Rectangle.  Here are the steps necessary to make a Golden Rectangle:

•  Draw a square using a straightedge and a measuring device. The square can be any size, but leave enough room on one side of your paper for extending the square up into a Golden Rectangle.  Here is a link to an image that shows the next few steps in drawing a Golden Rectangle:  ??http://www.wetcanvas.com/ArtSchool/Hagan/mean.gif

•  Now draw a vertical line through your square that divides it into two equal sized portions.

•  Now draw a line from the bottom of your new vertical drawing up to the upper right corner of your square.  This is figure 3 in these pictures:  ??http://www.wetcanvas.com/ArtSchool/Hagan/mean.gif

•  Now draw an arching line down from the right corner of your square and extend the bottom line of the square to meet it. 

•  Next draw a vertical line up from where your arching line from the right hard corner came and connect this with a horizontal line to make a closed rectangle.  This is the bottom shape in this image: ??http://www.wetcanvas.com/ArtSchool/Hagan/mean.gif

Now it’s time to add the golden spiral.  In geometry, a golden spiral is a logarithmic spiraldictated by the golden ratio.  That is, a golden spiral gets wider by a factor of Phi for every quarter turn it makes.

Using the Golden rectangle that you have already constructed, you will now add the golden spiral by following these steps.  Here is what your golden spiral should look like when it is done: ??http://linke.com.ve/content/wp-content/uploads/2009/05/golden-ratio-spiral.jpg

Using your ruler, copy the spiral you see in the image above over the golden rectangle drawing you had previously made.

A golden spiral is a shape that is often found in nature.  ??Here are some examples:

http://www.flickr.com/photos/lmingl/4223670815/

http://www.goldenmuseum.com/0602001.jpg

http://www.goldenmuseum.com/0602003.jpg

http://www.nautilusdivingbali.com/wp-content/uploads/2008/11/swimming-nautilus.jpg

Your homework for this class is to find and sketch five examples of the golden spiral that you can find where you live.  This might include shells, leaves, animals or anything else that you can find which is an example of the Golden Spiral (an expanding spiral found in nature).

After you have done your 5 drawings you should scan your work and upload it onto your Flickr page.  In the Flickr “Description” panel be sure to include the name of this assignment, which you can call “Golden Spiral”.  All drawings should be uploaded to your Flickr page by the evening of June 18.  Any work that is not uploaded by Oct. 3 will be marked down one letter grade for each week day it is late.

This concludes the seventh class for AR101.  If you have any questions regarding the material or assignment please email your instructor at art101@comfsm.fm