| National Museum of American History

This painting illustrates two different kinds of mathematical progressions, the geometric (on the top) and the arithmetic (on the bottom). Going across the top from left to right each section is twice as wide as the previous one, as in a geometric progression.

Description: This painting illustrates two different kinds of mathematical progressions, the geometric (on the top) and the arithmetic (on the bottom). Going across the top from left to right each section is twice as wide as the previous one, as in a geometric progression. Going across the bottom from right to left, each section is 1 unit wider than the previous one, as in an arithmetic progression.; If the width of the top sections, considered going from left to right, represents the numbers a, 2a, 4a, and 8a in a geometric progression, then the width of the bottom sections, going right to left, can represent logarithms of these numbers, b = log a, 2b =2 log a, 3b = 3 log a, and 4b =4 log a. Crockett Johnson may have sought to illustrate an account of logarithms given in an article by H. W. Turnbull in Newman's Men of Mathematics. This painting does not represent the traditional divisions of either a slide rule or a ruler.; The Scottish nobleman John Napier published his discovery of logarithms in 1614. The painting suggests how logarithms allow one to reduce multiplication (as in the terms of a geometric progression) to addition (as in the terms of an arithmetic progression). As addition is far simpler than multiplication, logarithms were widely used by people carrying out calculations from the seventeenth century onward.; The painting is #37 in the series. It is in oil or acrylic on masonite, and is signed: CJ66. There is a gray wooden frame.; Reference: H. W. Turnbull, “The Great Mathematicians,” in James R. Newman, The World of Mathematics, (1956), p. 124. This volume was in Crockett Johnson's library, but the figure is not annotated.
Location: Currently not on view

date made: 1966

referenced: Napier, John
painter: Johnson, Crockett

ID Number: 1979.1093.25
catalog number: 1979.1093.25
accession number: 1979.1093

In 1966, Crockett Johnson carefully read Nathan A. Court's book College Geometry, selecting diagrams that he thought would be suitable for paintings. In the chapter on harmonic division, he annotated several figures that relate to this painting.

Description: In 1966, Crockett Johnson carefully read Nathan A. Court's book College Geometry, selecting diagrams that he thought would be suitable for paintings. In the chapter on harmonic division, he annotated several figures that relate to this painting. The work shows two orthoganol circles, that is to say two circles in which the square of the line of centers equals the sum of the squares of the radii. A right triangle formed by the line of centers and two radii that intersect is shown. The small right triangle in light purple in the painting is this triangle.; Crockett Johnson's painting combines a drawing of this triangle with a more complex figure used in a discussion of further properties of lines drawn in orthoganal circles. In particular, suppose that one draws a line segment from a point outside a circle that intersects it in two points, and selects a fourth point on the line that divides the segment harmonically. For a single exterior point, all these such points lie on a single line, perpendicular to the line of centers of the two circles, which is called the polar line.; The painting is #38 in the series. It has a background in two shades of cream, and a light tan wooden frame. It shows two circles that overlap slightly and have various sections. The circles are in shades of blue, purple and cream. The painting is signed: CJ66.; References: Nathan A. Court, College Geometry (1964 printing), p. 175–78. This volume was in Crockett Johnson's library.; T. L. Heath, ed., Apollonius of Perga: Treatise on Conic Sections (1961 reprint). This volume was not in Crockett Johnson's library.
Location: Currently not on view

date made: 1966

referenced: Apollonius of Perga
painter: Johnson, Crockett

ID Number: 1979.1093.26
catalog number: 1979.1093.26
accession number: 1979.1093

Mathematicians have explored the properties of polyhedra since ancient times, but it was the Swiss scholar Leonhard Euler (1707–1783) who proved the formula V-E+F = 2. That is, for a simple convex polyhedron (e.g.

