##
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 7 of 8

## Painting -

*Biblical Squared Circles*- Description
- This painting, #92 in the series, relates to a verse in the Old Testament (I Kings, Chapter VII, Verse 23) which states, "Also he made a molten sea of ten cubits brim to brim, round in compass, . . . and a line of thirty cubits did compass it round about." This verse tells us that the circular sea had a circumference of 30 cubits and a diameter of 10 cubits. Because the value of pi is defined as the ratio of a circle’s circumference to its diameter (pi = c/d), the ancient Hebrew text uses 30/10 = 3 as the value for pi.

- To illustrate this value of pi, Crockett Johnson inscribes the six-pointed Star of David within a circle. The curve joining two opposite points of the star (point C and point F in his figure) serves as a reminder of how to construct a six-pointed figure inside a circle. Furthermore, he inscribes a second, smaller circle inside the hexagon created by the six-pointed star.

- In this painting, it is assumed that the value of pi is 3. There are several relationships in the painting that involve this number. The inner circle has radius 1/2 and the outer circle has radius 1. Thus, the smaller circle has circumference pi and the larger circle has area pi. Triangle ABC in Crockett Johnson's figure is a 30-60-90 triangle with AC = 1, AB = 2, and CB equals the square root of 3. It follows that CD, BD, EA, EF, and AF also equal the square root of 3. The Star of David is composed of two overlapping equilateral triangles (triangles AEF and BCD in the figure). Triangle AEF has altitude AH = 3/2 and triangle BCD has altitude BG = 3/2. Thus, the sum of their altitudes is AH + BG = 3. It is also interesting to note that, although the dotted lines in the accompanying figure are not present in the painting, the area of the square created by the dotted corners equals three.

- In reference to this painting, Crockett Johnson wrote, "Each of the six sides of the two equilateral triangles equaled the square root of the area of the outer circle and the square root of the circumference of the inner circle; together the altitudes of the male and female triangles equaled the area of the outer circle and the circumference of the inner circle. Of course both of these circular dimensions are pi, but ecclesiastically pi equaled 3."

- The artist chose several tints and shades of blue for this painting. The illustration is darker underneath the curve from C to F than it is above, and the transition between each tint and shade is subtle. The choice of this one, “cool” color evokes a feeling of tranquility.

- This work was painted in oil on masonite, and has a wood and metal frame. It is unsigned and its date of completion is unknown.

- Reference: Biblical Squared Circles, 1979.3083.02.09, Crockett Johnson Collection.

- Location
- Currently not on view

- date made
- ca 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.61

- catalog number
- 1979.1093.61

- accession number
- 1979.1093

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

## Painting -

*Squared Rectangle and Euler Line*- Description
- Crockett Johnson had a longstanding interest in squaring figures, that is to say, constructing squares equal in area to other plane figures. Euclid had shown in his
*Elements*(Book II, Proposition 14) how to construct a square equal in area to a given rectangle. Crockett Johnson developed his own construction, one case of which served as the basis of this painting. The rectangle, the square of equal area, and a circle used in the demonstration are shown in various shades of pink.

- Two drawings from Crockett Johnson’s papers illustrate his ideas. The one that relates most closely to this painting is labeled A in his figure. In it, the given rectangle is ABED. The angles at the corner A and D are bisected, and the bisectors extended to meet at point C. The line from corner B through C meets side DE at point X. Line segments CL and XS are constructed parallel to AD. By this construction, the segment DL is half the length of AD. From center X, one may draw a line segment of length DL that intersects CL at point O. The figure and painting then show a circle of radius OX and center O that intersected side AD at V (where OV equals DL and is perpendicular to AD), and side BE at F. The point Y on the circle is on OV extended. As Crockett Johnson states in his notes, XY squared equals the product of AB and AD.

- The Euler line of a triangle includes three points. These are the intersections of the altitudes, of the perpendicular bisectors (lines perpendicular to the sides at their midpoints), and of the medians (lines drawn from a vertex to the midpoint of the opposite side). For an inscribed right triangle, both the perpendicular bisectors and the medians intersect in the center of the inscribing circle, while the altitudes meet at the right angle of the triangle. In the painting there are three right triangles inscribed in the circle. These are triangles XEF, XYF, and VXY in the diagram. The Euler line for the first two triangles is XOF, the Euler line for the third is VOY. The colors of Crockett Johnson's painting draws special attention to XOF, and it is this line he mentions in his figure for the painting.

- The painting is on masonite, and is #94 in the series. It has a blue-black background and a black wooden frame. It is signed on the back: SQUARED RECTANGLE AND EULER LINE (/) Crockett Johnson 1972.

- Location
- Currently not on view

- date made
- 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.62

- catalog number
- 1979.1093.62

- accession number
- 1979.1093

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

## Painting -

*Area and Perimeter of a Squared Circle*- Description
- To "square” a figure, according to the classical Greek tradition, means to construct, with the aid of only straightedge and compass, a square equal in area to that of the figure. The Greeks could square numerous figures, but were unsuccessful in efforts to square a circle. It was not until the nineteenth century that the impossibility of squaring a circle was demonstrated.

