# Golden rectangle

A golden rectangle 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 $\frac{a+b}{a} = \frac{a}{b} \equiv \varphi\,.$

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

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.

According to astrophysicist and mathematics popularizer Mario Livio, since the publication of Luca Pacioli's Divina Proportione in 1509,[1] 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,"[2] 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.[3]

## 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 4 simple steps:

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