Golden rectangle
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, formula_1, which is formula_2 (the Greek letter phi), where formula_3 is approximately 1.618. Golden rectangles exhibit a special form of self-similarity: All rectangles created by adding or removing a square from an end are golden rectangles as well. Construction. A golden rectangle can be constructed with only a straightedge and compass in four simple steps: A distinctive feature of this shape is that when a square section is added—or removed—the product is another golden rectangle, having the same aspect  ratio as the first. Square addition or 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. Diagonal lines drawn between the first two orders of embedded golden rectangles will define the intersection point of the diagonals of all the embedded golden rectangles; Clifford A. Pickover referred to this point as "the Eye of God". History. The proportions of the golden rectangle have been observed as early as the Babylonian "Tablet of Shamash" (c. 888–855 BC), though Mario Livio calls  any knowledge of the golden ratio before the Ancient Greeks "doubtful". According to Livio, since the publication of Luca Pacioli's "Divina proportione" in 1509, "the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use." The 1927 Villa Stein designed by Le Corbusier, some of whose architecture utilizes the golden ratio, features dimensions that closely approximate golden rectangles. Relation to regular polygons and polyhedra. Euclid gives an alternative construction of the golden rectangle using 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 "a" + "b" = "c", 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. 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.