Where Art Meets Math
In the late 1960s and early 1970s, the American cartoonist Crockett Johnson created a series of paintings on mathematical subjects. They’re based on theorems, laws, and mathematical figures, but viewers don’t need a background in math to appreciate the completed pieces. Other mathematical works and figures from across the Smithsonian are in the mix.
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Painting  Law of Orbiting Velocities
 Description
 This creation, similar to works #22 (1979.1093.16) and #76 (1979.1093.50), is a further example of Crockett Johnson's work relating to Kepler's first two laws of planetary motion. The ellipse represents the path of a planet and the white sections represent equal areas swept out in equal times. This work is a silk screen on paper. It is number 99 in the series, and is signed in the right corner: Crockett Johnson (/) 67. It draws on a figure from The World of Mathematics by James R. Newman.
 Location
 Currently not on view
 date made
 1967
 referenced
 Kepler, Johannes
 painter
 Johnson, Crockett
 ID Number
 1979.1093.66
 catalog number
 1979.1093.66
 accession number
 1979.1093
 Data Source
 National Museum of American History

Painting  Fluxions (Newton)
 Description
 In the 17th century, the natural philosophers Isaac Newton and Gottfried Liebniz developed much of the general theory of the relationship between variable mathematical quantities and their rates of change (differential calculus), as well as the connection between rates of change and variable quantities (integral calculus).
 Newton called these rates of change "fluxions." This painting is based on a diagram from an article by H. W. Turnbull in Newman's The World of Mathematics. Here Turnbull described the change in the variable quantity y (OM) in terms of another variable quantity, x (ON). The resulting curve is represented by APT.
 Crockett Johnson's painting is based loosely on these mathematical ideas. He inverted the figure from Turnbull. In his words: "The painting is an inversion of the usual textbook depiction of the method, which is one of bringing together a fixed part and a ‘moving’ part of a problem on a cartesian chart, upon which a curve then can be plotted toward ultimate solution."
 The arc at the center of this painting is a circular, with a tangent line below it. The region between the arc and the tangent is painted white. Part of the tangent line is the hypotenuse of a right triangle which lies below it and is painted black. The rest of the lower part of the painting is dark purple. Above the arc is a dark purple area, above this a gray region. The painting has a wood and metal frame.
 This oil painting on pressed wood is #20 in the series. It is unsigned, but inscribed on the back: Crockett Johnson 1966 (/) FLUXIONS (NEWTON).
 References: James R. Newman, The World of Mathematics (1956), p. 143. This volume was in the library of Crockett Johnson. The figure on this page is annotated.
 Crockett Johnson, "On the Mathematics of Geometry in My Abstract Paintings," Leonardo, 5 (1972): pp. 97–8.
 Location
 Currently not on view
 date made
 1966
 referenced
 Newton, Isaac
 painter
 Johnson, Crockett
 ID Number
 1979.1093.14
 catalog number
 1979.1093.14
 accession number
 1979.1093
 Data Source
 National Museum of American History

Painting  Mystic Hexagon (Pascal)
 Description
 This painting is based on a theorem generalized by the French mathematician Blaise Pascal in 1640, when he was sixteen years old. When the opposite sides of a irregular hexagon inscribed in a circle are extended, they meet in three points. Pappus, writing in the 4th century AD, had shown in his Mathematical Collections that these three points lie on the same line. In the painting, the circle and creamcolored hexagon are at the center, with the sectors associated with different pairs of lines shown in green, blue and gray. The three points of intersection are along the top; the line that would join them is not shown. Pascal generalized the theorem to include hexagons inscribed in any conic section, not just a circle. Hence the figure came to be known as "Pascal’s hexagon" or, to use Pascal’s terminology, the "mystic hexagon." Pascal’s work in this area is known primarily from notes on his manuscripts taken by the German mathematician Gottfried Leibniz after his death.
 There is a discussion of Pascal’s hexagon in an article by Morris Kline on projective geometry published in James R. Newman's World of Mathematics (1956). A figure shown on page 629 of this work may have been the basis of Crockett Johnson's painting, although it is not annotated in his copy of the book.
 The oil or acrylic painting on masonite is signed on the bottom right: CJ65. It is marked on the back: Crockett Johnson (/) "Mystic" Hexagon (/) (Pascal). It is #10 in the series.
 References: Carl Boyer and Uta Merzbach, A History of Mathematics (1991), pp. 359–62.
 Florian Cajori, A History of Elementary Mathematics (1897), 255–56.
 Morris Bishop, Pascal: The Life of a Genius (1964), pp. 11, 81–7.
 Location
 Currently not on view
 date made
 1965
 referenced
 Pascal, Blaise
 painter
 Johnson, Crockett
 ID Number
 1979.1093.05
 catalog number
 1979.1093.05
 accession number
 1979.1093
 Data Source
 National Museum of American History

