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 here, with related diagrams from the artist’s library and papers.

Painting - Nine-Point Circle

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.
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

Painting -Law of Orbiting Velocity (Kepler)

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.
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

Painting - Numbers in a Spiral

Some of Crockett Johnson's paintings reflect relatively recent research. Mathematicians had long been interested in the distribution of prime numbers.
Description
Some of Crockett Johnson's paintings reflect relatively recent research. Mathematicians had long been interested in the distribution of prime numbers. At a meeting in the early 1960s, physicist Stanislaw Ulam of the Los Alamos Scientific Laboratory in New Mexico passed the time by jotting down numbers in grid. One was at the center, the digits from 2 to 9 around it to form a square, the digits from 10 to 25 around this, and the spiral continued outward.
Circling the prime numbers, Ulam was surprised to discover that they tended to lie on lines. He and several colleagues programmed the MANIAC computer to compute and plot a much larger number spiral, and published the result in the American Mathematical Monthly in 1964. News of the event also created sufficient stir for Scientific American to feature their image on its March 1964 cover. Martin Gardner wrote a related column in that issue entitled “The Remarkable Lore of the Prime Numbers.”
The painting is #77 in the series. It is unsigned and undated, and has a wooden frame painted white.
Location
Currently not on view
date made
ca 1965
painter
Johnson, Crockett
ID Number
1979.1093.51
catalog number
1979.1093.51
accession number
1979.1093

Painting - Parabolic Triangles (Archimedes)

According to the classical Greek tradition, the quadrature or squaring of a figure is the construction, with the aid of only straight edge and compass, of a square equal in area to that of the figure. But finding the area bounded by curved surfaces was not an easy task.
Description
According to the classical Greek tradition, the quadrature or squaring of a figure is the construction, with the aid of only straight edge and compass, of a square equal in area to that of the figure. But finding the area bounded by curved surfaces was not an easy task. The parabola and other conic sections had been known for almost a century before Archimedes wrote a short treatise called Quadrature of the Parabola in about 240 BC. This was the first demonstration of the area bounded by a conic section. In his proof, Archimedes first constructed a triangle whose sides consisted of two tangents of a parabola and the chord connecting the points of tangency. He then showed that the area under the parabola (shown in gray and black in the painting) is two thirds of the area of the triangle which circumscribes it. Once the area bounded by the tangent could be expressed in terms of the area of a triangle, it was easy to construct the corresponding square. Crockett Johnson’s painting follows two diagrams illustrating a discussion of Archimedes’s proof given by Heinrich Dorrie (Figure 54).
This oil or acrylic painting on masonite is #78 in the series and is signed “CJ67” in the bottom left corner. It has a gray wooden frame. For a related painting, see #43 (1979.1093.31).
References: Heinrich Dorrie, trans. David Antin, 100 Great Problems of Elementary Mathematics: Their History and Solution (1965), p. 239. This volume was in Crockett Johnson's library and the diagram in his copy is annotated.
James R. Newman, The World of Mathematics (1956), p. 105. This volume was in Crockett Johnson's library. The figure on this page (Figure 9) is annotated.
Location
Currently not on view
date made
1967
referenced
Archimedes
painter
Johnson, Crockett
ID Number
1979.1093.52
catalog number
1979.1093.52
accession number
1979.1093

Painting - Duality (Pascal-Brianchon)

As a 21-year-old student, the Frenchman Charles Jules Brianchon (1785–1864) discovered that in any hexagon circumscribed about a conic section (such as a circle), the three lines that join opposite diagonals meet in a single point.
Description
As a 21-year-old student, the Frenchman Charles Jules Brianchon (1785–1864) discovered that in any hexagon circumscribed about a conic section (such as a circle), the three lines that join opposite diagonals meet in a single point. He also pointed out connections between his result and Pascal's theorem concerning the points of intersection of opposite sides of a hexagon inscribed in a conic section.
In the painting, a hexagon (only the vertices are shown) is inscribed in a circle. Three diagonal lines (edges of the gray and black polygon) are collinear. The line in question is the line joining the points of intersection, white on one side and purple on the other. Crockett Johnson's painting closely resembles a diagram of A. S. Smogorzhevskii in which Brianchon's theorem is applied to a proof of Pascal's theorem.
The painting on masonite is #81 in the series. It has a purple background and a black wooden frame. It is signed: CJ66.
References: A. S. Smogorzhevskii, The Ruler in Geometrical Constructions (1961), p. 37. This volume was in Crockett Johnson's library. The figure is not annotated.
Carl Boyer and Uta Merzbach, A History of Mathematics (1991), p. 534.
Location
Currently not on view
date made
1966
referenced
Pascal, Blaise
Brianchon, Charles Julien
painter
Johnson, Crockett
ID Number
1979.1093.53
catalog number
1979.1093.53
accession number
1979.1093

