Golden Rectangle:

OUR EYES ARE VERY DISCERNING. Without much more than an impressionistic observation, we can quickly sense what looks right and what doesn't. For most woodworkers, this power of observation is sharpened from experience. When we complete a project, and we stand back to make a judgment of how it looks, our critical powers of perspective kick in: Is the finished project pleasing to our eye? Or, are the proportions not right?

What is it about a finished project that creates this immediate impression? Most often, it's the project's proportions, that is, length vs height vs depth. If those measurements aren't right, nothing else matters, because the rest of your design is -- more or less -- just window dressing.

The Golden Rectangle -- a mathematical formula of great visual elegance, but one that creates great challenge in giving a simple explanation -- is arrived at by bisecting a square and using the diagonal of one half of the square as a radius to extend the dimensions of the square to become a "Golden Rectangle", in the proportion arrived at, a:b=c:a." This will mean that the ratio AB/BC is equal to the ration AC/AB, and this ratio is phi, or 1.618.

Also known as the "Golden Box", "Golden Proportion", "Golden Ratio", "Golden Section" and "Golden Mean", it comes from Ancient Egypt and is carried on by the Greeks in their buildings, such as the Parthenon.

Regardless of whatever label it is given, it is a formulation that, when applied to diverse arts furniture, buildings, even sculptures and paintings (in fact, it is said to occur in nature's forms, too!) creates visually appealing designs through tension and asymmetry, rather than balance. Compositional variation and unevenness in design, in this sense, provoke an active, positive response in the viewer, while a design that is too balanced is simply unexciting. The Golden Ratio is significant because it demonstrates a mathematical equation for the concept of asymmetry; thus, odd as it may seem, one dynamic of successful composition has a precise mathematical explanation that is constant across all art forms and cultures.

The method of constructing these proportions is demonstrated in the figures below.

golden_ratio

When designing, integrating the ratio itself into the equation isn't difficult. For design purposes, the golden ratio basically defines the relationship between two sections of a line. Put simply, you divide the line into two parts so that the line's longer section is 1.618 times the length of the shorter one. In addition, the ratio between the line's longer section and the line's overall length is also 1.618. What these rules mean is that all you have need for the golden ratio to work is to comply is the formula: 1 to 1.618. Generally this formula is called phi.

golden_ratio

The golden ratio can also be used to create a golden rectangle. The length of the long side is 1.618 times the length of the short side. And likewise, a golden box can be constructed using phi. The depth of the box is dimensioned by dividing the short side by 1.618. The drawings below show all these relationships.

The Greeks used the "Golden Mean" to design what they built, from the parthenon to the floor plans of temples.

golden rectangle in Greek Parthenon

Jim Tolpin puts it elegantly:

"When designing temples for their gods, the craftsmen of ancient Greece used a proportional system that pleased the eye and nourished the soul."

Sources: Donis A Dondis, A Primer of Visual Literacy, Cambridge: MIT Press,, 1973; Matila Ghyka, The Geometry of Art and Life New York: Dover, 1977 (Ghyka's book is dedicated to a discussion of the concept, with the second chapter giving details about the concept's origin);
R J DeCristoforo, Woodworking Mistakes and Solutions, New York: Sterling, 1996, pages 135-138. (No preview available on Google Print.);
Jim Tolpin, Working Wood: A Complete Bench-Top Reference Worcester, MA: Davis Publications, 1997, page 120 (Using just words and pencil diagrams, in a single page, Tolpin captures the essence of the concept);
Paul Harrel, "Designing Along the Grain", Practical Design: Solutions and Strategies -- Key Advice for Sound Construction from Fine Woodworking Newtown, CT: Taunton, 2000; Mario Livio, The Golden Ratio New York: Broadway Books, 2002.