Description: Mathematicians have explored the properties of polyhedra since ancient times, but it was the Swiss scholar Leonhard Euler (1707–1783) who proved the formula V-E+F = 2. That is, for a simple convex polyhedron (e.g. one with no holes, so that it can be deformed into a sphere) the number of vertices minus the number of edges plus the number of faces is two. An equivalent formula had been presented by Descartes in an unpublished treatise on polyhedra. However, this formula was first proved and published by Euler in 1751 and bears his name.; Crockett Johnson's painting echoes a figure from a presentation of Euler's formula found in Richard Courant and Herbert Robbins's article “Topology,” which is in James R. Newman's The World the Mathematics (1956), p. 584. This book was in the artist’s library, but the figure that relates to this painting is not annotated.; To understand the painting we must understand the mathematical argument. It starts with a hexahedron, a simple, six-sided, box-shaped object. First, one face of the hexahedron is removed, and the figure is stretched so that it lies flat (imagine that the hexahedron is made of a malleable substance so that it can be stretched). While stretching the figure can change the length of the edges and the area and shape of the faces, it will not change the number of vertices, edges, or faces.; For the "stretched" figure, V-E+F = 8 - 12 + 5 = 1, so that, if the removed face is counted, the result is V-E+F = 2 for the original polyhedron. The next step is to triangulate each face (this is indicated by the diagonal lines in the third figure). If, in triangle ABC [C is not shown in Newman, though it is referred to], edge AC is removed, the number of edges and the number of faces are both reduced by one, so V-E+F is unchanged. This is done for each outer triangle.; Next, if edges DF and EF are removed from triangle DEF, then one face, one vertex, and two edges are removed as well, and V-E+F is unchanged. Again, this is done for each outer triangle. This yields a rectangle from which a right triangle is removed. Again, this will leave V-E+F unchanged. This last step will also yield a figure for which V-E+F = 3-3+1. As previously stated, if we count the removed face from the initial step, then V-E+F = 2 for the given polyhedron.; The “triangulated” diagram was the one Crockett Johnson chose to paint. Each segment of the painting is given its own color so as to indicate each step of the proof. Crockett Johnson executed the two right triangles that form the center rectangle in the most contrasting hues. This draws the viewer’s eyes to this section and thus emphasizes the finale of Euler's proof. This approach to the proof of Euler's polyhedral formula was pioneered by the French mathematician Augustin Louis Cauchy in 1813.; This oil painting on masonite is #39 in the series. It was completed in 1966 and is signed: CJ66. It is inscribed on the back: Crockett Johnson 1966 (/) POLYHEDRON FORMULA (EULER). It has a wood and chrome frame.; Reference:; David Richeson, “The Polyhedral Formula,” in Leonhard Euler: Life, Work and Legacy, editors R. E. Bradley and C. E. Sandifer (2007), pp. 431–34.
Location: Currently not on view

date made: 1966

referenced: Euler, Leonhard
painter: Johnson, Crockett

ID Number: 1979.1093.27
catalog number: 1979.1093.27
accession number: 1979.1093

This painting was inspired by ideas of Carl Friedrich Gauss (1777–1855). In his 1797 doctoral thesis, Gauss proved what is now called the fundamental theorem of algebra. He showed that every polynomial with real coefficients must have at least one real or complex root.

Description: This painting was inspired by ideas of Carl Friedrich Gauss (1777–1855). In his 1797 doctoral thesis, Gauss proved what is now called the fundamental theorem of algebra. He showed that every polynomial with real coefficients must have at least one real or complex root. A complex number has the form a+bi, where a and b are real numbers and i represents the square root of negative one. The French mathematician René Descartes (1596–1650) called such numbers "imaginary", which explains the reference in the title. Gauss demonstrated that, just as real numbers can be represented by points on a coordinate line, complex numbers can be represented by points in the coordinate plane.; The construction of this painting echoes a figure in an article on Gauss by Eric Temple Bell in J. R. Newman's The World of Mathematics that illustrates the representation of points on a plane. This book was in Crockett Johnson's library, and the figure is annotated.; In Bell's figure, real numbers c and -c are plotted on the x axis, the imaginary numbers ci and -ci are plotted on the y axis, and the complex number a+bi is shown in the first quadrant. The figure is meant to show that if a complex number a+bi is multiplied by the imaginary number i then the product is a complex number on the same circle but rotated ninety degrees counterclockwise. That is, i(a+bi) = ai+bi² = -b+ai. Thus, this complex number lies in the second quadrant. If this process is repeated the next product is -a-bi, which lies in the third quadrant. It is unknown why Johnson did not illustrate the fourth product.; The colors of opposite quadrants of the painting are related. Those in quadrant three echo those of quadrant one and those of quadrant four echo those of quadrant two.This oil painting is #40 in the series. It is signed: CJ67.; References:; James R. Newman, The World of Mathematics (1956), p. 308. This volume was in Crockett Johnson's library. The figure on this page is annotated.
Location: Currently not on view

date made: 1967

painter: Johnson, Crockett

ID Number: 1979.1093.28
catalog number: 1979.1093.28
accession number: 1979.1093

The locus of the midpoints of the chords of a given circle that pass through a fixed point is a circle when the point lies inside of or on the circle.