Painting - Pi Squared and Its Square Root

This painting is part of Crockett Johnson's exploration of constructions that might take place if one could draw squares equal in area to circles. It is based on a figure that includes two squares and a rectangle.
Description
This painting is part of Crockett Johnson's exploration of constructions that might take place if one could draw squares equal in area to circles. It is based on a figure that includes two squares and a rectangle. The smaller square (ABDX in Crockett Johnson's figure) is defined as having the same area as the circle circle with center O and diameters the diagonals of the rectangle with sides CE and EX. This circle also appears in his other diagram, although it does not appear in the painting. Other assumptions concerning the upper diagram are that the rectangle has area the square root of the area of the circle and that the triangles with sides CX and PX are isosceles and congruent.
If the small circle has radius one.and the are of the rectangle is assumed to be the square root of the area of that circle and the small square, the area of the rectangle is the square root of pi.
The painting is #83 in the series. It is in oil or acrylic on masonite. There is a black wooden frame. The work is unsigned and undated.
Location
Currently not on view
date made
1970-1975
painter
Johnson, Crockett
ID Number
1979.1093.54
catalog number
1979.1093.54
accession number
1979.1093

Painting - Doubled Cube (Newton)

Two paintings in the Crockett Johnson collection concern the ancient problem of doubling the volume of a given cube, or the Problem of Delos.
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 . . . . It could not be done with the compass and an unmarked straightedge."
(p. 99).
Crockett Johnson's paintings follow a construction proposed by the eminent English mathematician Isaac Newton. As Lucasian professor of mathematics at Cambridge University, Newton was required to deposit copies of his lectures in the university library. In 1683, after he had taught a course in algebra for 11 years, he finally deposited the notes for it. After Newton left Cambridge in 1696, his successor, William Whiston, arranged to have the lectures published in a book with the short title Arithmetica Universalis. Latin editions of the book appeared in 1707, 1722, 1732, and 1761; and English translations in 1720, 1728, and 1769.
In an appendix to this book, Newton discussed ways of finding the roots of numbers through geometric constructions. One problem was that of finding two mean proportions between given numbers. One case of this problem gives the cube root of a number. [Suppose the numbers are a and b and the proportionals x and y. Then a / x = x / y = y /b). Squaring the first and last term, a² / x² = y² / b². But, from the first equation, one also has x = y² / b. By substitution, a² / x² = x / b, or x³ = a² b. If a is 1, x is the cube root of b, as desired.]
Newton and Crockett Johnson represented the quantities involved as lengths of the sides of triangles. Newton’s figure is #99 in his Arithmetica Universalis. Crockett Johnson's figure is differently lettered, and the mirror image of that of Newton.
Following the artist's notation (figure 1979.3083.04.05), suppose AB = 1, bisect it at M, and construct an equilateral triangle MBX on MB. Draw AX and MX extended. Using a marked straightedge, construct line segment BZY, intersecting AX at Z and MX at Y in such a way that XY = AM = MB = 1/2. Then the distance BZ will have a length of one half the cube root of 2, that is to say the length of the side of a cube of side 1/2.
A proof of Newton’s construction is given in Dorrie. Crockett Johnson's copy of a drawing in this volume is annotated. The duplication of the cube also was discussed in at least two other books in Crockett Johnson's library. One is a copy of the 1764 edition of an English translation of the Arithmetica Universalis, which Crockett Johnson purchased in January of 1972. The second is W. W. Rouse Ball’s Mathematical Recreations and Essays, which also discusses Newton's solution.
Crockett Johnson's painting emphasized doubled lines in the construction, building on the theme of the painting. His diagram for the painting is oriented differently from the painting itself.
This oil painting on masonite is #85 in the series. It depicts overlapping blue, pink and gray circular segments in two adjacent rectangles. These rectangles are divided by various lines into gray and black sections. A lighter gray border goes around the edge. There is a metal and wooden frame. The painting is unsigned. For a mathematically related painting, see #56 (1979.1093.36).
References: Crockett Johnson, "On the Mathematics of Geometry in My Abstract Paintings," Leonardo 5 (1972): pp. 98–100. This specific painting is not discussed in the article.
Heinrich Dorrie, trans. David Antin, 100 Great Problems of Elementary Mathematics: Their History and Solution (1965) p. 171. The figure on this page, figure 27, is annotated.
Isaac Newton, Universal Arithmetick, (1769), esp. pp. 486–87, figure 99. This volume was in Crockett Johnson's library. It is not annotated.
W. W. Rouse Ball, rev. H. S. M. Coxeter, Mathematical Essays and Recreations, (1962 printing), pp. 327–33. This is a slightly different construction. The volume was in Crockett Johnson's library.
Isaac Newton, The Mathematical Works of Isaac Newton, assembled by Derek T. Whiteside, vol. 2, (1967). This includes a reprint of the 1728 English translation of the Arithmetica Universalis.
Location
Currently not on view
date made
ca 1970
referenced
Newton, Isaac
painter
Johnson, Crockett
ID Number
1979.1093.55
catalog number
1979.1093.55
accession number
1979.1093