- This painting is an original construction by Crockett Johnson. It begins with the assumption that the circle has been squared, the area of the larger square equals that of the circle. Crockett Johnson then constructed a smaller square so that it has perimeter equal to the circumference of the circle. His diagram for the painting is shown, with the large square having side AB and the small one side of length AC.

- The painting is #95 in the series. It has a black background. There is a rose circle superimposed on two gray squares. The painting is unsigned and has a metal frame.

- Reference: Carl B. Boyer and Uta C. Merzbach,
*A History of Mathematics*(1991), pp. 65-7, pp. 71–2.

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.63

- catalog number
- 1979.1093.63

- accession number
- 1979.1093

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

## Painting -

*Velocities and Right Triangles (Galileo)*- Description
- This is the third painting by Crockett Johnson to represent the motion of bodies released from rest from a common point and moving along different inclined planes. In the
*Dialogues Concerning Two New Sciences*(1638), Galileo argued that the points reached by the balls at a given time would lie on a circle. Two such circles and three inclined planes, as well as a vertical line of direct fall, are indicated in the painting. One circle has half the diameter of the other. Crockett Johnson also joins the base of points on the inclined planes to the base of the diameters of the circles, forming two sets of right triangles.

- This oil painting on masonite is #96 in the series. It has a black background and a wooden and metal frame. It is signed on the back: VELOCITIES AND RIGHT TRIANGLES (GALILEO) (/) Crockett Johnson 1972. Compare to paintings #42 (1979.1093.30) and #71 (1979.1093.46), as well as the figure from Valens,
*The Attractive Universe: Gravity and the Shape of Space*(1969), p. 135.

- Location
- Currently not on view

- date made
- 1972

- referenced
- Galilei, Galileo

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.64

- catalog number
- 1979.1093.64

- accession number
- 1979.1093

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

## Painting -

*Squares of 1, 2, 3, 4 and Square Roots to 8*- Description
- This painting reflects Crockett Johnson's enduring fascination with square roots and squaring. As the title suggests, it includes four squares whose areas are 1, 2, 3, and 4 square units, and seven line segments whose lengths are the square roots of 2, 3, 4, 5, 6, 7, and 8.

- One may construct these squares and square roots by alternate applications of the Pythagorean theorem to squares running along the diagonal of the painting, and to rectangles running across the top (not all the rectangles are shown). More specifically, assume that the light-colored square in the upper left corner of the painting has side of length 1 (which equals the square root of 1). Then the diagonal is the square root of two, and a quarter circle with this radius centered at upper left corner cuts the sides of the square extended to determine two sides of a second, larger square. The area of this square (shown in the painting) is the square of the square root of 2, or two.

- One can then consider the rectangle with side one and base square root of two that is in the upper left of the painting. It will have sides one and the square root of 2, and hence diagonal of length equal to the square root of three. The diagonal is not shown, but an circular arc with this radius forms the second arc in the painting. It determines the sides of a square with side equal to the square root of three and area 3. It also forms a rectangle with sides of length one and the square root of 4 (or two). This gives the third arc and the largest square in the painting.

- By continuing the construction (further squares and rectangles are not shown), Crockett Johnson arrived at portions of circular arcs that cut the diameter at distances of the square roots of 5, 6, 7, and 8. Only one point on the last arc is shown. It is at the lower right corner of the painting.

- Crockett Johnson executed the work in various shades and tints from his starting point at the white and pale-blue triangle to darker blues at the opposite corner.

- This oil painting on masonite is not signed and its date of completion is unknown. It is #97 in the series.

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.65

- catalog number
- 1979.1093.65

- accession number
- 1979.1093

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

## Painting -

*Law of Orbiting Velocities*- Description
- This creation, similar to works #22 (1979.1093.16) and #76 (1979.1093.50), is a further example of Crockett Johnson's work relating to 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 is a silk screen on paper. It is number 99 in the series, and is signed in the right corner: Crockett Johnson (/) 67. It draws on a figure from
*The World of Mathematics*by James R. Newman.

- Location
- Currently not on view

- date made
- 1967

- referenced
- Kepler, Johannes

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.66

- catalog number
- 1979.1093.66

- accession number
- 1979.1093

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

## Painting -

*Square Root of Pi*- Description
- This oil painting is an original construction of Crockett Johnson, and proceeds from the assumption that the circle has been "squared." If the circle has radius one, and if square with the same center has the same area, Crockett Johnson argued that the inscribed rectangle shown, which has a diagonal that meets opposite points of intersection of the square and circle, has an area equal to the square root of pi.

