Silver Ratio - Paper Sizes and Silver Rectangles

Paper Sizes and Silver Rectangles

The paper sizes under ISO 216 are rectangles in the proportion 1:√2 sometimes called "A4 rectangles". Removing a largest possible square from a sheet of such paper leaves a rectangle with proportions 1:√2−1 which is the same as 1+√2:1, the silver ratio. Removing a largest square from one of these sheets leaves one again with aspect ratio 1:√2. A rectangle whose aspect ratio is the silver ratio is sometimes called a silver rectangle by analogy with golden rectangles. Confusingly, "silver rectangle" can also refer to the paper sizes specified by ISO 216.

Removing the largest possible square from either kind yields a silver rectangle of the other kind, and then repeating the process once more gives a rectangle of the original shape but smaller by a linear factor of 1+√2.

However, only the Lichtenberg ratio rectangles (rectangles with the shape of ISO 216 paper) have the property that by cutting the rectangle in half across its long side produces two smaller rectangles of the same aspect ratio.

The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δ_S, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δ_S. If the edge length of a regular octagon is t, then the inradius of the octagon (the distance between opposite sides) is δ_St, and the area of the octagon is 2δ_St2.