Mathematical Paintings of Crockett Johnson - Introduction
"Mathematical Paintings of Crockett Johnson - Introduction" showing 2 items.
Painting - Square Root of Two (Descartes)
- Description
- La Géométrie, one of the most important works published by the mathematician and philosopher René Descartes (1596–1650), includes a discussion of methods for performing algebraic operations using a straight edge and compass. One of the first is a way to determine square roots. This construction is the subject of Crockett Johnson's painting. Descartes explained: "If the square root of GH is desired, I add, along the same straight line, FG equal to unity, then bisecting FH at K, I describe the circle FIH about K as a center, and draw from G a perpendicular and extend it to I, and GI is the required root." (this is a translation of portion of La Géométrie, as published by J. R. Newman, The World of Mathematics (1956), p. 241)
- To understand Descartes' description and the title of this painting, consider the diagram. An angle inscribed in a semicircle is a right angle, thus triangle FGI is similar to triangle IGH. Because this two triangles are similar, their corresponding sides are proportional. Thus, G/IFG = GH/GI. But FG is equal to one, so GH is the square of GI, and GI the square root of GH desired.
- In his painting, Crockett Johnson has flipped the image from La Géométrie found in his copy of The World of Mathematics. This figure is not annotated. The artist divided his painting into squares of area one, suggesting what came to be called Cartesian coordinates. The division indicates that the GH chosen has length two.
- Johnson chose white for the section of the semicircle that contains the edge of length equal to the square root of GH. This section provides a vivid contrast against the dull, surrounding colors. Crockett Johnson purposefully creates this area of interest to draw focus to the result of Descartes' construction.
- Square Root of Two is painting #19 in the series. It was painted in oil or acrylic on masonite, completed in 1965, and is signed: CJ65. The wooden frame is painted black.
- Location
- Currently not on view
- date made
- 1965
- referenced
- Descartes, Rene
- painter
- Johnson, Crockett
- ID Number
- 1979.1093.13
- catalog number
- 1979.1093.13
- accession number
- 1979.1093
- Data Source
- National Museum of American History, Kenneth E. Behring Center
Painting - Simple Equation (Descartes)
- Description
- In a pathbreaking book La Géométrie, René Descartes (1596–1650) described how to perform algebraic operations using geometric methods. One such explanation is the subject of this Crockett Johnson painting. More specifically, Descartes described geometrical methods for finding the roots of simple polynomials. He wrote (as translated from the original French): "Finally, if I have z² = az -b², I make NL equal to (1/2)a and LM equal to b as before: then, instead of joining the points M and N, I draw MQR parallel to LN, and with N as center describe a circle through L cutting MQR in the points Q and R; then z, the line sought, is either MQ or MR, for in this way it can be expressed in two ways, namely: z = (1/2)a + √((1/4)a² - b²) and z = (1/2)a - √((1/4)a² - b²)."
- To verify that z = MR is a solution to the equation z²= az - b², note that the square of the length of the tangent ML equals the product of the two line segments MQ and MR. As ML is defined to equal b, its square is b squared. The length of MR is z, and the length of MQ is the difference between the diameter of the circle (length a) and the segment MR, that is to say (a – z) . Hence b squared equals z (a – z) which, on rearrangement of terms, gives the result desired.
- Crockett Johnson's painting directly imitates Descartes's figure found in Book I of La Géométrie. A translation of part of Book I is found in the artist’s copy of James R. Newman's The World of Mathematics. The figure on page 250 is annotated.
- This oil or acrylic painting on masonite is #36 in the series. It was completed in 1966 and is signed: CJ66. It has a wooden frame.
- Location
- Currently not on view
- date made
- 1966
- referenced
- Descartes, Rene
- painter
- Johnson, Crockett
- ID Number
- 1979.1093.24
- catalog number
- 1979.1093.24
- accession number
- 1979.1093
- Data Source
- National Museum of American History, Kenneth E. Behring Center