Painting Construction of Heptagon
 Description
 This painting represents one of Crockett Johnson's early constructions of a heptagon. It shows a large purple circle, a pink triangle superimposed, and two smaller circles. Crockett Johnson's diagram for the painting is shown. Two equal circles are constructed, with the center of the first on the second and conversely (circles with centers C and D in the diagram), and a line segment drawn that includes their points of intersection. Then, in Crockett Johnson's words, "Against a straight edge controlling their alignment the sought points B, U, and E, are determined by the adjustment of compass arcs BC from U and EC from B. Angles FBC, CBD, DBE, and BAF are π/ 7." Detailed examination of the triangles in the drawing shows that this is indeed the case.
 The colors of the painting highlight the circles, lines, and arcs central to the construction, and the largest of the resulting isosceles triangles with vertex angle π/7 is shown in bold shades of pink. The short line called CF in the drawing (as well as line segments CD and DE, which are not shown), is the length of the side of a heptagon inscribed in a circle centered at B with radius BF.
 The oil on masonite work is #116 in the series. It has a gray background and a wood and metal frame. It is inscribed on the back: CONSTRUCTION OF HEPTAGON (/) . . .(8) (/) Crockett Johnson 1973.
 Location
 Currently not on view
 date made
 1973
 painter
 Johnson, Crockett
 ID Number
 1979.1093.78
 accession number
 1979.1093
 catalog number
 1979.1093.78
 Data Source
 National Museum of American History

Painting  Every Positive Integer (Gauss)
 Description
 This painting is loosely based on a theorem proven by the German mathematician Carl Friedrich Gauss (1777–1855) in 1776 when he was just nineteen years old. The proposition, one of Gauss’s many contributions to the branch of mathematics called number theory, states that every positive integer is the sum of three triangular numbers. The concept of triangular numbers dates to antiquity. Suppose one arranges dots in rows, with one in the first row, two in the second, three in the first and so forth. Three dots form a triangle, as do 6 dots, 10 dots, and 15 dots. The numbers 3, 6, 10, 15, and so forth are called "triangular numbers." The integers 0 and 1 are thought of as special cases of triangular numbers.
 Crockett Johnson derived his painting from an entry in Gauss's diary published in an article by Eric Temple Bell included by James R. Newman in his book The World of Mathematics (1956), p. 304. The entry includes the phrase EUREKA in Greek, and indicates that any positive integer is the sum of three triangular numbers.
 Crockett Johnson’s painting abstractly represents this theorem through the juxtaposition of three triangles. The triangles are equal, but each figure is painted a different color. It is possible that the artist chose to illustrate each triangle in its own color to demonstrate that each triangle generally represents its own triangular number when computing a positive integer. However, the triangles are congruent, which reminds the viewer that the triangles are related because they all represent a triangular number.
 This work was painted in oil on masonite, completed in 1966, and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) EVERY POSITIVE INTEGER (/) (GAUSS). It is painting #29 in the series, and has a wooden frame.
 Reference: J. R. Newman, The World of Mathematics, 1956, p. 304.
 Location
 Currently not on view
 date made
 1966
 referenced
 Gauss, Carl Friedrich
 painter
 Johnson, Crockett
 ID Number
 1979.1093.21
 catalog number
 1979.1093.21
 accession number
 1979.1093
 Data Source
 National Museum of American History

Painting  Logarithms
 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
 Data Source
 National Museum of American History

Painting  Parabolic Triangles (Archimedes)
 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
 Data Source
 National Museum of American History

