#
Art

The National Museum of American History is not an art museum. But works of art fill its collections and testify to the vital place of art in everyday American life. The ceramics collections hold hundreds of examples of American and European art glass and pottery. Fashion sketches, illustrations, and prints are part of the costume collections. Donations from ethnic and cultural communities include many homemade religious ornaments, paintings, and figures. The Harry T Peters "America on Stone" collection alone comprises some 1,700 color prints of scenes from the 1800s. The National Quilt Collection is art on fabric. And the tools of artists and artisans are part of the Museum's collections, too, in the form of printing plates, woodblock tools, photographic equipment, and potters' stamps, kilns, and wheels.

"Art - Overview" showing 3328 items.

Page 2 of 333

## Painting -

*Seventeen Sides (Gauss)*- 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

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

## 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, Kenneth E. Behring Center

## Painting -

*Equal Areas, Their Triangular Square Root and 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

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

## Painting -

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

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

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

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

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.68

- catalog number
- 1979.1093.68

- accession number
- 1979.1093

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

## Painting -

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

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

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

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

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.70

- catalog number
- 1979.1093.70

- accession number
- 1979.1093

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

## Painting -

*Heptagon from Its Seven Sides*- Description
- Toward the end of his life, Crockett Johnson took up the problem of constructing a regular seven-sided polygon or heptagon. This construction, as Gauss had demonstrated, requires more than a straight edge and compass. Crockett Johnson used compass and a straight edge with a unit length marked on it. Archimedes and Newton had suggested that constructions of this sort could be used to trisect the angle and to find a cube with twice the volume of a given cube, and Crockett Johnson followed their example.

- One may construct a heptagon given an angle of pi divided by seven. If an isosceles triangle with this vertex angle is inscribed in a circle, the base of the triangle will have the length of one side of a regular heptagon inscribed in that circle. According to Crockett Johnson's later account, in the fall of 1973, while having lunch in the city of Syracuse on Sicily during a tour of the Mediterranean, he toyed with seven toothpicks, arranging them in various patterns. Eventually he created an angle with his menu and wine list and arranged the seven toothpicks within the angle in crisscross patterns until his arrangement appeared as is shown in the painting.

- Crockett Johnson realized that the vertex angle of the large isosceles triangle shown is exactly π/7 radians, as desired. The argument suggested by his diagram is more complex than what he later published. The numerical results shown in the figure suggest his willingness to carry out detailed calculations.

*Heptagon from its Seven Sides*, painted in 1973 and #107 in the series, shows a triangle with purple and white sections on a navy blue background. This oil or acrylic painting on masonite is signed on its back : HEPTAGON FROM (/) ITS SEVEN SIDES (/) (Color sketch for larger painting) (/) Crockett Johnson 1973. No larger painting on this pattern is at the Smithsonian.

- Reference: Crockett Johnson, "A Construction for a Regular Heptagon," Mathematical Gazette, 1975, vol. 59, pp. 17–21.

- Location
- Currently not on view

- date made
- 1973

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.74

- catalog number
- 1979.1093.74

- accession number
- 1979.1093

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

## Painting -

*Hippias' Curve*- Description
- This painting is a construction of Crockett Johnson, relating to a curve attributed to the ancient Greek mathematician Hippias. This was one of the first curves, other than the straight line and the circle, to be studied by mathematicians. None of Hippias's original writings survive, and the curve is relatively little known today. Crockett Johnson may well have followed the description of the curve given by Petr Beckmann in his book
*The History of Pi*(1970). Crockett Johnson's copy of Beckmann’s book has some light pencil marks on his illustration of the theorem on page 39 (see figure).

- Hippias envisioned a curve generated by two motions. In Crockett Johnson's own drawing, a line segment equal to OB is supposed to move uniformly leftward across the page, generating a series of equally spaced vertical line segments. OB also rotates uniformly about the point O, forming the circular arc BQA. The points of intersection of the vertical lines and the arc are points on Hippias's curve. Assuming that the radius OK has a length equal to the square root of pi, the square AOB (the surface of the painting) has area equal to pi. Moreover, the height of triangle ASO, OS, is √(4 / pi), so that the area of triangle ASO is 1.

