#
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 80 items.

Page 3 of 8

## Painting -

*Every Positive Integer (Gauss)*- Description
- This painting is loosely based on a theorem proven by the German mathematician Carl Friedrich Gauss (1777–1855) in 1776 when he was just nineteen years old. The proposition, one of Gauss’s many contributions to the branch of mathematics called number theory, states that every positive integer is the sum of three triangular numbers. The concept of triangular numbers dates to antiquity. Suppose one arranges dots in rows, with one in the first row, two in the second, three in the first and so forth. Three dots form a triangle, as do 6 dots, 10 dots, and 15 dots. The numbers 3, 6, 10, 15, and so forth are called "triangular numbers." The integers 0 and 1 are thought of as special cases of triangular numbers.

- Crockett Johnson derived his painting from an entry in Gauss's diary published in an article by Eric Temple Bell included by James R. Newman in his book
*The World of Mathematics*(1956), p. 304. The entry includes the phrase EUREKA in Greek, and indicates that any positive integer is the sum of three triangular numbers.

- Crockett Johnson’s painting abstractly represents this theorem through the juxtaposition of three triangles. The triangles are equal, but each figure is painted a different color. It is possible that the artist chose to illustrate each triangle in its own color to demonstrate that each triangle generally represents its own triangular number when computing a positive integer. However, the triangles are congruent, which reminds the viewer that the triangles are related because they all represent a triangular number.

- This work was painted in oil on masonite, completed in 1966, and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) EVERY POSITIVE INTEGER (/) (GAUSS). It is painting #29 in the series, and has a wooden frame.

- Reference: J. R. Newman,
*The World of Mathematics*, 1956, p. 304.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Gauss, Carl Friedrich

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.21

- catalog number
- 1979.1093.21

- accession number
- 1979.1093

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

## Painting -

*Transversals (Ceva)*- Description
- A transversal is a line that intersects a system of other lines or line segments. Here Crockett Johnson explores the properties of certain transversals of the sides of a triangle. The Italian mathematician Giovanni Ceva showed in 1678 that lines drawn from a point to the vertices of a triangle divide the edges of the triangle into six segments such that the product of the length of three nonconsecutive segments equals the product of the remaining three segments.

- This painting shows a triangle (in white), lines drawn from a point inside the triangle to the three vertices, and a line drawn from a point outside the triangle (toward the bottom of the painting) to the three vertices. Segments of the sides of the triangle to be multiplied together are of like color. Crockett Johnson's painting combines two diagrams on page 159 of Nathan Court's
*College Geometry*(1964 printing). These diagrams are annotated in his copy of the volume. Several of the triangles adjacent to the central triangle were used by Court in his proof of Ceva's theorem.

- The painting is #31 in the series. It is signed: CJ66. There is a wooden frame painted off-white.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Ceva, Giovanni

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.22

- catalog number
- 1979.1093.22

- accession number
- 1979.1093

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

## Painting -

*One Surface and One Edge (Möbius)*- Description
- Most geometric surfaces have a distinct inside and outside. This painting shows one that doesn’t. Take a strip of material, give it a half-twist, and attach the ends together. The result is a band with only one surface and one edge. Mathematicians began to explore such surfaces in the nineteenth century. In 1858 German astronomer and mathematician August Ferdinand Möbius (1790–1868), who had studied theoretical astronomy under Carl Friedrich Gauss at the University of Goettingen, discovered the one-sided surface shown in the painting. It has come to be known by his name. As often happens in the history of mathematics, another scholar, Johann Benedict Listing, had found the same result a few months earlier. Listing did not publish his work until 1861.

- If one attaches the ends of a strip of paper without a half twist, the resulting figure is a cylinder. The cylinder has two sides such that one can paint the outside surface red and the inside surface green. If you try to paint the outside surface of a Möbius band red you will paint the entire band red without crossing an edge. Similarly, if you try to paint the inside surface of a Möbius band green you will paint the entire surface green. A cylinder has an upper edge and a lower edge. However, if you start at a point on the edge of a Möbius band you will trace out its entire edge and return to the point at which you began. Since Möbius's time, mathematicians have discovered and explored many other one-sided surfaces.

