##
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 5 of 8

## Painting -

*Square Roots of One, Two and Three*- Description
- Crockett Johnson much enjoyed constructing square roots of numbers geometrically. He offered the following account of this painting, as well as the figure shown: "Let AN and BN be 1. Then the diagonal AB is the square root of 2, because it is the hypotenuse of a right triangle with sides of length √1 and √1. The large right triangle √1 plus √2 adds up to a hypotenuse of √3. The compass traces pronounce a statement and also declare its proof. The square root of 2 is 1.4142 . . . and the square root of 3 is 1.7321 . . . Their decimals run on and on but as produced by the compass and blind straightedge both numbers are quite as finite as 1. The triangle embodies three dimensions of the cube. CB is any edge, AB is a face diagonal, and AC is an internal diagonal." Crockett-Johnson described the source of the painting as "Artist's Construction, or Anybody's."

- The triangle with three sides equal to the lengths of interest is painted white. Remaining segments of the construction are in dark gray and purple, with a black background. The painting has a brown wooden frame.

- The painting is #66 in the series and is signed: CJ69. For a related painting, see #45 (1979.1093.32).

- Reference: "Geometric Geometric [sic] Paintings by Crockett Johnson" NMAH Collections.

- Location
- Currently not on view

- date made
- 1969

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.41

- accession number
- 1979.1093

- catalog number
- 1979.1093.41

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

## Painting -

*Squared Lunes (Hippocrates Of Chios)*- Description
- Classical Greek mathematicians were able to square all convex polygons. That is, given any polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straight edge. Figures with curved sides proved more difficult. However, as this painting suggests, the mathematician Hippocrates of Chios (5th century BC) squared a lune, a figure bounded by arcs of two circle with different radii (lunes resemble quarter moons, hence the name). Finding the area of a lune in terms of a square might seem more difficult than squaring a circle, but the latter problem would prove intractable.

- The painting follows annotated figures in Evans G. Valens's
*The Number of Things*(1964), p.103, which was part of Crockett Johnson's mathematical library. It corresponds to an early diagram in Valens's discussion of squaring the circle. According to Valens, Hippocrates began by arguing that the areas of similar segments of different circles are in the same ratio as the squares of their bases. Suppose an isosceles right triangle is inscribed in a semicircle of diameter c. Construct smaller semicircles of diameter a and b on the sides of the inscribed triangle. As the square of a plus the square of b equals the square of c, the area of the two smaller semicircles equals that of the large one. The proof goes on to consider the area of the two crescents and the triangle.

- In this version of
*Squared Lunes*Crockett Johnson uses brown, black, red, and white against a gray background. This oil painting is #67 in the series, and the first in the series with the title "Squared Lunes." It was completed in 1968 and is signed: CJ68. It is inscribed on the back: SQUARED LUNES (/) (HIIPPOCRATES OF CHIOS) (/) Crockett Johnson 1968. A related painting is #68 (1979.1093.43).

- Location
- Currently not on view

- date made
- 1968

- referenced
- Hippocrates of Chios

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.42

- accession number
- 1979.1093

- catalog number
- 1979.1093.42

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

## Painting -

*Squared Lunes (Hippocrates Of Chios)*- Description
- The title of this painting refers to Hippocrates of Chios (5th century BC), one of the greatest geometers of antiquity. Classical Greek mathematicians were able to square convex polygons. That is, given a polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straightedge. They were unable to square a circle. This painting is based on the earliest known squaring of a figure bounded by curves rather than straight lines. The mathematician Hippocrates squared a lune, a figure bounded by arcs of two circles with different radii. This achievement might seem more difficult than squaring a circle.

- Crockett Johnson's painting follows two annotated figures in Evans G. Valens's
*The Number of Things*(1964), pp. 103–104, a book in the artist’s mathematical library. The finished piece shows isosceles triangles T, and a second congruent triangle connected to it base to base to form a square. Also present in the painting are three lunes, two small and one large. The area of triangle T is equal to the sum of the areas of lunes A and B (see figures). The area of triangle T is also equal to the area of a lune composed of X, Y, and the area T-C. Furthermore, because triangle T is congruent to the triangle below it, triangle T is equal to the area of this lune. Thus, the area of the square is equal to the sum of the areas of the three lunes. In summary, Johnson pictorially represented a "squared" curvilinear region; that is, he successfully constructed a square with the same area as that of the region of three lunes bounded by curves.

- Crockett Johnson executed this painting in 4 tints and darker shades of purple upon a black background. The center triangle is the darkest shade of purple. As one moves outward, the colors grow lighter. This allows a dramatic distinction to be seen between the figure and the background, and thus puts a greater emphasis on the lunes.

- This oil painting on masonite is #68 in Crockett Johnson's series. Its date of completion is unknown and the work is unsigned. It is closely related to painting #67 (1979.1093.42).