Description: The locus of the midpoints of the chords of a given circle that pass through a fixed point is a circle when the point lies inside of or on the circle. The small circle painted white is the locus of the midpoints of chords drawn in the large circle that pass through a point toward the top left of the inside of the circle. Three chords of the large circle are suggested. These are the diameter, whose midpoint is the center of the circle, a vertical chord through the point, and a horizontal chord through the point (only a small part of this chord is indicated). The painting is based on a diagram from College Geometry by Nathan Court. It is unclear why Crockett Johnson associated this painting with Plato.; The oil painting on masonite is #41 in the series. It has a background of two purple and gray rectangles. It has a metal and wooden frame. It shows a circle with a smaller circle inside it. The smaller circle is in two shades of white, the larger one in orange, black, gray and light purple. The painting is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) LOCUS OF POINT ON CHORD (PLATO).; Reference: Nathan Court, College Geometry, (1964 printing), p. 13. This figure is annotated in Crockett Johnson's copy of this volume.

date made: 1966

referenced: Plato
painter: Johnson, Crockett

ID Number: 1979.1093.29
catalog number: 1979.1093.29
accession number: 1979.1093

This oil painting is based on a figure from Galileo Galilee's Dialogues Concerning Two New Sciences (1638), Book 3. Here Galileo discussed the time of descent of bodies rolling without friction along inclined planes.

Description: This oil painting is based on a figure from Galileo Galilee's Dialogues Concerning Two New Sciences (1638), Book 3. Here Galileo discussed the time of descent of bodies rolling without friction along inclined planes. He argued that if from the highest point in a vertical circle there be drawn any inclined planes meeting the circumference of the circle, the times of descent along these chords are equal to one another. This painting shows two inclined planes drawn from the highest point of a vertical circle, with a ball moving along each chord. Crockett Johnson probably became familiar with Galileo's figure by examining the translation of part of his book published in James R. Newman, The World of Mathematics, vol. 2, New York: Simon and Schuster, 1956, p. 751–52. This volume was in Crockett Johnson's library. The figure on p. 752 is annotated.; The painting has a gray background and a metal and wooden frame. It shows two superimposed triangles (inclined planes), one reddish purple, and the other smaller one blue. Both of these triangles are inscribed in the same white circular arc. A light purple circle is shown near the bottom of the purple triangle, and a light blue circle near the bottom of the blue triangle.; The work is # 42 in the series. It is signed: CJ66. Compare to paintings #96 (1979.1093.64) and #71 (1979.1093.46).
Location: Currently not on view

date made: 1966

referenced: Galilei, Galileo
painter: Johnson, Crockett

ID Number: 1979.1093.30
catalog number: 1979.1093.30
accession number: 1979.1093

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. 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. 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 white and light green in the painting) is two thirds of the area of the triangle that 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 is based on diagrams illustrating a discussion of Archimedes’s proof given by H. Dorrie (Figure 54) or J. R. Newman (Figure 9).; This oil painting is #43 in the series, and is signed: CJ69. It has a gray background and a gray frame. It shows a triangle that circumscribes a portion of a parabola. The large triangle is divided into a triangle in shades of light green, which touches a triangle in shades of dark green. The region between the triangles is divided into black and white areas. A second painting in the series, #78 (1979.1093.52) illustrates the same theorem.; 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 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 is annotated.
Location: Currently not on view

date made: 1969

referenced: Archimedes
painter: Johnson, Crockett

ID Number: 1979.1093.31
catalog number: 1979.1093.31
accession number: 1979.1093

Greek mathematicians knew that numbers could not always be represented as simple ratios of whole numbers. They devised ways to describe them geometrically.