Painting  Morley Triangle
 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 Englishborn mathematician Frank Morley (1860–1937). Morley emigrated to the United States and taught at Haverford College and Johns Hopkins University.
 The painting illustrates his bestknown 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.
 Location
 Currently not on view
 Date made
 1969
 painter
 Johnson, Crockett
 ID Number
 1979.1093.48
 catalog number
 1979.1093.48
 accession number
 1979.1093
 Data Source
 National Museum of American History

Sun Mathematics
 Date
 1974
 Artist
 Agnes Denes, American, b. Budapest, Hungary, 1938
 Accession Number
 04.21
 Data Source
 Hirshhorn Museum and Sculpture Garden

Homage to the SquareWhite LineBlack
 Artist
 Josef Albers, born Bottrop, Germany 1888died New Haven, CT 1976
 Object number
 1976.108.13
 Data Source
 Smithsonian American Art Museum

Painting Law of Orbiting Velocity (Kepler)
 Description
 This piece is a further example of Crockett Johnson's exploration of Kepler’s first two laws of planetary motion. The ellipse represents the path of a planet, and the white sections represent equal areas swept out in equal times. This work, a silk screen inked on paper board, is signed: CJ66. It is #76 in the series, and it echoes painting #22 (1979.1093.16) and painting #99 (1979.1093.66).
 Location
 Currently not on view
 date made
 1966
 referenced
 Kepler, Johannes
 painter
 Johnson, Crockett
 ID Number
 1979.1093.50
 catalog number
 1979.1093.50
 accession number
 1979.1093
 Data Source
 National Museum of American History

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

Key Blue (from series, the Mathematical Basis of the Arts)
 Date
 ca. 1934
 Artist
 Joseph Schillinger, born Kharkov, Russia 1895died New York City 1943
 Object number
 1966.39.3
 Data Source
 Smithsonian American Art Museum

Painting  Pencil of Ratios (Monge)
 Description
 The history of projective geometry begins with the work of the French mathematician Gerard Desargues (1591–1661). During his lifetime his work was well known in some mathematical circles, but after his death, his contributions to the field were largely forgotten. When Gaspard Monge (1746–1818) and his student, JeanVictor Poncelet (1788–1867) began their studies of projective geometry, they were largely unaware of the work of Desargues. This may be why Crockett Johnson included Monge's name as opposed to Desargues' in this painting's title.
 One of the fundamental concepts of projective geometry, which was touched upon, but not fully understood, by the Greeks, is that of a crossratio, or "ratio of ratios." It is the topic of Johnson's painting. If points A, B, C, and D on line l are projected from point O, and if the line l’ crosses the four projected line segments, then the ratio of ratios (A’B’/C’B’)/(A’D’/ D’B’) of the corresponding points A’,B’,C’, and D’ is the same as the ratio of ratios (AC/CB)/(AD/DB). Thus, a crossratio is a projective invariant for all line segments l’.
 The artist may have received inspiration for this painting from his copy of James R. Newman's The World of Mathematics (1956), p. 632. The figure is found there in an article by Morris Kilne entitled "Projective Geometry." This figure is not annotated, and the painting flips Kline's image.
 Crockett Johnson chose purple, white, black, and brown to color this work. He executed the projection in three tints of purple and one shade of white. The background, which is divided by line l’, was executed in black and brown.
 Pencil of Ratios, an oil painting on masonite, is #18 in the series. It was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) PENCIL OF RATIOS (MONGE). The painting is unframed.
 Location
 Currently not on view
 date made
 1966
 referenced
 Monge, Gaspard
 painter
 Johnson, Crockett
 ID Number
 1979.1093.12
 catalog number
 1979.1093.12
 accession number
 1979.1093
 Data Source
 National Museum of American History

