# Painting - Golden Rectangle (Pythagoras)

Description
The ancient Greek mathematician Euclid showed in his Elements that it is possible to divide a line segment into two smaller segments wherein the ratio of the whole length to the longer part equals the ratio of the longer part to the smaller. He used this theorem in his construction of a regular pentagon. The ratio came to be called the golden ratio. If the sides of a rectangle are in the golden ratio, it is called a golden rectangle. Several Crockett Johnson paintings explore the golden ratio and related geometric figures. This paintings suggest how a golden rectangle can be constructed, given the length of its shorter side. On the right in the painting is the golden rectangle that results. Lines in a triangle on the left indicate how the rectangle could have been constructed. Also included are the outlines of a hexagon and a five-pointed star constructed once the ratio had been found.
This painting follows a diagram on the top of page 131 in Evans G. Valens, The Number of Things. This diagram is annotated. Valens describes a geometrical solution to the two expressions f x f = e x c and f = e - c, and associates it with the Pythagoreans. The right triangle on the upper part of Valens's drawing, with the short side and part of the hypotenuse equal to f, is shown facing to the left in the painting. It can be constructed from a square with side equal to the shorter side of the rectangle. Two of the smaller rectangles in the painting are also golden rectangles. Crockett Johnson also includes in the background the star shown by Valens and related lines.
The painting on masonite is #46 in the series. It has a black and purple background and a black wooden frame. It is unsigned. The inscription on the back reads: GOLDEN RECTANGLE (/) (PYTHAGORAS) (/) Crockett Johnson 1968. Compare #103 (1979.1093.70) and #64 (1979.1093.39).
Location
Currently not on view
Object Name
painting
1968
referenced
Pythagoras
painter
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
Measurements
overall: 25 in x 25 in x 1 15/16 in; 63.5 cm x 63.5 cm x 4.99999 cm
ID Number
1979.1093.33
catalog number
1979.1093.33
accession number
1979.1093
subject
Science & Mathematics
Art
Crockett Johnson
See more items in
Medicine and Science: Mathematics
Crockett Johnson
Data Source
National Museum of American History, Kenneth E. Behring Center
Credit Line
Ruth Krauss in memory of Crockett Johnson