Description: Greek mathematicians knew that numbers could not always be represented as simple ratios of whole numbers. They devised ways to describe them geometrically. The title of this painting refers to Theodorus of Cyrene (about 465–398 BC), a Greek geometer who, according to the Greek mathematician Theaetetus (about 417–369 BC), constructed the square roots of the numbers from 3 through 17. Crockett Johnson's painting follows a diagram in Evans G. Valens's The Number of Things that stops with the square root of 16.; The construction of this oil or acrylic painting, #45 in the series, begins with a vertical line segment of length one. Crockett Johnson then drew a right angle at the base of the segment and an adjacent line with length one. From the Pythagorean theorem, it follows that a line from the center of the spiral has length equal to the square root of 2. The construction was continued until the last hypotenuse displayed length equal to the square root of 16.; The painting, which looks like a seashell, shows a specific color pattern. The three dark gray triangles have hypotenuses whose lengths are whole numbers (the square roots of 4, 9, and 16). The six white triangles have hypotenuses whose lengths are irrational and are square roots of even integers. Finally, the six tan triangles have hypotenuses whose lengths are irrational and the square roots of odd integers.; The painting dates from 1967 and is signed: CJ67. It is marked on the back: Crockett Johnson (/) SQUARE ROOTS TO SIXTEEN (/) (THEODORUS OF CYRENE).
Location: Currently not on view

date made: 1967

referenced: Theodorus of Cyrene
painter: Johnson, Crockett

ID Number: 1979.1093.32
catalog number: 1979.1093.32
accession number: 1979.1093

The ancient Greek mathematician 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.

Description: The ancient Greek mathematician 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. The ratio came to be called the golden ratio. If the sides of a rectangle are in the golden ratio, it is called a golden rectangle. Several Crockett Johnson paintings explore the golden ratio and related geometric figures. This paintings suggest how a golden rectangle can be constructed, given the length of its shorter side. On the right in the painting is the golden rectangle that results. Lines in a triangle on the left indicate how the rectangle could have been constructed. Also included are the outlines of a hexagon and a five-pointed star constructed once the ratio had been found.; This painting 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.; The painting on masonite is #46 in the series. It has a black and purple background and a black wooden frame. It is unsigned. The inscription on the back reads: GOLDEN RECTANGLE (/) (PYTHAGORAS) (/) Crockett Johnson 1968. Compare #103 (1979.1093.70) and #64 (1979.1093.39).
Location: Currently not on view

date made: 1968

referenced: Pythagoras
painter: Johnson, Crockett

ID Number: 1979.1093.33
catalog number: 1979.1093.33
accession number: 1979.1093

Crockett Johnson used a wide range of geometrical constructions as the basis for his paintings.

Description: Crockett Johnson used a wide range of geometrical constructions as the basis for his paintings. This painting is based on a method of constructing a rectangle equal in area to a given rectangle, given one side of the rectangle to be constructed.; In the painting, suppose that the cream-colored rectangle on the bottom left is given, as well as a line segment extending from the upper right corner of it. Construct the small triangle on the upper left. Draw the three horizontal lines shown, as well as the diagonal of the rectangle constructed. Extend this diagonal until it meets the bottom line, creating another triangle. The length of the base of this triangle will be the side of the rectangle desired. This rectangle is on the upper right in the painting.; This construction has been associated with the ancient Pythagoreans. Crockett Johnson may well have learned it from Evans G. Valens, The Number of Things. The drawing on page 121 of this book is annotated, although the annotations are faint.; The oil painting is #48 in the series. It has a black background and a black wooden frame, with the two equal triangles in light shades. The painting is signed on the front: CJ69. It is signed on the back: RECTANGLES OF EQUAL AREA (/) (PYTHAGORAS) (/) Crockett Johnson 1969.
Location: Currently not on view

date made: 1969

referenced: Pythagoras
painter: Johnson, Crockett

ID Number: 1979.1093.34
catalog number: 1979.1093.34
accession number: 1979.1093

This oil painting on pressed wood, #52 in the series, shows an original construction of Crockett Johnson. He executed this work in 1968, three years after he began creating mathematical paintings.