- This painting, #34 in the series, was executed in oil on masonite and is signed: CJ65. The strip is shown in three shades of gray based on the figure’s position. The shades of gray, especially the lightest shade, are striking against the rose-colored background, and this contrast allows the viewer to focus on the properties of the Möbius band. The painting has a wooden frame.

- Crockett Johnson's painting is similar to illustrations in James R. Newman's
*The World of Mathematics*(1956), p. 596. However, the figures are not annotated in the artist's copy of the book.

- Location
- Currently not on view

- date made
- 1965

- referenced
- Moebius, August Ferdinand

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.23

- catalog number
- 1979.1093.23

- 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

## Painting -

*Logarithms*- Description
- This painting illustrates two different kinds of mathematical progressions, the geometric (on the top) and the arithmetic (on the bottom). Going across the top from left to right each section is twice as wide as the previous one, as in a geometric progression. Going across the bottom from right to left, each section is 1 unit wider than the previous one, as in an arithmetic progression.

- If the width of the top sections, considered going from left to right, represents the numbers a, 2a, 4a, and 8a in a geometric progression, then the width of the bottom sections, going right to left, can represent logarithms of these numbers, b = log a, 2b =2 log a, 3b = 3 log a, and 4b =4 log a. Crockett Johnson may have sought to illustrate an account of logarithms given in an article by H. W. Turnbull in Newman's
*Men of Mathematics*. This painting does not represent the traditional divisions of either a slide rule or a ruler.

- The Scottish nobleman John Napier published his discovery of logarithms in 1614. The painting suggests how logarithms allow one to reduce multiplication (as in the terms of a geometric progression) to addition (as in the terms of an arithmetic progression). As addition is far simpler than multiplication, logarithms were widely used by people carrying out calculations from the seventeenth century onward.

- The painting is #37 in the series. It is in oil or acrylic on masonite, and is signed: CJ66. There is a gray wooden frame.

- Reference: H. W. Turnbull, “The Great Mathematicians,” in James R. Newman,
*The World of Mathematics*, (1956), p. 124. This volume was in Crockett Johnson's library, but the figure is not annotated.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Napier, John

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.25

- catalog number
- 1979.1093.25

- accession number
- 1979.1093

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

## Painting -

*Polar Line of a Point and a Circle (Apollonius)*- Description
- In 1966, Crockett Johnson carefully read Nathan A. Court's book
*College Geometry*, selecting diagrams that he thought would be suitable for paintings. In the chapter on harmonic division, he annotated several figures that relate to this painting. The work shows two orthoganol circles, that is to say two circles in which the square of the line of centers equals the sum of the squares of the radii. A right triangle formed by the line of centers and two radii that intersect is shown. The small right triangle in light purple in the painting is this triangle.

- Crockett Johnson's painting combines a drawing of this triangle with a more complex figure used in a discussion of further properties of lines drawn in orthoganal circles. In particular, suppose that one draws a line segment from a point outside a circle that intersects it in two points, and selects a fourth point on the line that divides the segment harmonically. For a single exterior point, all these such points lie on a single line, perpendicular to the line of centers of the two circles, which is called the polar line.

- The painting is #38 in the series. It has a background in two shades of cream, and a light tan wooden frame. It shows two circles that overlap slightly and have various sections. The circles are in shades of blue, purple and cream. The painting is signed: CJ66.

- References: Nathan A. Court,
*College Geometry*(1964 printing), p. 175–78. This volume was in Crockett Johnson's library.

- T. L. Heath, ed.,
*Apollonius of Perga: Treatise on Conic Sections*(1961 reprint). This volume was not in Crockett Johnson's library.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Apollonius of Perga

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.26

- catalog number
- 1979.1093.26

- 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

## Painting -

*Multiplication through Imaginary Numbers (Gauss)*- Description
- This painting was inspired by ideas of Carl Friedrich Gauss (1777–1855). In his 1797 doctoral thesis, Gauss proved what is now called the fundamental theorem of algebra. He showed that every polynomial with real coefficients must have at least one real or complex root. A complex number has the form a+bi, where a and b are real numbers and i represents the square root of negative one. The French mathematician René Descartes (1596–1650) called such numbers "imaginary", which explains the reference in the title. Gauss demonstrated that, just as real numbers can be represented by points on a coordinate line, complex numbers can be represented by points in the coordinate plane.

