The golden rectangle in life and in glass art

 

The Golden Rectangle, also called the perfect rectangle, is a rectangle in which the ratio of its length to its width is an exact representation of the Golden Ratio .

Many believe that this is one of the most visually pleasing of all geometric shapes. and it appears in many works of art and architecture.  The Parthenon of ancient Greece is the most famous example of the use of the golden rectangle. People also find the golden rectangle in the Mona Lisa, and many other Renaissance art works.

Now to be technical for a moment, in geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, which is to say 1 : 1 + 5 2 , which is 1 : φ (the Greek letter phi), where φ is approximately 1.618.

Got that? I’ll be running a quiz later.

Anyhow golden rectangles exhibit a special form of self-similarity: All rectangles created by adding or removing a square from an end are golden rectangles as well. If you start with a Golden Rectangle, you can divide it into a square and another Golden Rectangle.  You can then repeat this pattern endlessly with each succeeding smaller rectangle.

If arcs are drawn in the Golden Rectangle (as shown below), then you get a very graceful curve that is closely related to the spiral of a nautilus seashell. Indeed, this spiral can also be seen in many places in nature, such as flowers whose leaves spiral around the stem.

A golden rectangle can be constructed with only a straightedge and compass in four simple steps:

1. Draw a square.
2. Draw a line from the midpoint of one side of the square to an opposite corner.
3. Use that line as the radius to draw an arc that defines the height of the rectangle.
4. Complete the golden rectangle.

As I’ve said, a remarkable and distinctive feature of this shape is that when a square section is added — or removed — then the product is another golden rectangle, having the same aspect ratio as the first.

This “Square addition or removal” can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property. Diagonal lines drawn between the first two orders of embedded golden rectangles will define the intersection point of the diagonals of all the embedded golden rectangles; writer, editor, inventor and scientist Clifford A. Pickover referred to this point as “the Eye of God”.

The proportions of the golden rectangle have been observed as early as the Babylonian Tablet of Shamash (c. 888–855 BC), though some experts consider any knowledge of the golden ratio before the Ancient Greeks “doubtful”.

Then again, we simply don’t know everything there is to be known about the wisdom of the ancients, so who knows?

Certainly their building skills were already far advanced, but it’s difficult to know exactly where their mathematics was up to. Buildings and objects exhibiting the golden ratio might just have been happy accidents, created that way simply because they looked nice.

According to science populariser Mario Livio, since the publication of Luca Pacioli‘s Divina proportione in 1509, “the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use.”

Certainly it has become increasingly popular in both art and architecture, and is still seen in many buildings to this day.

A great example is the Villa Stein which was designed by Le Corbusier, some of whose architecture utilises the golden ratio, and features dimensions that closely approximate golden rectangles.

The beautiful villa was built for Gabrielle Colaco-Osorio de Monzie (1882–1961) and Sarah Stein, sister-in-law of American writer Gertrude Stein, between 1926 and 1928.

 

Let me show you some pieces of glass art I’ve made that show you how pleasing it is to look at something that’s created using the principles of the perfect rectangle.  These ones are a shape we call the Elegant Elongated Japanese Platter – I hope you like these designs as much as I enjoy making them.  Each one here is 35cms long by 20cms wide.

And my Giant Fruit Platter are almost exactly the measurements of the golden rectangle, each one you see below is 44cms in length by 33cms wide.

Do you love these rectangles too? Here are some sushi plates that are exactly the perfect rectangle.  I make these sushi plates for left handed people as well as right handed people.  Order as many as you like from me, I love making them and they make eating sushi even more fun!

If you want to work out if your works, canvasses, paintings, paper, whatever is a golden rectangle, then I’ve attached a link to the calculator here.  Let me know if you’ve found the calculator helpful.

And I hope you’ve enjoyed reading about the golden rectangle!

Talk soon.

jenie

Tags: art art glass compass geometry golden ratio golden rectangle rectangle renaissance art shape