Description: This oil painting on pressed wood, #52 in the series, shows an original construction of Crockett Johnson. He executed this work in 1968, three years after he began creating mathematical paintings. It is evident that the artist was very proud of this construction because he drew four paintings dealing with the problem of squaring the circle. The construction was part of Crockett Johnson's first original mathematical work, published in The Mathematical Gazette in early 1970. A diagram relating to the painting was published there.; To "square a circle," mathematically speaking, is to construct a square whose area is equal to that of a given circle using only a straightedge (an unmarked ruler) and a compass. It is an ancient problem dating from the time of Euclid and is one of three problems that eluded Greek geometers and continued to elude mathematicians for 2,000 years. In 1880, the German mathematician Ferdinand von Lindermann showed that squaring a circle in this way is impossible - pi is a transcendental number. Because this proof is complicated and difficult to understand, the problem of squaring a circle continues to attract amateur mathematicians like Crockett Johnson. Although he ultimately understood that the circle cannot be squared with a straightedge and compass, he managed to construct an approximate squaring.; Crockett Johnson began his construction with a circle of radius one. In this circle he inscribed a square. Therefore, in the figure, AO=OB=1 and OC=BC=√(2) / 2. AC=AO+OC=1 + √(2) / 2 and AB=√(AC² + BC²) which equals the square root of the quantity (2+√(2)). Crockett Johnson let N be the midpoint of OT and constructed KN parallel to AC. K is thus the midpoint of AB, and KN=AO - (AC)/2=1/2 - √(2) / 4. Next, he let P be the midpoint of OG, and he drew KP, which intersects AO at X. Crockett Johnson then computed NP=NO+OP=(√(2))/4+(1/2). Triangle POX is similar to triangle PNK, so XO/OP=KN/NP. From this equality it follows that XO=(3-2√(2))/2.; Also, AX=AO-XO=(2√(2)-1)/2 and XC=XO+OC=(3-√(2))/2. Crockett Johnson continued his approximation by constructing XY parallel to AB. It is evident that triangle XYC is similar to triangle ABC, and so XY/XC=AB/AC. This implies that XY=[√((2+√(2)) × (8-5√(2))]/2. Finally he constructed XZ=XY and computed AZ=AX+XZ=[2√(2)-1+(√(2+√(2)) × (8-5√(2))]/2 which approximately equals 1.7724386. Crockett Johnson knew that the square root of pi approximately equals 1.772454, and thus AZ is approximately equal to √(Π) - 0.000019. Knowing this value, he constructed a square with each side equal to AZ. The area of this square is (AZ)² = 3.1415258. This differs from the area of the circle by less than 0.0001. Thus, Crockett Johnson approximately squared the circle.; The painting is signed: CJ68. It is marked on the back: SQUARED CIRCLE* (/) Crockett Johnson 1968 (/) FLAT OIL ON PRESSED WOOD) (/) MATHEMATICALLY (/) DEMONSTRATED (/) TO √π + 0.000000001. It has a white wooden frame. Compare to painting #91 (1979.1093.60).; References: Crockett Johnson, “On the Mathematics of Geometry in My Abstract Paintings,” Leonardo 5 (1972): p. 98.; C. Johnson, “A Geometrical look at √π," Mathematical Gazette, 54 (1970): p. 59–60. the figure is from p. 59.
Location: Currently not on view

date made: 1968

painter: Johnson, Crockett

ID Number: 1979.1093.35
catalog number: 1979.1093.35
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. . . ."(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

This painting demonstrates a construction for finding the geometric mean of two line segments credited to the Greek mathematician Archytas (flourished 400–350 BC), an admirer of Pythagoras. Place the line segments end to end, and draw a circle with this length as diameter.

Description: This painting demonstrates a construction for finding the geometric mean of two line segments credited to the Greek mathematician Archytas (flourished 400–350 BC), an admirer of Pythagoras. Place the line segments end to end, and draw a circle with this length as diameter. Erect a perpendicular at the point where the line segments meet (d in the figure), and consider this to be the altitude of a right triangle inscribed in the semicircle. By similar triangles, the length of the perpendicular of a triangle inscribed in a semicircle is the geometric mean of the two lengths into which it divides the diameter of the circle. Hence the length of d is the mean of the segments e and f.; This painting an orange-red background, and shows a triangle inscribed in an orange semicircle. The perpendicular from the right angle of the triangle divides the triangle into triangles similar to it, painted in black and white.; The painting, and the attribution of the theorem to Archytas, are based on a passage from Evans G. Valens, The Number of Things: Pythagoras, Geometry and Humming Strings (1964), p. 118. The figure on this page of this book from Crockett Johnson's library is annotated.; This oil painting on masonite is #65 in the series. It is inscribed on the back: GEOMETRIC MEAN (ARCHYTAS) (/) Crockett Johnson 1968. It has a wooden frame.
Location: Currently not on view