- The painting has a gray border and a wood and metal frame. The sections of the square and of the regions under Hippias's curve are painted in various pastel shades, ordered after the order of a color wheel.

- This oil painting is #114 in the series. It is signed on the back: HIPPIAS' CURVE (/) SQUARE AREA = (/) TRIANGLE " = 1 = [ . .] (/) Crockett Johnson 1973.

- Location
- Currently not on view

- date made
- 1973

- referenced
- Hippias

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.76

- accession number
- 1979.1093

- catalog number
- 1979.1093.76

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

## Painting -

*Construction of a Heptagon*- Description
- This is one of three very similar Crockett Johnson paintings closely related to the construction of a side of an inscribed regular heptagon which the artist published in
*The Mathematical Gazette*in 1975. The paper presents a way of producing an isosceles triangle with angles in the ratio 3:3:1, so that the smallest angle in the triangle is π / 7. This angle is then inscribed in a large circle, and intercepts an arc length of π/7. A central angle of the same circle intercepts twice the angle, that is to say 2π/7, and the corresponding chord the side of an inscribed heptagon.

- Crockett Johnson described the construction of his isosceles triangle in the diagram shown. The horizontal line segment below the circle on the painting corresponds to unit length BF in the figure, and the largest triangle in the painting is triangle is ABF in the figure, with vertex angle equal to one seventh of pi. This angle is inscribed in the large circular arc KDC. The side of the heptagon is the chord KC.

- This version of Crockett Johnson's construction of a heptagon is #115 in the series. It has a dark blue background and a wood and metal frame. The painting is an oil or acrylic on masonite. The work is unsigned. See also #108 (335571) and #117 (1979.1093.79).

- References: Crockett Johnson, “A Construction for a Regular Heptagon,”
*Mathematical Gazette*, 1975, vol. 59, pp. 17–21.

- Location
- Currently not on view

- date made
- ca 1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.77

- accession number
- 1979.1093

- catalog number
- 1979.1093.77

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

## 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, Kenneth E. Behring Center

## Painting -

*Construction of Heptagon*- Description
- Three very similar paintings in the Crockett Johnson collection are closely related to the the construction of a side of an inscribed regular a heptagon which he published in
*The Mathematical Gazette*in 1975. The paper presents a way of producing an isosceles triangle with angles in the ratio 3:3:1, so that the smallest angle in the triangle is π/ 7. This angle is then inscribed in a large circle, and intercepts an arc length of π/7. A central angle of the same circle intercepts twice the angle, that is to say 2π/7, and the corresponding chord the side of an inscribed heptagon. Crockett Johnson described the construction of his isosceles triangle in the diagram reproduced. The horizontal line segment below the circle on the painting corresponds to unit length BF in the figure, and the triangle is ABF. Three of the four light-colored sections of the painting highlight important points in the construction. The critical steps are drawing a perpendicular bisector to the line segment BF, marking off an arc of radius equal to the √(2) with center F, and measuring the unit length AO along a marked straightedge that passes through B and intersects the perpendicular bisector at A. Finally, one finds the side of the regular inscribed heptagon.

*Construction of Heptagon*is #117 in the series. The oil painting on masonite is in shades of purple, cream, turquoise, and black. It has a black wood and metal frame. The work is unsigned. The surface appears damaged, perhaps from water. See also #115 (1979.1093.77) and #108 (335571).

- Reference: Crockett Johnson, “A Construction for a Regular Heptagon,”
*Mathematical Gazette*, 1975, vol. 59, pp. 17–21.

- Location
- Currently not on view

- date made
- ca 1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.79

- accession number
- 1979.1093

- catalog number
- 1979.1093.79

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

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