#
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 18 items.

Page 1 of 2

## Painting -

*Problem of Delos Constructed from a Solution by Isaac Newton (Arithmetica Universalis)*- Description
- Two paintings in the Crockett Johnson collection concern the ancient problem of doubling the volume of a given cube, or the problem of Delos. Crockett Johnson wrote of this problem: "Plutarch mentions it, crediting as his source a now lost version of the legend written by the third century BC Alexandrian Greek astronomer Eratosthenes, who first measured the size of the Earth. Suffering from plague, Athens sent a delegation to Delos, Apollo’s birthplace, to consult its oracle. The oracle’s instruction to the Athenians, to double the size of their cubical altar stone, presented an impossible problem. . . ."(p. 99). Hence the reference to the problem of Delos in the title of the painting.

- Isaac Newton suggested a solution to the problem in his book
*Arithmetica Universalis*, first published in 1707. His construction served as the basis of the painting. Newton’s figure, as redrawn by Crockett Johnson, begins with a base (OA), bisected at a point (B), with an equilateral triangle (OCB) constructed on one of the halves of the base. Newton then extended the sides of this triangle through one vertex. Placing a marked straightedge at one end of the base (O), he rotated the rule so that the distance between the two lines extended equaled the sides of the triangle (in the figure, DE = OB = BA = OC = BC). If these line segments are of length one, one can show that the line segment OD is of length equal to the cube root of two, as desired.

- In Crockett Johnson’s painting, the line OA slants across the bottom and the line ODE is vertical on the left. The four squares drawn from the upper left corner (point E) have sides of length 1, the cube root of 2, the cube root of 4, and two. The distance DE (1) represents the edge of the side and the volume of a unit cube, while the sides of three larger squares represent the edge (the cube root of 2), the side (the square of the cube root of 2) and the volume (the cube of the cube root of two) of the doubled cube.

- This oil painting on masonite is #56 in the series and dates from 1970. The work is signed: CJ70. It is inscribed on the back: PROBLEM OF DELOS (/) CONSTRUCTED FROM A SOLUTION BY (/) ISAAC NEWTON (ARITHMETICA UNIVERSALIS) (/) Crockett Johnson 1970. The painting has a wood and metal frame. For related documentation see 1979.3083.04.06. See also painting number 85 (1979.1093.55), with the references given there.

- Reference: Crockett Johnson, “On the Mathematics of Geometry in My Abstract Paintings,”
*Leonardo*5 (1972): pp. 98–9.

- Location
- Currently not on view

- date made
- 1970

- referenced
- Newton, Isaac

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.36

- catalog number
- 1979.1093.36

- accession number
- 1979.1093

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

## Painting -

*Division of the Square by Conic Rectangles*- Description
- This painting shows three rectangles of equal area, one in shades of blue, one in shades of purple, and one in shades of pink. The height of the middle rectangle equals the height of the first rectangle less its own width, while the height of the third rectangle equals the height of the first triangle less the width of the first triangle. Crockett Johnson associated these properties with conic curves. The construction is that of the artist. The coloring was suggested by a recently discovered French cave painting. The narrow rectangle on the left side and the dark, thin triangle at the base were also added to correspond to the cave painting.

- The oil painting on masonite is #60 in the series. It is signed: CJ70, and inscribed on the back: DIVISION OF THE SQUARE BY CONIC RECTANGLES (/) (GNOMON ADDED AT THE SUGGESTION OF A CRO-MAGNON (/) ARTIST OF LASCAUX (/) Crockett Johnson 1970. The painting is in a black wooden frame. For related documentation see 1979.3083.02.05.

- Reference: Crockett Johnson, "On the Mathematics of Geometry in My Abstract Paintings,"
*Leonardo*5 (1972): pp. 98–101.

- Location
- Currently not on view

- date made
- 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.37

- catalog number
- 1979.1093.37

- accession number
- 1979.1093

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

## Painting -

*Aligned Triangles (Desargues)*- Description
- In the 17th century, the French engineer and architect Girard Desargues (1591–1661) explored interconnections between extensions of the lines within a pencil of three line segments (a pencil of line segments consists of several line segments originating at a common point). His theorems, as published in his own extremely obscure work and also by his contemporary, Abraham Bosse, were extended in the 19th century, and proved of fundamental importance to projective geometry.

- Crockett Johnson's library contains discussions of Desargues' theorem by H. S. M. Coxeter, N. A. Court, Heinrich Dorrie, and William M. Ivins. This painting most resembles a figure from Coxeter, although the diagram is not annotated. Suppose that the vertices of two triangles (PQR and P'Q'R' in Figure 1.5B from Coxeter) lie on a pencil of three line segments emanating from the point O. Suppose that similarly situated sides of the two triangles can be extended to meet in the three points denoted by A, C and B in the figure. According to Desargues' theorem, A, C, and B are collinear.

- In the painting, the two concurrent triangles are shown in shades of gray and black, while the top of the pencil of three lines is in shades of gold. Extensions of the sides and their points of intersection are clearly shown. Both the figure and the background of the painting are divided by the line joining the points of intersection

- The painting is #63 in the series. It is painted in oil or acrylic on masonite, and has a brown wooden frame. The painting is signed: CJ70.

- References:

- Newman, J. R.,
*The World of Mathematics*, p. 133. Figure annotated.

- Court, N. A.,
*College Geometry*(1952), pp. 163–5. The figure is not annotated.

- Coxeter, H. S. M.,
*The Real Projective Plane*, (1955 edition), p. 7. The figure resembles the painting but is not annotated.

