Proportional rectangles

The rules it illustrates

Commensurability. Let every part of a building be measured by the same unit and relate to the whole by simple whole-number ratios, so that one module generates all its dimensions.

The diagonal of the square. To make a rectangle of 1 : √2, draw a square and swing its diagonal down to extend one side; the result cannot be measured in whole units of the square, but it can be drawn exactly.

The √3 rectangle and the equilateral triangle. Swing the diagonal of a √2 rectangle to obtain a rectangle of 1 : √3, the proportion of an equilateral triangle’s height to half its base, used for setting out by triangles (ad triangulum).

The golden section. To divide a line in extreme and mean ratio, make the whole stand to the larger part as the larger part to the smaller; the ratio is φ = (1 + √5)/2 ≈ 1.618. Construct the golden rectangle by swinging the line from the midpoint of a square’s side to its far corner.

Measure before you believe. Treat claims that a building embodies the golden section with suspicion unless the designer’s own writings or drawings say so and careful measurement agrees; a ratio near 1.6 is found somewhere in almost anything.

The Modulor. Take the height of a man (183 cm) and of his raised hand (226 cm) and divide each by the golden section, again and again, to obtain two interlocking series of dimensions for every part of a building.

Where it appears in the book

In the styles Neoclassical.

In the chapters Proportion and Number.