The Golden Rectangle — One Rectangle in a Continuum of Infinitely Many Similar Rectangles

The Rectangles Paper
Table 1. First Series Rectangles
Table 2. Second Series Rectangles

I've completed a draft of the paper, The Golden Rectangle — One Rectangle in a Continuum of Infinitely Many Similar Rectangles.

I'll now be preparing a tool to configure and print versions of the spirals.

Multiple Unit Rectangles, Orders 1, 2, and 3

Fractional Unit Rectangles, Orders 1, 2, 3, and 4

Second Series Rectangle, Spiral Example

This is the first example of a second series rectangle rendered with its "spiral". This is a "2nd order, fractional unit" rectangle. That means that the ratio of long side to short side is b/(a-b/2⁴). The ratio of a/b is (1 + √65)/8, or approximately 1.13278. A "zero order, fractional unit" rectangle is the same as the golden rectangle.

First Series Rectangle, Spiral Example

This is the first example of a new series rectangle rendered with its "spiral". This is a "1st order, multiple unit" rectangle. That means that the ratio of long side to short side is (a+b)/(a-b). The ratio of a/b is 1 + √2, or approximately 2.414213. A "zero order, multiple unit" rectangle is the same as the golden rectangle.

An Infinity of Rectangles

On the day before Thanksgiving, I finally posted "The Golden Rectangle — One Rectangle in a Continuum of Infinitely Many Similar Rectangles".

Now I'll be working on a Flash program to render the spirals that I can create with the new rectangles. The new spirals will be similar to the golden spiral. However, unlike approximations of a logarithmic spiral, they'll be elongated, or flattened looking.

Solution to fractional rectangles

In a previous blog I proposed a problem: to find the formulas (ratios for the long to short sides) for rectangles that are generated in a different series. That is, the golden rectangle is constructed by drawing a line across a half unit square. So, what are the formulas for a series of rectangles constructed by first drawing a line across a third, then a fourth, a fifth, a sixth, etcetera, of a unit square?

What I found was a series constructed by drawing a line across a fourth, eighth, sixteenth, etcetera, of a unit square. Here is a table of what I call the second through eighth order fractional rectangles, with the golden rectangle being the first order.

These rectangles have the following features in common with the golden rectangle:
1. Simple construction. The rectangles can be constructed with a straightedge and compass. This is done by starting with a square and drawing a line across a fraction of the square. That line is used as a radius to draw an arc that defines the height of the rectangle.
2. Simple formula. Maybe not as simple as the golden ratio. The golden ratio is expressed algebraically as:
(a+b)/a = a/b
The second order fractional rectangle ratio is expressed as:
b/(a – b/2) = a/b
Each successive fractional rectangle in the series has a similar formula, the 2 being increased to 4, 8, 16, 32, etcetera.
3. Simple generation of rectangles spirally in from an original rectangle. When a simple fraction if a unit square is removed, the remainder is another rectangle with the same proportions as the first. Removal of the same fraction of a unit square can be repeated infinitely, which leads to spiral-like image.

West Coast art history

Portland was an excellent place to be this weekend for expanding my knowledge of West Coast art history. I saw Morgan Neville's Cool School on Friday at the Northwest Film Center. Neville was there to comment on the film. Tom Marioni gave a talk at Reed College on Saturday. I was very pleasantly surprised that much of the work he showed clicked with me. Finally, Kathan Brown of Crown Point Press gave a talk at PAM on Sunday.