date made: 1968

referenced: Archytas
painter: Johnson, Crockett

ID Number: 1979.1093.40
accession number: 1979.1093
catalog number: 1979.1093.40

Crockett Johnson much enjoyed constructing square roots of numbers geometrically. He offered the following account of this painting, as well as the figure shown: "Let AN and BN be 1.

Description: Crockett Johnson much enjoyed constructing square roots of numbers geometrically. He offered the following account of this painting, as well as the figure shown: "Let AN and BN be 1. Then the diagonal AB is the square root of 2, because it is the hypotenuse of a right triangle with sides of length √1 and √1. The large right triangle √1 plus √2 adds up to a hypotenuse of √3. The compass traces pronounce a statement and also declare its proof. The square root of 2 is 1.4142 . . . and the square root of 3 is 1.7321 . . . Their decimals run on and on but as produced by the compass and blind straightedge both numbers are quite as finite as 1. The triangle embodies three dimensions of the cube. CB is any edge, AB is a face diagonal, and AC is an internal diagonal." Crockett-Johnson described the source of the painting as "Artist's Construction, or Anybody's."; The triangle with three sides equal to the lengths of interest is painted white. Remaining segments of the construction are in dark gray and purple, with a black background. The painting has a brown wooden frame.; The painting is #66 in the series and is signed: CJ69. For a related painting, see #45 (1979.1093.32).; Reference: "Geometric Geometric [sic] Paintings by Crockett Johnson" NMAH Collections.
Location: Currently not on view

date made: 1969

painter: Johnson, Crockett

ID Number: 1979.1093.41
accession number: 1979.1093
catalog number: 1979.1093.41

Classical Greek mathematicians were able to square all convex polygons. That is, given any polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straight edge. Figures with curved sides proved more difficult.

Description: Classical Greek mathematicians were able to square all convex polygons. That is, given any polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straight edge. Figures with curved sides proved more difficult. However, as this painting suggests, the mathematician Hippocrates of Chios (5th century BC) squared a lune, a figure bounded by arcs of two circle with different radii (lunes resemble quarter moons, hence the name). Finding the area of a lune in terms of a square might seem more difficult than squaring a circle, but the latter problem would prove intractable.; The painting follows annotated figures in Evans G. Valens's The Number of Things (1964), p.103, which was part of Crockett Johnson's mathematical library. It corresponds to an early diagram in Valens's discussion of squaring the circle. According to Valens, Hippocrates began by arguing that the areas of similar segments of different circles are in the same ratio as the squares of their bases. Suppose an isosceles right triangle is inscribed in a semicircle of diameter c. Construct smaller semicircles of diameter a and b on the sides of the inscribed triangle. As the square of a plus the square of b equals the square of c, the area of the two smaller semicircles equals that of the large one. The proof goes on to consider the area of the two crescents and the triangle.; Although Valens called the crescent moon shape a crescent, Crockett Johnson used the term lune. This probably indicates that he also read Herbert Westren Turnball “The Great Mathematicians” in The World of Mathematics, edited by James R. Newman (1956), where the term lune is used. Also, on page page 91 of Turnball’s article there is a diagram on which the painting could have been based.; In this version of Squared Lunes Crockett Johnson uses brown, black, red, and white against a gray background. This oil painting is #67 in the series, and the first in the series with the title "Squared Lunes." It was completed in 1968 and is signed: CJ68. It is inscribed on the back: SQUARED LUNES (/) (HIIPPOCRATES OF CHIOS) (/) Crockett Johnson 1968. A related painting is #68 (1979.1093.43).
Location: Currently not on view

date made: 1968

referenced: Hippocrates of Chios
painter: Johnson, Crockett

ID Number: 1979.1093.42
accession number: 1979.1093
catalog number: 1979.1093.42

The title of this painting refers to Hippocrates of Chios (5th century BC), one of the greatest geometers of antiquity. Classical Greek mathematicians were able to square convex polygons.

