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 VE+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 An equivalent formula had been presented by Descartes in an unpublished treatise on polyhedra.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, sixsided, boxshaped 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, VE+F = 8  12 + 5 = 1, so that, if the removed face is counted, the result is VE+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 VE+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 VE+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 VE+F unchanged. This last step will also yield a figure for which VE+F = 33+1. As previously stated, if we count the removed face from the initial step, then VE+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
 Object Name

painting
 date made

1966
 referenced

Euler, Leonhard
 painter

Johnson, Crockett
 Physical Description

masonite (substrate material)

wood (frame material)

chrome (frame material)

canvas (overall material)
 Measurements

overall: 64 cm x 79 cm x 3.4 cm; 25 3/16 in x 31 1/8 in x 1 5/16 in
 ID Number

1979.1093.27
 catalog number

1979.1093.27
 accession number

1979.1093
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Crockett Johnson
 Data Source

National Museum of American History, Kenneth E. Behring Center