Wednesday, August 19, 2009

The Wooden Ratio

You've heard about the Golden Ratio, 1.618..., computed as the mean of 1 and sqrt(5), or as the limit of the ratios of successive terms in the Fibonacci Sequence.  This ratio rears its head in the natural world, music, art, and in the proportions within our own bodies.

I have recently become aware of another ratio with similar properties.  Computed as the mean of 1 and 3, or as the ratio of successive terms in the sequence {2^n}, I have dubbed this number the Wooden Ratio (after the two-by-four).  It begins 2.000... and we can represent it with the symbol þ (thorn).

While the Golden Ratio appears in architecture such as the Parthenon, just think about how many Wooden Ratios appear in all of the brick buildings made by humanity.


In each of these bricks, the length is þ times the width, and the width is þ times the thickness. Incidentally, these bricks are what we might call Wooden Rectangles.  Like the Golden Rectangle, they appear in beautiful works of art, as well as in the proportions of the beautiful face.


Notice how Da Vinci placed Mona's Left eye at the position w/þ, where w is the width of the canvas.


Tyra Banks, aguably America's top model, has features which perfectly fit inside a series of Wooden Rectangles.  Notice how her mouth and eyes both show the ratio þ.

We often see the Golden Ratio in flowers and seeds.  But look at these trees.  Enough said.


While the Golden Ratio is tied to the self-similar logarithmic spiral of a nautilus, the Wooden Ratio turns up in any natural form with bilateral symmetry.  It is also evident in the fractal formed by successive divisions of a segment into þ segments of equal length (affectionately dubbed "abacaba"), and in the size of computer storage by repeated multiplications of the byte by þ.

Where else might the Wooden Ratio turn up?


  1. [...] in plants, and doesn’t go overboard with finding the Golden Ratio everywhere, as I have been annoyed to see some do. Share and [...]

  2. If the Length is 3" and Height is 2" what is the Thickness of Depth of brick if in the Phi Ratio? What is the formula?

  3. I am not a math student so I have some difficulty with the calculations. I want to make a brick that is 2" in Height, which makes the Width 2" and 15/16th, (in Phi dimensions. What is the "thickness" in the same Phi Ratio? Also what is a "simple" formula if I change the Height of the brick? Thank you!

  4. Well, if Length is 3 and Height is 2, then the brick does not have the Phi ratio of 1.618..., but instead has the ratio 1.5. In that case, the depth would be 2/1.5 = 4/3 or about 1.3.