Description: The title of this painting refers to Hippocrates of Chios (5th century BC), one of the greatest geometers of antiquity. Classical Greek mathematicians were able to square convex polygons. That is, given a polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straightedge. They were unable to square a circle. This painting is based on the earliest known squaring of a figure bounded by curves rather than straight lines. The mathematician Hippocrates squared a lune, a figure bounded by arcs of two circles with different radii. This achievement might seem more difficult than squaring a circle.; Crockett Johnson's painting follows two annotated figures in Evans G. Valens's The Number of Things (1964), pp. 103–104, a book in the artist’s mathematical library. The finished piece shows isosceles triangles T, and a second congruent triangle connected to it base to base to form a square. Also present in the painting are three lunes, two small and one large. The area of triangle T is equal to the sum of the areas of lunes A and B (see figures). The area of triangle T is also equal to the area of a lune composed of X, Y, and the area T-C. Furthermore, because triangle T is congruent to the triangle below it, triangle T is equal to the area of this lune. Thus, the area of the square is equal to the sum of the areas of the three lunes. In summary, Johnson pictorially represented a "squared" curvilinear region; that is, he successfully constructed a square with the same area as that of the region of three lunes bounded by curves.; Although Valens called the crescent moon shape a crescent, Crockett Johnson used the term lune. This probably indicates that he also read Herbert Westren Turnball “The Great Mathematicians” in The World of Mathematics, edited by James R. Newman (1956), where the term lune is used. Also, on page page 91 of Turnball’s article there is a diagram on which the painting could have been based.; Crockett Johnson executed this painting in 4 tints and darker shades of purple upon a black background. The center triangle is the darkest shade of purple. As one moves outward, the colors grow lighter. This allows a dramatic distinction to be seen between the figure and the background, and thus puts a greater emphasis on the lunes.; This oil painting on masonite is #68 in Crockett Johnson's series. Its date of completion is unknown and the work is unsigned. It is closely related to painting #67 (1979.1093.42).
Location: Currently not on view

date made: ca 1965; ca 1966

referenced: Hippocrates of Chios
painter: Johnson, Crockett

ID Number: 1979.1093.43
accession number: 1979.1093
catalog number: 1979.1093.43

From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle. Each of these curves may be considered as a projection of the circle.

Description: From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle. Each of these curves may be considered as a projection of the circle. Nineteenth-century mathematicians were much interested in the properties of conics that were preserved under projection. They knew from the work of the ancient mathematician Pappus that the cross ratio of line segments created by two straight lines cutting the same pencil of lines was a constant (a pencil of lines is a set of lines emanating from a common point). In the drawing, which is Figure 5 from an article by Morris Kline in James R. Newman's The World of Mathematics (1956), if line segment l’ crosses lines emanating from the point O at points A’, B’, C’, and D’; and line segment l croses the same lines at points A, B, C, and D, the cross ratio:; (A’C’/C’B’) / (A’D’/D’B’) = (AC/BC) / (AD/DB), in other words it is independent of the cutting line. (see Crockett Johnson's painting Pencil of Ratios (Monge)).; The French mathematician Michel Chasles introduced a related result, which is the subject of this painting. He considered two points on a conic section (such as an ellipse) that were both linked to the same four other points on the conic. He found that lines crossing both pencils of rays had the same cross ratio. Moreover, a conic section could be characterized by its cross ratio. This opened up an entirely different way of describing conic sections. Crockett Johnson associated this painting with both Chasles and another French advocate of projective geometry, Victor Poncelet.; The painting, in oil or acrylic on masonite, is #69 in the series. It has a dark gray or blue background and a black wooden frame. It shows a white ellipse, two points on the ellipse (on the left side of the painting), and two pencils of lines that produce the same cross ratio. The painting is not signed. It is inscribed on the back, in Crockett Johnson’s hand: CROSS RATIO IN AN ELLIPSE (PONCELET) (/) Crockett Johnson 1968. Compare #21 (1979.1093.15).; Reference: Morris Kline in James R. Newman, The World of Mathematics (1956), p. 634. This volume was in Crockett Johnson's library. The figure on this page is annotated.
Location: Currently not on view

date made: 1968

referenced: Poncelet, Jean-Victor
painter: Johnson, Crockett

ID Number: 1979.1093.44
accession number: 1979.1093
catalog number: 1979.1093.44

Those making mathematical instruments for surveying, navigation, or the classroom have long been interested in creating equal divisions of the circle. Ancient geometers knew how to divide a circle into 2, 3, or 5 parts, and as well as into multiples of these numbers.