Homage to the Square: Glow
 Provenance
 Unknown Source, Krefeld, Germany (via Galerie Denise Rene, Paris, 5 October 1967)
 Joseph H. Hirshhorn, New York, 5 October 19671972
 Gift of Joseph H. Hirshhorn, 1972
 Exhibition History
 SIDNEY JANIS GALLERY, New York. Paintings by Josef Albers, 531 October 1970.
 ROYAL DUBLIN SOCIETY, Ireland. "ROSC '71: An International Art Exhibition," 24 October29 December 1971, no. 3, ill. p. 23.
 HIRSHHORN MUSEUM AND SCULPTURE GARDEN, Smithsonian Institution, Washington, DC. "Inaugural Exhibition," 4 October 197415 September 1975.
 HIRSHHORN MUSEUM AND SCULPTURE GARDEN, Smithsonian Institution, Washington, DC. "The Golden Door: ArtistsImmigrants of America 18761976," 20 May20 October 1976, no. 125, ill. p. 277.
 WALKER ARTS CENTER, Minneapolis. "Vice President's House," 15 December 197715 March 1978.
 HIRSHHORN MUSEUM AND SCULPTURE GARDEN, Smithsonian Institution, Washington, DC. "Josef Albers," 27 September 197923 March 1980, no cat.
 LINGOTTO, Turin, Italy. "Arte Americana 19301970," 11 January 1992 31 March 1992, p. 116.
 HIRSHHORN MUSEUM AND SCULPTURE GARDEN, Smithsonian Institution, Washington, DC. "Josef Albers: Innovation and Inspiration," 13 February11 April 2010, no cat.
 FUNDACIÓN JUAN MARCH, Madrid. "Josef Albers: Minimal Means, Maximum Effect," 28 March6 July 2014, no. 94, color ill. pg. 150.
 Published References
 LERNER, ABRAM et al. The Hirshhorn Museum and Sculpture Garden (New York: Abrams, 1974) no. 866, p. 569658.
 PIPER, DAVID ed. Encyclopedia of Painting and Sculpture (London: Mitchell Beazley, 1979).
 American Education Commemorative Postage Stamp, unveiled 28 April 1980, issued 12 September 1980.
 AMERICAN ASSOCIATION OF MUSEUMS. Aviso (Monthly Dispatch) no. 9 (September 1980) p. 1.
 HUMAN SERVICES DEVELOPMENT. Human Services Development Catalog (Boston, MA: Human Services Development, 1981) cover ill.
 THE URBAN INSTITUTE, Washington D.C. 1982 Annual Report.
 BRITSCH, RALPH A. and TODD A. BRITSCH. The Arts in Western Culture (Englewood Cliffs, NJ: PrenticeHall, Inc., 1985) fig. 1632, p. 421.
 WESTVACO PEOPLE. Excellence: An American Treasury (Westvaco Corporation, 1988) frontispiece II, p. 251252.
 UNSIGNED. "Where & When," The Washingtonian (December 1990) p. 47.
 BURCHARD, HANK. "Now Everyone's a Critic," Washington Post Weekend (14 December 1990).
 WELZENBACH, MICHAEL. "How to be an Art Critic," Washington Post (13 January 1991).
 WHEELER, DANIEL. Art Since MidCentury: 1945 to the Present (New York: The Vendome Press, 1991) fig. 423, p. 230.
 KILIAN, MICHAEL. "Advance and Retreat," Chicago Tribune (10 January 1991) p. 12, sec. 5.
 HOBBS, JACK and RICHARD SALOME. The Visual Experience (Worcester, MA: Davis Publications Inc., 1995) fig. 16 and fig.216, p. 7, 17, 301.
 RAGANS, ROSALIND. Arttalk 3rd ed. (New York: Glencoe/McGrawHill, 2000).
 Date
 1966
 Artist
 Josef Albers, American, b. Bottrop, Germany, 1888–1976
 Accession Number
 72.3
 Data Source
 Hirshhorn Museum and Sculpture Garden