- Dorrie, Heinrich,
*100 Great Problems of Elementary Mathematics: Their History and Solution*(1965), p. 267. There is an annotated figure here for another theorem of Desargues, the theorem of involution.

- Field, J. V.,
*The Invention of Infinity: Mathematics and Art in the Renaissance*(1997), pp. 190–206.

- Ivins, William M. Jr.,
*Art & Geometry: A Study in Space Intuitions*(1946), pp. 87–94.

- Location
- Currently not on view

- date made
- 1970

- referenced
- Desargues, Girard

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.38

- accession number
- 1979.1093

- catalog number
- 1979.1093.38

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

## Painting -

*Golden Rectangle*- Description
- Crockett Johnson annotated several diagrams in his copy of Valens’s book
*The Number of Things*, and used a few of them as the basis of paintings. This is one example. It shows three golden rectangles, the curves from a compass used to construct the rectangles, and a section of a five-pointed Pythagorean star.

- Euclid showed in his
*Elements*that it is possible to divide a line segment into two smaller segments wherein the ratio of the whole length to the longer part equals the ratio of the longer part to the smaller. He used this theorem in his construction of a regular pentagon. This ratio came to be called the “golden ratio.”

- A golden rectangle is a rectangle whose sides adhere to the golden ratio (in modern terms, the ratio of its length to its width equals (1 + √(5) ) /2, or about 1.62). The golden rectangle is described as the rectangle whose proportions are most pleasing to the eye.

- This painting shows the relationship between a golden rectangle and a five-pointed Pythagorean star by constructing the star from the rectangle. It follows a diagram on the top of page 131 in Evans G. Valens,
*The Number of Things*. This diagram is annotated. Valens describes a geometrical solution to the two expressions f x f = e x c and f = e - c, and associates it with the Pythagoreans. The right triangle on the upper part of Valens's drawing, with the short side and part of the hypotenuse equal to f, is shown facing to the left in the painting. It can be constructed from a square with side equal to the shorter side of the rectangle. Two of the smaller rectangles in the painting are also golden rectangles. Crockett Johnson also includes in the background the star shown by Valens and related lines.

- This painting on masonite, #64 in the series, dates from 1970 and is signed: CJ70. It also is marked on the back: ”GOLDEN RECTANGLE (/) Crockett Johnson 1970. It is executed in two hues of gold to emphasize individual sections. While this method creates a detailed and organized contrast, it disguises the three rectangles and the star. Compare paintings 1979.1093.33 (#46) and 1979.1093.70 (#103).

- Reference: Evans G. Valens,
*The Number of Things*(1964), p. 131.

- Location
- Currently not on view

- date made
- 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.39

- accession number
- 1979.1093

- catalog number
- 1979.1093.39

- 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 -

*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 -

*Equal Triangles*- Description
- In this painting, based on his own construction, Crockett Johnson continued his exploration of rectilinear figures that could be made with a ruler and compass, assuming that one could construct a square of area equal to pi (e.g. if one could square the circle). More specifically, he assumed that he could construct a square of that area (the square in blue and dark blue in the painting) and found four triangles, also shown in shades of blue, that would be of equal area. A sheet from his papers presents his argument (1979.3083.04.02)

- The oil painting on masonite is #86 in the series. It is signed on the back: EQUAL TRIANGLES (/) Crockett Johnson 1972. There is a wood and chrome frame.

- Location
- Currently not on view

- date made
- 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.56

- catalog number
- 1979.1093.56

- accession number
- 1979.1093

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

## Painting -

*Squares of 2, 4, 16 from Square Root of x*- Description
- In this painting, Crockett Johnson supposed that one was given two lengths, one the square root of the second. Although no numerical values were given, he sought to construct three squares, one the square root of the second and the second the square root of the third, and to give their values numerically. His solution is represented in the painting, and described in his notes as work from 1972.

- The three squares are visible, one the entire surface of the painting and the two others within it. The vertical lines point to the starting point of the painting, a line segment along the base and its square root. From here, Crockett Johnson constructed the elaborate geometrical argument illustrated by the painting. He claimed that he had constructed squares of area 2, 4, and 16. The ratios of the areas are as he describes, but the absolute numerical values depend on the units of measure.

- This oil painting on masonite is #88 in the series. It is unsigned. There is an inset metal strip in the wooden frame.

- Location
- Currently not on view

- date made
- 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.57

- catalog number
- 1979.1093.57

- accession number
- 1979.1093

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

## Painting -

*Square Root Of Pi - 0.00001*- Description
- This oil painting on masonite, #91 in the series, uses the same construction as that of painting #52 (see 1979.1093.35). Crockett Johnson's construction leads to a square with side approximately equal to 1.772435, which differs from the square root of pi by less than 0.00001, as the title states. Thus, a square with this side would have an area approximately equal to 3.1415258.

- Unlike painting #52 (1979.1093.35), the circle of this work is divided into four quadrants. Crockett Johnson chose darker shades and lighter tints of pink to illustrate his figure, which appear bold juxtaposed against the black background. The triangle executed in the lightest tint of pink and the shape executed in white with a pink tip adjoin the horizontal line segment that has an approximate length of the square root of pi.

- This painting was completed in 1972, is unsigned, and has a wooden frame accented with chrome. On the back is an inscription, partly obscured, that reads: - 0.00001 (/) Crockett Johnson 1972.

- Location
- Currently not on view

- date made
- 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.60

- catalog number
- 1979.1093.60

- accession number
- 1979.1093

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