Description: Those making mathematical instruments for surveying, navigation, or the classroom have long been interested in creating equal divisions of the circle. Ancient geometers knew how to divide a circle into 2, 3, or 5 parts, and as well as into multiples of these numbers. For them to draw polygons with other numbers of sides required more than a straightedge and compass.; In 1796, as an undergraduate at the University of Göttingen, Friedrich Gauss proposed a theorem severely limiting the number of regular polygons that could be constructed using ruler and compass alone. He also found a way of constructing the 17-gon.; Crockett Johnson, who himself would develop a great interest in constructing regular polygons, drew this painting to illustrate Gauss's discovery. His painting follows a somewhat later solution to the problem presented by Karl von Staudt in 1842, modified by Heinrich Schroeter in 1872, and then published by the eminent mathematician Felix Klein. Klein's detailed account was in Crockett Johnson's library, and a figure from it is heavily annotated.; This oil painting on masonite is #70 in the series. It is signed: CJ69. The back is marked: SEVENTEEN SIDES (GAUSS) (/) Crockett Johnson 1969. The painting has a black background and a wood and metal frame. There are two adjacent purple triangles in the center, with a white circle inscribed in them. The triangles have various dark gray regions, and the circle has various light gray regions and one dark gray segment. The length of the top edge of this segment is the chord of the circle corresponding to length of the side of an inscribed 17-sided regular polygon.; Reference: Felix Klein, Famous Problems of Elementary Geometry (1956), pp. 16–41, esp. 41.
Location: Currently not on view

date made: 1969

referenced: Gauss, Carl Friedrich
painter: Johnson, Crockett

ID Number: 1979.1093.45
accession number: 1979.1093
catalog number: 1979.1093.45

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

In the late 1960s and early 1970s, the American cartoonist Crockett Johnson created a series of paintings on mathematical subjects.

Description: In the late 1960s and early 1970s, the American cartoonist Crockett Johnson created a series of paintings on mathematical subjects. This oil painting, #74 in the series, dates from 1969 and is signed "CJ69." It is based on a theorem in plane geometry proved by the English-born mathematician Frank Morley (1860–1937). Morley emigrated to the United States and taught at Haverford College and Johns Hopkins University.; The painting illustrates his best-known result. It shows lines that divide the three angles of the large triangle into three equal parts. Lines coming from different vertices of the triangle meet in points. The triangle formed by joining the intersections of the trisectors, which lie nearest to the three sides of the triangle, is shown in white in the painting. According to Morley's theorem, this is an equilateral triangle.

Date made: 1969

painter: Johnson, Crockett

ID Number: 1979.1093.48
catalog number: 1979.1093.48
accession number: 1979.1093

Mathematical Paintings of Crockett Johnson

Painting - Logarithms

Painting - Polar Line of a Point and a Circle (Apollonius)

Painting - Polyhedron Formula (Euler)

Painting - Multiplication through Imaginary Numbers (Gauss)

Painting - Locus of Point on Chord (Plato)

Painting - Velocity on Inclined Planes (Galileo)

Painting - Parabolic Triangles (Archimedes)

Painting - Square Roots to Sixteen (Theodorus of Cyrene)

Painting - Golden Rectangle (Pythagoras)

Painting - Rectangles of Equal Area (Pythagoras)

Painting - Squared Circle

Painting - Problem of Delos Constructed from a Solution by Isaac Newton (Arithmetica Universalis)

Painting - Division of the Square by Conic Rectangles

Painting - Aligned Triangles (Desargues)

Painting - Golden Rectangle

Painting - Geometric Mean (Archytas)

Painting - Square Roots of One, Two and Three

Painting - Squared Lunes (Hippocrates Of Chios)

Painting - Squared Lunes (Hippocrates Of Chios)

Painting - Cross Ratio in an Ellipse (Poncelet)

Painting -Seventeen Sides (Gauss)

Painting - Law of Motion (Galileo)

Painting - Rotated Triangle and Reflections

Painting - Morley Triangle