- Location
- Currently not on view

- date made
- ca 1965

- referenced
- Hippocrates of Chios

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.43

- accession number
- 1979.1093

- catalog number
- 1979.1093.43

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

## Painting -

*Cross Ratio in an Ellipse (Poncelet)*- Description
- From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle. Each of these curves may be considered as a projection of the circle. Nineteenth-century mathematicians were much interested in the properties of conics that were preserved under projection. They knew from the work of the ancient mathematician Pappus that the cross ratio of line segments created by two straight lines cutting the same pencil of lines was a constant (a pencil of lines is a set of lines emanating from a common point). In the drawing, which is Figure 5 from an article by Morris Kline in James R. Newman's
*The World of Mathematics*(1956), if line segment l’ crosses lines emanating from the point O at points A’, B’, C’, and D’; and line segment l croses the same lines at points A, B, C, and D, the cross ratio:

- (A’C’/C’B’) / (A’D’/D’B’) = (AC/BC) / (AD/DB), in other words it is independent of the cutting line. (see Crockett Johnson's painting
*Pencil of Ratios (Monge)*).

- The French mathematician Michel Chasles introduced a related result, which is the subject of this painting. He considered two points on a conic section (such as an ellipse) that were both linked to the same four other points on the conic. He found that lines crossing both pencils of rays had the same cross ratio. Moreover, a conic section could be characterized by its cross ratio. This opened up an entirely different way of describing conic sections. Crockett Johnson associated this painting with both Chasles and another French advocate of projective geometry, Victor Poncelet.

- The painting, in oil or acrylic on masonite, is #69 in the series. It has a dark gray or blue background and a black wooden frame. It shows a white ellipse, two points on the ellipse (on the left side of the painting), and two pencils of lines that produce the same cross ratio. The painting is not signed. It is inscribed on the back, in Crockett Johnson’s hand: CROSS RATIO IN AN ELLIPSE (PONCELET) (/) Crockett Johnson 1968. Compare #21 (1979.1093.15).

- Reference: Morris Kline in James R. Newman,
*The World of Mathematics*(1956), p. 634. This volume was in Crockett Johnson's library. The figure on this page is annotated.

- Location
- Currently not on view

- date made
- 1968

- referenced
- Poncelet, Jean-Victor

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.44

- accession number
- 1979.1093

- catalog number
- 1979.1093.44

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

## Painting -

*Seventeen Sides (Gauss)*- Description
- Those making mathematical instruments for surveying, navigation, or the classroom have long been interested in creating equal divisions of the circle. Ancient geometers knew how to divide a circle into 2, 3, or 5 parts, and as well as into multiples of these numbers. For them to draw polygons with other numbers of sides required more than a straightedge and compass.

- In 1796, as an undergraduate at the University of Göttingen, Friedrich Gauss proposed a theorem severely limiting the number of regular polygons that could be constructed using ruler and compass alone. He also found a way of constructing the 17-gon.

- Crockett Johnson, who himself would develop a great interest in constructing regular polygons, drew this painting to illustrate Gauss's discovery. His painting follows a somewhat later solution to the problem presented by Karl von Staudt in 1842, modified by Heinrich Schroeter in 1872, and then published by the eminent mathematician Felix Klein. Klein's detailed account was in Crockett Johnson's library, and a figure from it is heavily annotated.

- This oil painting on masonite is #70 in the series. It is signed: CJ69. The back is marked: SEVENTEEN SIDES (GAUSS) (/) Crockett Johnson 1969. The painting has a black background and a wood and metal frame. There are two adjacent purple triangles in the center, with a white circle inscribed in them. The triangles have various dark gray regions, and the circle has various light gray regions and one dark gray segment. The length of the top edge of this segment is the chord of the circle corresponding to length of the side of an inscribed 17-sided regular polygon.

- Reference: Felix Klein,
*Famous Problems of Elementary Geometry*(1956), pp. 16–41, esp. 41.

- Location
- Currently not on view

- date made
- 1969

- referenced
- Gauss, Carl Friedrich

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.45

- accession number
- 1979.1093

- catalog number
- 1979.1093.45

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

## Painting -

*Law of Motion (Galileo)*- Description
- Crockett Johnson based this painting on the discussion of motion along inclined planes by Galileo Galilee in his
*Dialogues Concerning Two New Sciences*(1638). Here Galileo showed that if from a fixed point straight lines be extended indefinitely downwards and a point be imagined to move along each line at a constant speed, all starting from the fixed point at the same time and moving with equal speeds, the locus of the moving points will be an expanding circle.

- This painting shows four superimposed circles in various shades of gray, white and black. These circles all have a common point at the center top, and differ in radius. They are shaded into several regions which are divided by lines originating at the common point. The work has an orange background and a black wooden frame. It is probably based on a drawing in E. G. Valens,
*The Attractive Universe*(1969). This volume is in Crockett Johnson's library, annotated on the page indicated.

- The painting is #71 in the series. It is signed: CJ70.

- References: Galileo Galilee,
*Dialog Concerning Two New Sciences, Third Day*(Figure 59 in the Dover edition).

- E. G. Valens,
*The Attractive Universe: Gravity and the Shape of Space*, Cleveland and New York: World Publishing Company, 1969, p. 135.