Painting - Equal Triangles

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).
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

Painting - Squares of 2, 4, 16 from Square Root of x

In this painting, Crockett Johnson supposed that one was given two lengths, one the square root of the second.
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

Painting - Square Root of Pi

This oil painting is an original construction of Crockett Johnson. Since there are no notes on this construction, there is no way to tell what, if any, geometrical statement Johnson was representing in this painting.
Description
This oil painting is an original construction of Crockett Johnson. Since there are no notes on this construction, there is no way to tell what, if any, geometrical statement Johnson was representing in this painting. It may be the case that he merely thought of a more artistic way to portray the rectangles with area the square root of pi that appear in notes used for another painting, “Pi Squared and its Square Root” (1979.1093.54).
This painting has at its center a circle with center O and area pi. Also in the painting there are two rectangles, each of area the square root of pi, that share a diagonal that is the diameter of the circle with one end at point E. The purple rectangle in the painting has sides CE and EX and the white rectangle has sides DE and EF. The square in the painting is congruent to the square BDXA so it also has area pi, but it has been translated so its center is the same as the center of the circle, i.e. at O.
This is one of two paintings in the collection with this same title referring to the area the rectangles shown in the paintings. The geometry of the two is identical (see painting #100 - 1979.1093.67) but the dimensions and colors are different. The method of the color scheme of this painting, #89 in the series, is similar to painting #100 because, like the electric blue rectangle in the other painting, the white color of the rectangle against the purple background creates a dramatic contrast that highlights a rectangle with area the title of the painting.
This painting was executed in oil on masonite and has a black wooden frame. It is unsigned and undated.
Location
Currently not on view
date made
1970-1975
painter
Johnson, Crockett
ID Number
1979.1093.58
catalog number
1979.1093.58
accession number
1979.1093

Painting - Equal Areas, Their Triangular Square Root and Pi

This painting, based on a construction of Crockett Johnson, shows a central brown circle, a blue square, and a pink rectangle of equal area. Assuming the radius ot the circle is one, this area equals pi.
Description
This painting, based on a construction of Crockett Johnson, shows a central brown circle, a blue square, and a pink rectangle of equal area. Assuming the radius ot the circle is one, this area equals pi. The blue triangle has an approximate area of square root of pi, presenting the "triangular square root" in the title.
The diagram is from Crockett Johnson's papers. It begins with construction of a circle of radius one (the smaller circle with center X in the figure) and assumes he could find the square root of pi and construct the line XC equal to this as a side of the square shown. Assuming he can do this, the area of the square is pi. He then draws a circle of radius 2 centered at X , which intersects the square at F and extensions of the line XC at A and at N. Bisecting FX at O, he can draw a second unit circle centered at O. He joined A to B and F to N to obtain triangles XAB and XNF. Next, the artist constructed the semicircle with that intersects circle O at point I and the larger circle at point K. He then drew diameter KP and extended FI to H with IH = 1. To complete the illustration, Crockett Johnson outlined rectangle with sides HI and IP.
To show that the construction is correct, note that XC = JF = √(pi) because the square with side XC and circle O both have area pi. Triangle XNF = (1/2)(XN)(JF) = (1/2)(2)(√(pi)) = √(pi). To show that the rectangle with sides PI and HI has area pi observe that right triangle PIF is congruent to right triangle PFK. Thus P/IPF = PF/PK and PI = (PF)²/(PK) = (2JF)²/PK = 4(JF)²/PK = 4(√((pi))²)/4 = pi. So, the rectangle has area (HI)(PI) = (1)(pi) = pi, and the demonstration is complete.
This painting is executed in oil on masonite and is #90 in the series. The figures of the painting that display the painting’s title are colored in bright, bold colors while those shapes that constitute the background are less drastically highlighted. Thus, Crockett Johnson uses color to distinguish the important features of his construction.
This painting is unsigned and its date of completion is unknown.
Location
Currently not on view
date made
1970-1975
painter
Johnson, Crockett
ID Number
1979.1093.59
catalog number
1979.1093.59
accession number
1979.1093

Painting - Square Root Of Pi - 0.00001

This oil painting on masonite, #91 in the series, uses the same construction as that of painting #52 (see 1979.1093.35).
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.
Some sources refer to this painting as Circle Squared to 0.0001.
date made
1972
painter
Johnson, Crockett
ID Number
1979.1093.60
catalog number
1979.1093.60
accession number
1979.1093

