Painting - Square Roots to Sixteen (Theodorus of Cyrene)

Greek mathematicians knew that numbers could not always be represented as simple ratios of whole numbers. They devised ways to describe them geometrically. The title of this painting refers to Theodorus of Cyrene (about 465–398 BC), a Greek geometer who, according to the Greek mathematician Theaetetus (about 417–369 BC), constructed the square roots of the numbers from 3 through 17. Crockett Johnson's painting follows a diagram in Evans G. Valens's The Number of Things that stops with the square root of 16.
The construction of this oil or acrylic painting, #45 in the series, begins with a vertical line segment of length one. Crockett Johnson then drew a right angle at the base of the segment and an adjacent line with length one. From the Pythagorean theorem, it follows that a line from the center of the spiral has length equal to the square root of 2. The construction was continued until the last hypotenuse displayed length equal to the square root of 16.
The painting, which looks like a seashell, shows a specific color pattern. The three dark gray triangles have hypotenuses whose lengths are whole numbers (the square roots of 4, 9, and 16). The six white triangles have hypotenuses whose lengths are irrational and are square roots of even integers. Finally, the six tan triangles have hypotenuses whose lengths are irrational and the square roots of odd integers.
The painting dates from 1967 and is signed: CJ67. It is marked on the back: Crockett Johnson (/) SQUARE ROOTS TO SIXTEEN (/) (THEODORUS OF CYRENE).
Currently not on view
Object Name
date made
Theodorus of Cyrene
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
overall: 108 cm x 84 cm x 3.2 cm; 42 1/2 in x 33 1/16 in x 1 1/4 in
ID Number
catalog number
accession number
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Crockett Johnson
Data Source
National Museum of American History, Kenneth E. Behring Center


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