Golden Rectangle

Given a rectangle having sides in the ratio , the golden ratio is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio . Such a rectangle is called a golden rectangle.

Euclid used the following construction to construct them. Draw the square , call the midpoint of , so that . Now draw the segment , which has length

(1)

and construct with this length. Now complete the rectangle , which is golden since

(2)

Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.

The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated above.

If the top left corner of the original square is positioned at (0, 0), the center of the spiral occurs at the position

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

and the parameters of the spiral are given by

			(12)
			(13)
			(14)
			(15)

A rectangle with length-to-height ratio can be dissected into three similar rectangles as illustrated above (Pegg 2019).