Painting - Biblical Squared Circles

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, . . .
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.
date made
ca 1972
painter
Johnson, Crockett
ID Number
1979.1093.61
catalog number
1979.1093.61
accession number
1979.1093

Painting - Squared Rectangle and Euler Line

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.
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

Painting - Area and Perimeter of a Squared Circle

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.
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

Painting - Velocities and Right Triangles (Galileo)

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.
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

Painting - Squares of 1, 2, 3, 4 and Square Roots to 8

This painting reflects Crockett Johnson's enduring fascination with square roots and squaring.
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

Painting - Law of Orbiting Velocities

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.
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

Painting - Square Root of Pi

This oil painting is an original construction of Crockett Johnson. Since there are no notes on this construction, there is no way to tell what, if any, geometrical statement Johnson was representing in this painting.
Description
This oil painting is an original construction of Crockett Johnson. Since there are no notes on this construction, there is no way to tell what, if any, geometrical statement Johnson was representing in this painting. It may be the case that he merely thought of a more artistic way to portray the rectangles with area the square root of pi that appear in notes used for another painting, “Pi Squared and its Square Root” (#83 - 1979.1093.54).
This painting has at its center a circle with center O and area pi. Also in the painting there are two rectangles, each of area the square root of pi, that share a diagonal that is the diameter of the circle with one end at point E. The black rectangle in the painting has sides CE and EX and the blue rectangle has sides DE and EF. The square in the painting is congruent to the square BDXA so it also has area pi, but it has been translated so its center is the same as the center of the circle, i.e. at O.
This is one of two paintings in the collection with this same title referring to the area the rectangles shown in the paintings. The geometry of the two is identical but 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. Its partner, #89 in the series (1979.1093.58), displays the same rectangle in white, which contrasts brilliantly with its black and purple surroundings.
The painting is unsigned and its precise date is unknown. It has a plain wooden frame.
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

Painting -Approximation of Pi to .0001

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.
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 right side, 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 tleft side 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

Painting - Euclidian Values of a Squared Circle

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.
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

Painting - Star Construction

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.
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

Painting - Archimedes Transversal

The construction of regular polygons using straightedge and compass alone is a problem that has intrigued mathematicians from ancient times.
Description
The construction of regular polygons using straightedge and compass alone is a problem that has intrigued mathematicians from ancient times. Crockett Johnson was particularly interested in the construction of regular seven-sided figures or heptagons, which require not only a compass but a marked straight edge. The mathematician Archimedes reportedly proposed such a construction, which was included in a treatise now lost. Relying heavily on Thomas Heath's Manual of Greek Mathematics, Crockett Johnson prepared this painting.
Archimedes had reduced the problem of finding a regular hexagon to that of finding two points that divided a line segment into two mean proportionals. He then used a construction somewhat like that of the painting to find a line segment divided as desired. Crockett Johnson's papers include not only photocopies of the relevant portion of Heath, but his own diagrams.
The painting is #104 in the series. It is in acrylic or oil on masonite., and has purple, yellow, green and blue sections. There is a black wooden frame. The painting is unsigned and undated. Relevant correspondence in the Crockett Johnson papers dates from 1974.
References: Heath, Thomas L., A Manual of Greek Mathematics (1963 edition), pp. 340–2.
Crockett Johnson, "A construction for a regular heptagon," Mathematical Gazette, 59 (March 1975): pp. 17–18.
Location
Currently not on view
date made
ca 1974
referenced
Archimedes
painter
Johnson, Crockett
ID Number
1979.1093.71
catalog number
1979.1093.71
accession number
1979.1093

Painting - Heptagon 1:3:3 Triangle

This painting is part of Crockett Johnson's exploration of the properties of the heptagon, extended to include a 14-sided regular polygon. The design of the painting is shown in his figure, which includes many of the line segments in the painting.
Description
This painting is part of Crockett Johnson's exploration of the properties of the heptagon, extended to include a 14-sided regular polygon. The design of the painting is shown in his figure, which includes many of the line segments in the painting. Here Crockett Johnson argues that the triangle ABF in the figure is the one he sought, with angle FAB being one seventh of pi. Segment CD in the figure, which appears in the painting, is the length of the edge of a regular 14-sided figure inscribed in a portion of the larger circle shown.
The painting, of oil or acrylic on masonite, is number 105 in the series. It is drawn in shades of cream, blue, and purple on a light purple background. It has a metal frame and is unsigned.
Location
Currently not on view
date made
ca 1973
painter
Johnson, Crockett
ID Number
1979.1093.72
catalog number
1979.1093.72
accession number
1979.1093

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