#
Mathematical Paintings of Crockett Johnson

Inspired by the allure of the space age, many Americans of the 1960s took great interest in mathematics and science. One of them was the cartoonist, book illustrator, and children’s author David Crockett Johnson. From 1965 until his death in 1975 Crockett Johnson painted over 100 works relating to mathematics and mathematical physics. Of these paintings, eighty are found in the collections of the National Museum of American History. We present them her, with related diagrams from the artist’s library and papers.

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

## Painting -

*Point Collineation in the Triangle (Euler)*- Description
- Leonhard Euler (1707–1783) was the most prolific mathematician of the eighteenth century. He made significant contributions to geometry, calculus, mechanics, and number theory. He produced more than 800 publications during his lifetime, almost half of which were dictated after his eyesight failed in 1766. While Euler is best remembered for his contributions to analysis and mechanics, his interests included geometry. This figure illustrates a theorem about triangles associated with his name.

- Euler showed that three points related to a triangle lie on a common line. The first is the circumcenter (point O in the figure), the intersection of the perpendicular bisectors of the three sides. This point is the center of the circle which passes through the vertices of the triangle. Johnson also constructed the three medians of the triangle and the three altitudes of the triangle. The medians intersect in a common point (point N in the figure) and the altitudes meet at a third point (H in the figure). These three points, Euler showed, lie on the same line. In the painting, Crockett Johnson also constructed the circle that circumscribes the triangle, as well as a circle of half the radius known as the nine-point circle. For a full description of this circle, see painting #75 (1979.1093.49).

- In the painting, the circumcircle is centered exactly on the backing, and the Euler line extends from the lower right corner to the upper left corner. This divides the work into two triangles of equal area. The right half of the painting was executed in shades of red and purple, while the left half of the painting was executed in shades of gray and black. Crockett Johnson also joined the nine points of the nine-point circle to form an irregular polygon.

- This oil painting on masonite is #28 in the series. There is a wooden frame painted black. The work was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) POINT COLLINEATION IN THE TRIANGLE (/) (EULER). For a related painting, see #75 (1979.1093.49).

- Reference: Nathan A. Court,
*College Geometry*(1964 printing), p. 103, cover. The figure on p. 103 is annotated.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Euler, Leonhard

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.20

- catalog number
- 1979.1093.20

- accession number
- 1979.1093

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

## Painting -

*Polyhedron Formula (Euler)*- Description
- Mathematicians have explored the properties of polyhedra since ancient times, but it was the Swiss scholar Leonhard Euler (1707–1783) who proved the formula V-E+F = 2. That is, for a simple convex polyhedron (e.g. one with no holes, so that it can be deformed into a sphere) the number of vertices minus the number of edges plus the number of faces is two. An equivalent formula had been presented by Descartes in an unpublished treatise on polyhedra. However, this formula was first proved and published by Euler in 1751 and bears his name.

- Crockett Johnson's painting echoes a figure from a presentation of Euler's formula found in Richard Courant and Herbert Robbins's article “Topology,” which is in James R. Newman's
*The World the Mathematics*(1956), p. 584. This book was in the artist’s library, but the figure that relates to this painting is not annotated.

- To understand the painting we must understand the mathematical argument. It starts with a hexahedron, a simple, six-sided, box-shaped object. First, one face of the hexahedron is removed, and the figure is stretched so that it lies flat (imagine that the hexahedron is made of a malleable substance so that it can be stretched). While stretching the figure can change the length of the edges and the area and shape of the faces, it will not change the number of vertices, edges, or faces.

- For the "stretched" figure, V-E+F = 8 - 12 + 5 = 1, so that, if the removed face is counted, the result is V-E+F = 2 for the original polyhedron. The next step is to triangulate each face (this is indicated by the diagonal lines in the third figure). If, in triangle ABC [C is not shown in Newman, though it is referred to], edge AC is removed, the number of edges and the number of faces are both reduced by one, so V-E+F is unchanged. This is done for each outer triangle.

- Next, if edges DF and EF are removed from triangle DEF, then one face, one vertex, and two edges are removed as well, and V-E+F is unchanged. Again, this is done for each outer triangle. This yields a rectangle from which a right triangle is removed. Again, this will leave V-E+F unchanged. This last step will also yield a figure for which V-E+F = 3-3+1. As previously stated, if we count the removed face from the initial step, then V-E+F = 2 for the given polyhedron.

- The “triangulated” diagram was the one Crockett Johnson chose to paint. Each segment of the painting is given its own color so as to indicate each step of the proof. Crockett Johnson executed the two right triangles that form the center rectangle in the most contrasting hues. This draws the viewer’s eyes to this section and thus emphasizes the finale of Euler's proof. This approach to the proof of Euler's polyhedral formula was pioneered by the French mathematician Augustin Louis Cauchy in 1813.

- This oil painting on masonite is #39 in the series. It was completed in 1966 and is signed: CJ66. It is inscribed on the back: Crockett Johnson 1966 (/) POLYHEDRON FORMULA (EULER). It has a wood and chrome frame.

- Reference:

- David Richeson, “The Polyhedral Formula,” in
*Leonhard Euler: Life, Work and Legacy*, editors R. E. Bradley and C. E. Sandifer (2007), pp. 431–34.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Euler, Leonhard

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.27

- catalog number
- 1979.1093.27

- accession number
- 1979.1093

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