Sunday, July 19, 2009

Harmonic Proportions and Ancient Design Systems

Old masters made use of classical geometry to find the best way to arrange figures and objects within a given space. The composition scheme based on harmonic proportions is the most common one (and it derives from Pythagorus's discovery of harmonic proportions -- musical root harmonies -- that explain order and beauty in sound).

I created two samples below, the first on what's called the "armature of the rectangle" and the second on the "root 2 rectangle system" (a rectangle whose dimensions are based on the ratio of the square root of the number 2)

Example 1: Composition Based on the Rectangle's Armature.
I first created a rectangle based on the proportion of my chosen canvas ("30 x 24"). Then I sketched a series of diagonals from the corners and the middle of the sides. Where they intersect, the natural harmonies are created. I've highlighted these with circles. They represent where the eye of the viewer naturally moves to. The goal is to frame subjects or objects at certain harmonic divisions, to rest or place parts of a figure along a diagonal, or to run a harmonic division through key places in the design (typically an eye of a figure).

How did I do? Pretty well, considering I didn't think of the compositon of the surroundings when taking the reference photo. The dots on the photo below represent the natural "eyes" of my rectangle, situated in the natural centre (half point), thirds, and quarters. On my computer, I arranged the photo under the rectangle's armatures diagonals so that the bottom of the book's spine would meet the absolute centre. The remaining eyes are in unusual places but at least I can see some parts of the figure lining up with the diagonals, and Wayne is centered. (If I had asked Wayne to lower and close the book slightly, it would have been perfectly placed)... The grey strip on the far left represents how far I will have to extend the window in the painting to keep Wayne in the centre.

Example 2: Composition Based on the Root 2 Rectangle.

Here, I've taken a Root 2 Rectangle (height 1.41 x width) and imposed it on the image again. This rectangle is sitting over the larger grey rectangle which represnts my 30z24 canvas. The crossing of the main diagonals with the secondary diagonals creates four 90 degree angles and divides the rectangle into thirds. These crossings become 4 "eyes" (the equivalent of the green eyes in the previous picture).

Again, after centering the figure, I see it's not a perfect fit. BUT now I know how big to make the figure and where to position it on the canvas... With this Root 2 rectangle, my figure would be smaller (59.62 cm tall) than with the Rectangle Armature design (66.98 cm tall) -- my preference is the first.


  1. I am currently learning about root 2 rectangles in my 2D design class and I am working on my second project using one. Anyway, your post and pictures really helped me this second time around. I'm glad I found this site!

  2. Thanks so much for this explanation of a practical way to use the root rectangles!!