- Location
- Currently not on view

- date made
- 1970

- referenced
- Galilei, Galileo

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.46

- catalog number
- 1979.1093.46

- accession number
- 1979.1093

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

## Painting -

*Rotated Triangle and Reflections*- Description
- Plane figures of the same size and shape can be moved about in several ways and preserve their size and form. Such congruent transformations, as they are called, are combinations of rotations about a point or a line, reflections about a line, or translations in which the figure moves about the plane but the directions of the sides is unchanged.

- This painting, which closely follows a diagram from a book by H. S. M. Coxeter, illustrates two properties of congruent transformations. First, a transformation in which only one point remains unchanged is a rotation. In the figure, the triangle PQR passes through a congruent transformation into the triangle PQ'R'. Suppose that the transformation consisted of a reflection. Then triangle PQR could be rotated about the line m to another triangle, PRR[1]. However, these two triangles have a line, and not simply a point, in common. Coxeter went on to argue that any congruent transformation can be constructed as the product of reflections, the number of which can be reduced to three.

- In the painting, as in the diagram, there are three congruent triangles. One light blue and gray triangle rotates into another light blue triangle above it to the right (the axis of rotation is perpendicular to the painting). The blue and blue-gray triangle is a rotation of the first triangle about the axis m, and a reflection of the other. The background is in two shades of gray, divided by this line of rotation.

- The painting is #73 in the series and signed: CJ70. It has a metal frame.

- Reference: H. S. M. Coxeter,
*The Real Projective Plane*, p. 153.

- Location
- Currently not on view

- date made
- 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.47

- catalog number
- 1979.1093.47

- accession number
- 1979.1093

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

## Painting -

*Morley Triangle*- Description
- In the late 1960s and early 1970s, the American cartoonist Crockett Johnson created a series of paintings on mathematical subjects. This oil painting, #74 in the series, dates from 1969 and is signed "CJ69." It is based on a theorem in plane geometry proved by the English-born mathematician Frank Morley (1860–1937). Morley emigrated to the United States and taught at Haverford College and Johns Hopkins University.

- The painting illustrates his best-known result. It shows lines that divide the three angles of the large triangle into three equal parts. Lines coming from different vertices of the triangle meet in points. The triangle formed by joining the intersections of the trisectors, which lie nearest to the three sides of the triangle, is shown in white in the painting. According to Morley's theorem, this is an equilateral triangle.

- Location
- Currently not on view

- Date made
- 1969

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.48

- catalog number
- 1979.1093.48

- accession number
- 1979.1093

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

## Painting -

*Nine-Point Circle*- Description
- Although 18th- and 19th-century mathematicians were much interested in analysis and algebra, they continued to explore geometrical constructions. In 1765, the eminent Swiss-born mathematician Leonhard Euler showed that nine points constructed from a triangle lie on a circle. This circle would come to be called the Feuerbach circle after Karl Wilhelm Feuerbach, a professor at the gymnasium in Erlangen, Germany. In 1822, he published a paper explaining and proving the theorem.

- It seems likely that the direct inspiration for this painting was a figure in H. S. M. Coxeter’s
*The Real Projective Plane*(1955). A diagram on p. 143 of this book shows a triangle with its respective nine points. In his copy of the book, Crockett Johnson connected the points himself, thereby completing the circle (see the annotated figure). In addition, Johnson also annotated a figure in Nathan A. Court’s*College Geometry*(1964 printing), p. 103. Crockett Johnson's painting does not directly imitate either drawing, but it is evident that he studied each figure in creating his own construction.

- The first three points of the nine-point circle are the midpoints of the sides of triangle QRP (points L, M, and N in the annotated drawing). The second three points are the bases of the altitudes of the triangle (points A, B, C). These altitudes meet at a point (S). The midpoints of the lines joining the vertices of the triangle to the intersection of the altitudes create the last three points that indicate the nine-point circle (L’, M’, N’).

- The segments of the triangle that are not part of the circle are colored in shades of blue and gray. Those segments that are part of the circle are white and various shades of pink and yellow. The painting has a background defined by two shades of gray.

- This oil painting on masonite, #75 in the series, dates from 1970, is signed in the upper left corner : CJ70. It is inscribed on the back: NINE-POINT CIRCLE (/) Crockett Johnson 1970. There is a metal frame.

- Location
- Currently not on view

- date made
- 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.49

- catalog number
- 1979.1093.49

- accession number
- 1979.1093

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

## Painting -

*Law of Orbiting Velocity (Kepler)*- Description
- This piece is a further example of Crockett Johnson's exploration of Kepler’s first two laws of planetary motion. The ellipse represents the path of a planet, and the white sections represent equal areas swept out in equal times. This work, a silk screen inked on paper board, is signed: CJ66. It is #76 in the series, and it echoes painting #22 (1979.1093.16) and painting #99 (1979.1093.66).

- Location
- Currently not on view

- date made
- 1966

- referenced
- Kepler, Johannes

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.50

- catalog number
- 1979.1093.50

- accession number
- 1979.1093

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