- The verification of Crockett Johnson's construction is straightforward. The circle has radius one so that its area is pi. Because it is assumed that the circle has been "squared," the area of the square is also pi, and the length of one of its sides equals the square root of pi. The area of the rectangle is equal to the sum of the area of the two triangles formed by the diagonal. These triangles have bases equal to the diameter of the circle (2) and height equal to half the length of the side of the square (half of the square root of two). Hence each triangle has area half of the square root of pi, and the entire rectangle has area equal to the square root of pi. There is a second rectangle in the painting of the same area.

- There are two paintings in the collection with this title. The geometry of the two is identical; only the dimensions and colors are different. For this painting, #100 in the series, Johnson illustrates the subject vividly through the electric blue color of the rectangle with area equal to the square root of pi. Its partner, #89 in the series (1979.1093.58), displays the same rectangle in white, which contrasts brilliantly with its black and purple surroundings.

- This painting is unsigned and its precise date is unknown. It has a plain wooden frame.

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.67

- catalog number
- 1979.1093.67

- accession number
- 1979.1093

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

## Painting -

*Approximation of Pi to .0001*- Description
- In this painting, Crockett Johnson continued his exploration of ways to find rectilinear figures of area approximately equal to pi with another of his own constructions. He took advantage of the fact that the square root of two is 1.414214, while pi is approximately 3.141597. By constructing a length of one tenth the √2 and adding it to length three, he had a length 3.1414214 which, in his language, is an approximation of pi to .0001.

- Here he assumed that the two large overlapping circles both have diameter two, and the smaller circle diameter one. The three blue and white squares then have sides of length one and diagonals of length √2. Suppose (as Crockett Johnson does) that one marks off a length of 1/10 along the side of the rightmost square, and erects a perpendicular. It will cut the diagonal of the small square to form a right triangle that has hypotenuse of length equal to one tenth √2, as desired. This then serves as the radius of a small circular arc, and is added on to the length of the sides of the three unit squares to form an approximate value of pi.

- A diagram from Crockett Johnson's papers presents the mathematics of his construction.

- The painting is #101 in the series. It has a black border and is unframed. It shows two overlapping circles of the same size, a smaller of half the diameter, and the arc of a still smaller circle. The circles are divided by straight lines into turquoise and white sections on the bottom, which form the area approximately equal in area to one of the large circles. The length approximately equal to pi is across the bottom. Sections at the top are in dark purple and black.

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.68

- catalog number
- 1979.1093.68

- accession number
- 1979.1093

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

## Painting -

*Euclidian Values of a Squared Circle*- Description
- To "square" a figure, according to the classical Greek tradition, means to construct, with the aid of only straightedge and compass, a square equal in area to that of the figure. The Greeks could square numerous figures, but were unsuccessful in efforts to square a circle. It was not until the 19th century that the impossibility of squaring a circle was demonstrated.

- This painting is an original construction by Crockett Johnson. It begins with the assumprion that the circle has been squared. In this case, Crockett Johnson performed a sequence of constructions that produce several additional squares, rectangles, and circles whose areas are geometrically related to that of the original circle. These figures are produced using traditional Euclidean geometry, and require only straightedge and compass.

- The painting on masonite is #102 in the series. It has a blue-black background and a metal frame. It shows various superimposed sections of circles, squares, and rectangles in shades of light blue, dark blue, purple, white and blue-black. It is unsigned. See 1979.3083.02.13.

- References: Carl B. Boyer and Uta C. Merzbach,
*A History of Mathematics*(1991), Chapter 5.

- Crockett Johnson, "A Geometrical Look at the Square Root of Pi,"
*Mathematical Gazette*54 (February, 1970): pp. 59–60.

- Location
- Currently not on view

- date made
- ca 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.69

- catalog number
- 1979.1093.69

- accession number
- 1979.1093

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

## Painting -

*Star Construction*- Description
- Crockett Johnson's interest in regular polygons included the pentagram, or five-pointed star. The relation between the pentagon and the star is simple. If each side of a regular pentagon is extended, a regular five-pointed star results. Similarly, connecting each diagonal of a regular pentagon creates a regular five-pointed star. The star will have a pentagon in it, so the method is self-perpetuating.

- A method for a pentagram's construction in described in Book IV, Proposition II of Euclid's
*Elements*, but the construction illustrated in this painting is the artist's own creation. It builds on the relationship between the sides of a regular five-pointed star and the golden ratio. As Crockett Johnson may have recalled from his earlier paintings, the five rectangles that surround the central pentagon of the star are golden, that is to say the ratio of the length of the two equal sides of the triangle to the side of the enclosed pentagon is (1 + √5) / 2. Hence one can construct the star by finding a line segment divided in this ratio. No figure by Crockett Johnson showing his construction has been found.

- The pentagram, executed appropriately enough in hues of gold, contrasts vividly with the purple background in
*Star Construction*.

- The painting is #103 in the series. It is in oil or acrylic on pressed wood and has a gold-colored metal frame. The painting is unsigned and undated. Compare #46 (1979.1093.33) and #64 (1979.1093.39).

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.70

- catalog number
- 1979.1093.70

- accession number
- 1979.1093

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