Golden rectangle

A golden rectangle (in pink) with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship ${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\equiv \varphi \,.}$

In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, ${\displaystyle 1:{\tfrac {1+{\sqrt {5}}}{2}}}$, which is ${\displaystyle 1:\varphi }$ (the Greek letter phi), where ${\displaystyle \varphi }$ is approximately 1.618.

Construction

A method to construct a golden rectangle. The square is outlined in red. The resulting dimensions are in the golden ratio.

A golden rectangle can be constructed with only straightedge and compass by four simple steps:

1. Construct a simple square.
2. Draw a line from the midpoint of one side of the square to an opposite corner.
3. Use that line as the radius to draw an arc that defines the height of the rectangle.
4. Complete the golden rectangle.

Relation to regular polygons and polyhedra

A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same aspect ratio as the first. Square removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property.

Three golden rectangles in an icosahedron

An alternative construction of the golden rectangle uses three polygons circumscribed by congruent circles: a regular decagon, hexagon, and pentagon. The respective lengths a, b, and c of the sides of these three polygons satisfy the equation a2 + b2 = c2, so line segments with these lengths form a right triangle (by the converse of the Pythagorean theorem). The ratio of the side length of the hexagon to the decagon is the golden ratio, so this triangle forms half of a golden rectangle.[1]

The convex hull of two opposite edges of a regular icosahedron forms a golden rectangle. The twelve vertices of the icosahedron can be decomposed in this way into three mutually-perpendicular golden rectangles, whose boundaries are linked in the pattern of the Borromean rings.[2]

Applications

According to astrophysicist and mathematics popularizer Mario Livio, since the publication of Luca Pacioli's Divina Proportione in 1509,[3] when "with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use,"[4] many artists and architects have been fascinated by the presumption that the golden rectangle is considered aesthetically pleasing. The proportions of the golden rectangle have been observed in works predating Pacioli's publication.[5]