- The construction of this painting echoes a figure in an article on Gauss by Eric Temple Bell in J. R. Newman's
*The World of Mathematics*that illustrates the representation of points on a plane. This book was in Crockett Johnson's library, and the figure is annotated.

- In Bell's figure, real numbers c and -c are plotted on the x axis, the imaginary numbers ci and -ci are plotted on the y axis, and the complex number a+bi is shown in the first quadrant. The figure is meant to show that if a complex number a+bi is multiplied by the imaginary number i then the product is a complex number on the same circle but rotated ninety degrees counterclockwise. That is, i(a+bi) = ai+bi² = -b+ai. Thus, this complex number lies in the second quadrant. If this process is repeated the next product is -a-bi, which lies in the third quadrant. It is unknown why Johnson did not illustrate the fourth product.

- The colors of opposite quadrants of the painting are related. Those in quadrant three echo those of quadrant one and those of quadrant four echo those of quadrant two.This oil painting is #40 in the series. It is signed: CJ67.

- References:

- James R. Newman,
*The World of Mathematics*(1956), p. 308. This volume was in Crockett Johnson's library. The figure on this page is annotated.

- Location
- Currently not on view

- date made
- 1967

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.28

- catalog number
- 1979.1093.28

- accession number
- 1979.1093

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

## Painting -

*Locus of Point on Chord (Plato)*- Description
- The locus of the midpoints of the chords of a given circle that pass through a fixed point is a circle when the point lies inside of or on the circle. The small circle painted white is the locus of the midpoints of chords drawn in the large circle that pass through a point toward the top left of the inside of the circle. Three chords of the large circle are suggested. These are the diameter, whose midpoint is the center of the circle, a vertical chord through the point, and a horizontal chord through the point (only a small part of this chord is indicated). The painting is based on a diagram from
*College Geometry*by Nathan Court. It is unclear why Crockett Johnson associated this painting with Plato.

- The oil painting on masonite is #41 in the series. It has a background of two purple and gray rectangles. It has a metal and wooden frame. It shows a circle with a smaller circle inside it. The smaller circle is in two shades of white, the larger one in orange, black, gray and light purple. The painting is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) LOCUS OF POINT ON CHORD (PLATO).

- Reference: Nathan Court,
*College Geometry*, (1964 printing), p. 13. This figure is annotated in Crockett Johnson's copy of this volume.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Plato

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.29

- catalog number
- 1979.1093.29

- accession number
- 1979.1093

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

## Painting -

*Velocity on Inclined Planes (Galileo)*- Description
- This oil painting is based on a figure from Galileo Galilee's
*Dialogues Concerning Two New Sciences*(1638), Book 3. Here Galileo discussed the time of descent of bodies rolling without friction along inclined planes. He argued that if from the highest point in a vertical circle there be drawn any inclined planes meeting the circumference of the circle, the times of descent along these chords are equal to one another. This painting shows two inclined planes drawn from the highest point of a vertical circle, with a ball moving along each chord. Crockett Johnson probably became familiar with Galileo's figure by examining the translation of part of his book published in James R. Newman,*The World of Mathematics*, vol. 2, New York: Simon and Schuster, 1956, p. 751–52. This volume was in Crockett Johnson's library. The figure on p. 752 is annotated.

- The painting has a gray background and a metal and wooden frame. It shows two superimposed triangles (inclined planes), one reddish purple, and the other smaller one blue. Both of these triangles are inscribed in the same white circular arc. A light purple circle is shown near the bottom of the purple triangle, and a light blue circle near the bottom of the blue triangle.

- The work is # 42 in the series. It is signed: CJ66. Compare to paintings #96 (1979.1093.64) and #71 (1979.1093.46).

- Location
- Currently not on view

- date made
- 1966

- referenced
- Galilei, Galileo

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.30

- catalog number
- 1979.1093.30

- accession number
- 1979.1093

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