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.

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. . . ."(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.
date made
1970
referenced
Newton, Isaac
painter
Johnson, Crockett
ID Number
1979.1093.36
catalog number
1979.1093.36
accession number
1979.1093
This painting shows three rectangles of equal area, one in shades of blue, one in shades of purple, and one in shades of pink.
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
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).
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
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.
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
Crockett Johnson based this painting on the discussion of motion along inclined planes by Galileo Galilee in his Dialogues Concerning Two New Sciences (1638).
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
Plane figures of the same size and shape can be moved about in several ways and preserve their size and form.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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

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