##
Mathematical Paintings of Crockett JohnsonResources

### Selected Works of David Crockett Johnson

*Barnaby*, New York, NY: Henry Holt and Company, 1943.

*Barnaby and Mr. O’Malley*, New York: Henry Holt and Company, 1944.

*Harold and the Purple Crayon*, New York: Harper 7 Row, 1955.

“A Geometrical Look at vp,”

*Mathematical Gazette*, 54 (Feb 1970): 59-60.

“On the Mathematics of Geometry in My Abstract Paintings,”

*Leonardo*, 5 (1972): 97-101.

“A construction for a regular heptagon,”

*Mathematical Gazette*, 17 (March 1975): 17-21.

Papers of Crockett Johnson, Mathematics Collections, National Museum of American History, Smithsonian Institution.

Correspondence in the Harley Flanders Papers, Mathematics Collections, National Museum of American History.

Correspondence in the Ad Reinhardt Papers, Archives of American Art, Smithsonian Institution.

### Selected Works about Crockett Johnson

Stephanie Cawthorne and Judy Green, “Cubes, Conic Sections, and Crockett Johnson,” *Convergence*, vol. 11, 2014. http://www.maa.org/publications/periodicals/convergence/cubes-conic-sections-and-crockett-johnson

Stephanie Crawthorne and Judy Green, “Harold and the Purple Heptagon,” *Math Horizons* (September 2009): 5-9.

Philip Nel, “Crockett Johnson and the Purple Crayon: A Life in Art,” *Comic Art*, 5 (2004): 2-18.

Philip Nel. *Crockett Johnson and Ruth Krauss: A Biography*, Jackson: University Press of Mississippi, in preparation.

James B. Stroud, “Crockett Johnson's Geometric Paintings,” *Journal of Mathematics and the Arts*, 2 #2 (June 2008): 77-99.

For a more detailed bibliography and further information, see the Crockett Johnson Web site created and maintained by Philip Nel.

For a description of American mathematics and science education at the time of Crockett Johnson’s paintings, see the Museum's Web site: “Mobilizing Minds: Teaching Math and Science in the Age of Sputnik.”

### Credits

This introduction and the accounts of Crockett Johnson paintings given below have benefited from insights of Uta C. Merzbach, Judy Green, J. B. Stroud, Philip Nel, Mark Kidwell, Emmy Scandling, and Joan Krammer.

"Mathematical Paintings of Crockett Johnson - Resources" showing 2 items.

## Painting -

*Proof of the Pythagorean Theorem (Euclid)*- Description
- The Pythagorean theorem states that in any right triangle, the square of the side opposite the right angle (the hypotenuse), is equal to the sum of the squares of the other two sides. This painting depicts the “windmill” figure found in Proposition 47 of Book I of Euclid’s
*Elements*. Although the method of the proof depicted was written about 300 BC and is credited to Euclid, the theorem is named for Pythagoras, who lived 250 years earlier. It was known to the Babylonians centuries before then. However, knowing a theorem is different from demonstrating it, and the first surviving demonstration of this theorem is found in Euclid’s*Elements*.

- Crockett Johnson based his painting on a diagram in Ivor Thomas’s article on Greek mathematics in
*The World of Mathematics*, edited by James R. Newman (1956), p. 191. The proof is based on a comparison of areas. Euclid constructed a square on the hypotenuse BΓ of the right triangle ABΓ. The altitude of this triangle originating at right angle A is extended across this square. Euclid also constructed squares on the two shorter sides of the right triangle. He showed that the square on side AB was of equal area to the rectangle of sides BΔ and Δ;Λ. Similarly, the area of the square on side AΓ was of equal area to the rectangle of sides EΓ and EΛ. But then the square of the hypotenuse of the right triangle equals the sum of the squares of the shorter sides, as desired.

- Crockett Johnson executed the right triangle in the neutral, yet highly contrasting, hues of white and black. Each square area that rests on the sides of the triangle is painted with a combination of one primary color and black. This draws the viewer’s attention to the areas that complete Euclid’s proof of the Pythagorean theorem.

*Proof of the Pythagorean Theorem*, painting #2 in the series, is one of Crockett Johnson’s earliest geometric paintings. It was completed in 1965 and is marked: CJ65. It also is signed on the back: Crockett Johnson 1965 (/) PROOF OF THE PYTHAGOREAN THEOREM (/) (EUCLID).

- Location
- Currently not on view

- date made
- 1965

- referenced
- Euclid

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.01

- catalog number
- 1979.1093.01

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Bouquet of Triangle Theorems (Euclid)*- Description
- The mathematician Euclid lived around 300 BC, probably in Alexandria in what is now Egypt. Like most western scholars of his day, he wrote in Greek. Euclid prepared an introduction to mathematics known as
*The Elements*. It was an eminently successful text, to the extent that most of the works he drew on are now lost. Translations of parts of*The Elements*were used in geometry teaching well into the nineteenth century in both Europe and the United States.

- Euclid and other Greek geometers sought to prove theorems from basic definitions, postulates, and previously proven theorems. The book examined properties of triangles, circles, and more complex geometric figures. Euclid's emphasis on axiomatic structure became characteristic of much later mathematics, even though some of his postulates and proofs proved inadequate.

- To honor Euclid's work, Crockett Johnson presented not a single mathematical result, but what he called a bouquet of triangular theorems. He did not state precisely which theorems relating to triangles he intended to illustrate in his painting, and preliminary drawings apparently have not survived. At the time, he was studying and carefully annotating Nathan A. Court's book
*College Geometry*(1964). Court presents several theorems relating to lines through the midpoints of the side of a triangle that are suggested in the painting. The midpoints of the sides of the large triangle in the painting are joined to form a smaller one. According to Euclid, a line through two midpoints of sides of a triangle is parallel to the third side. Thus the construction creates a triangle similar to the initial triangle, with one fourth the area (both the height and the base of the initial triangle are halved). In the painting, triangles of this smaller size tile the plane. All three of the lines joining midpoints create triangles of this small size, and the large triangle at the center has an area four times as great.

- The painting also suggests properties of the medians of the large triangle, that is to say, the lines joining each midpoint to the opposite vertex. The three medians meet in a point (point G in the figure from Court). It is not difficult to show that point G divides each median into two line segments, one twice as long as the other.

- To focus attention on the large triangle, Crockett Johnson executed it in shades of white against a background of smaller dark black and gray triangles.

*Bouquet of Triangle Theorems*apparently is the artist's own construction. It was painted in oil or acrylic and is #26 in the series. It was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) BOUQUET OF TRIANGLE THEOREMS (/) (EUCLID).

- Reference: Nathan A. Court,
*College Geometry*, (1964 printing), p. 65. The figure on this page is not annotated.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Euclid

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.19

- catalog number
- 1979.1093.19

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center