Painting  Square Root of Two (Descartes)
 Description
 La Géométrie, one of the most important works published by the mathematician and philosopher René Descartes (1596–1650), includes a discussion of methods for performing algebraic operations using a straight edge and compass. One of the first is a way to determine square roots. This construction is the subject of Crockett Johnson's painting. Descartes explained: "If the square root of GH is desired, I add, along the same straight line, FG equal to unity, then bisecting FH at K, I describe the circle FIH about K as a center, and draw from G a perpendicular and extend it to I, and GI is the required root." (this is a translation of portion of La Géométrie, as published by J. R. Newman, The World of Mathematics (1956), p. 241)
 To understand Descartes' description and the title of this painting, consider the diagram. An angle inscribed in a semicircle is a right angle, thus triangle FGI is similar to triangle IGH. Because this two triangles are similar, their corresponding sides are proportional. Thus, G/IFG = GH/GI. But FG is equal to one, so GH is the square of GI, and GI the square root of GH desired.
 In his painting, Crockett Johnson has flipped the image from La Géométrie found in his copy of The World of Mathematics. This figure is not annotated. The artist divided his painting into squares of area one, suggesting what came to be called Cartesian coordinates. The division indicates that the GH chosen has length two.
 Johnson chose white for the section of the semicircle that contains the edge of length equal to the square root of GH. This section provides a vivid contrast against the dull, surrounding colors. Crockett Johnson purposefully creates this area of interest to draw focus to the result of Descartes' construction.
 Square Root of Two is painting #19 in the series. It was painted in oil or acrylic on masonite, completed in 1965, and is signed: CJ65. The wooden frame is painted black.
 Location
 Currently not on view
 date made
 1965
 referenced
 Descartes, Rene
 painter
 Johnson, Crockett
 ID Number
 1979.1093.13
 catalog number
 1979.1093.13
 accession number
 1979.1093
 Data Source
 National Museum of American History

Painting  Cross Ratio in an Ellipse (Poncelet)
 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. Nineteenthcentury 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, JeanVictor
 painter
 Johnson, Crockett
 ID Number
 1979.1093.44
 accession number
 1979.1093
 catalog number
 1979.1093.44
 Data Source
 National Museum of American History

Painting  Archimedes Transversal
 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 sevensided 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
 Data Source
 National Museum of American History

Painting  Doubled Cube (Newton)
 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
 Data Source
 National Museum of American History

Painting  Aligned Triangles (Desargues)
 Description
 In the 17th century, the French engineer and architect Girard Desargues (1591–1661) explored interconnections between extensions of the lines within a pencil of three line segments (a pencil of line segments consists of several line segments originating at a common point). His theorems, as published in his own extremely obscure work and also by his contemporary, Abraham Bosse, were extended in the 19th century, and proved of fundamental importance to projective geometry.
 Crockett Johnson's library contains discussions of Desargues' theorem by H. S. M. Coxeter, N. A. Court, Heinrich Dorrie, and William M. Ivins. This painting most resembles a figure from Coxeter, although the diagram is not annotated. Suppose that the vertices of two triangles (PQR and P'Q'R' in Figure 1.5B from Coxeter) lie on a pencil of three line segments emanating from the point O. Suppose that similarly situated sides of the two triangles can be extended to meet in the three points denoted by A, C and B in the figure. According to Desargues' theorem, A, C, and B are collinear.
 In the painting, the two concurrent triangles are shown in shades of gray and black, while the top of the pencil of three lines is in shades of gold. Extensions of the sides and their points of intersection are clearly shown. Both the figure and the background of the painting are divided by the line joining the points of intersection
 The painting is #63 in the series. It is painted in oil or acrylic on masonite, and has a brown wooden frame. The painting is signed: CJ70.
 References:
 Newman, J. R., The World of Mathematics, p. 133. Figure annotated.
 Court, N. A., College Geometry (1952), pp. 163–5. The figure is not annotated.
 Coxeter, H. S. M., The Real Projective Plane, (1955 edition), p. 7. The figure resembles the painting but is not annotated.
 Dorrie, Heinrich, 100 Great Problems of Elementary Mathematics: Their History and Solution (1965), p. 267. There is an annotated figure here for another theorem of Desargues, the theorem of involution.
 Field, J. V., The Invention of Infinity: Mathematics and Art in the Renaissance (1997), pp. 190–206.
 Ivins, William M. Jr., Art & Geometry: A Study in Space Intuitions (1946), pp. 87–94.
 Location
 Currently not on view
 date made
 1970
 referenced
 Desargues, Girard
 painter
 Johnson, Crockett
 ID Number
 1979.1093.38
 accession number
 1979.1093
 catalog number
 1979.1093.38
 Data Source
 National Museum of American History
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 Art 81
 Science & Mathematics 81
 Crockett Johnson 80
 Medicine and Science: Mathematics 80
 Smithsonian American Art Museum Collection 23
 Graphic Arts 19
 Hirshhorn Museum and Sculpture Garden Collection 12
 Geometric Abstraction 10
 Cooper Hewitt, Smithsonian Design Museum Collection 4
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