#
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 188 items.

Page 1 of 19

## Painting -

*Proof of the Pythagorean Theorem (Euclid)*- Description
- The Pythagorean theorem states that in any right triangle, the square of the side opposite the right angle (the hypotenuse), is equal to the sum of the squares of the other two sides. This painting depicts the “windmill” figure found in Proposition 47 of Book I of Euclid’s
*Elements*. Although the method of the proof depicted was written about 300 BC and is credited to Euclid, the theorem is named for Pythagoras, who lived 250 years earlier. It was known to the Babylonians centuries before then. However, knowing a theorem is different from demonstrating it, and the first surviving demonstration of this theorem is found in Euclid’s*Elements*.

- Crockett Johnson based his painting on a diagram in Ivor Thomas’s article on Greek mathematics in
*The World of Mathematics*, edited by James R. Newman (1956), p. 191. The proof is based on a comparison of areas. Euclid constructed a square on the hypotenuse BΓ of the right triangle ABΓ. The altitude of this triangle originating at right angle A is extended across this square. Euclid also constructed squares on the two shorter sides of the right triangle. He showed that the square on side AB was of equal area to the rectangle of sides BΔ and Δ;Λ. Similarly, the area of the square on side AΓ was of equal area to the rectangle of sides EΓ and EΛ. But then the square of the hypotenuse of the right triangle equals the sum of the squares of the shorter sides, as desired.

- Crockett Johnson executed the right triangle in the neutral, yet highly contrasting, hues of white and black. Each square area that rests on the sides of the triangle is painted with a combination of one primary color and black. This draws the viewer’s attention to the areas that complete Euclid’s proof of the Pythagorean theorem.

*Proof of the Pythagorean Theorem*, painting #2 in the series, is one of Crockett Johnson’s earliest geometric paintings. It was completed in 1965 and is marked: CJ65. It also is signed on the back: Crockett Johnson 1965 (/) PROOF OF THE PYTHAGOREAN THEOREM (/) (EUCLID).

- Location
- Currently not on view

- date made
- 1965

- referenced
- Euclid

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.01

- catalog number
- 1979.1093.01

- accession number
- 1979.1093

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

## Painting -

*Squares of a 3-4-5 Triangle in Scalene Perspective (Dürer)*- Description
- This painting, while similar in subject to the painting entitled
*Perspective (Alberti)*, depicts three planes perpendicular to the canvas. These three planes provide a detailed, three-dimensional view of space through the use of perspective. Three vanishing points are implied (though not shown) in the painting, one in each of the three planes.

- The painting shows a 3-4-5 triangle surrounded by squares proportional in number to the square of the side. That is, the horizontal plane contains nine squares, the vertical plane contains sixteen squares, and the oblique plane, which represents the hypotenuse of the 3-4-5 triangle, contains twenty-five squares. This explains the extension of the vertical and oblique planes and reminds the viewer of the Pythagorean theorem. Thus, Crockett Johnson has cleverly shown the illustration of two of his other paintings;
*Squares of a 3-4-5 Triangle (Pythagoras)*and*Proof of the Pythagorean Theorem (Euclid)*, in perspective; hence the title of the painting.

- The title of this painting points to the role of the German artist Albrecht Dürer (1471–1528) in creating ways of representing three-dimensional figures in a plane. Dürer is particularly remembered for a posthumously published treatise on human proportion. In his book entitled
*The Life and Art of Albrecht Dürer*, art historian Erwin Panofsky explains that the work of Dürer with perspective demonstrated that the field was not just an element of painting and architecture, but an important branch of mathematics.

- This construction may well have originated with Crockett Johnson. The oil painting was completed in 1965 and is signed: CJ65. It is #8 in his series of mathematical paintings.

- Location
- Currently not on view

- date made
- 1965

- referenced
- Duerer, Albrecht

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.04

- catalog number
- 1979.1093.04

- accession number
- 1979.1093

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

## Painting -

*Conic Curve (Apollonius)*- Description
- In ancient times, the Greek mathematician Apollonius of Perga (about 240–190 BC) made extensive studies of conic sections, the curves formed when a plane slices a cone. Many centuries later, the French mathematician and philosopher René Descartes (1596–1650) showed how the curves studied by Apollonius might be related to points on a straight line. In particular, he introduced an equation in two variables expressing points on the curve in terms of points on the line. An article by H. W. Turnbull entitled "The Great Mathematicians" found in
*The World of Mathematics*by James R. Newman discussed the interconnections between Apollonius and Descartes, and apparently was the basis of this painting. The copy of this book in Crockett Johnson's library is very faintly annotated on this page. Turnbull shows variable length ON, with corresponding points P on the curve.

- The analytic approach to geometry taken by Descartes would be greatly refined and extended in the course of the seventeenth century.

- Johnson executed his painting in white, purple, and gray. Each section is painted its own shade. This not only dramatizes the coordinate plane but highlights the curve that extends from the middle of the left edge to the top right corner of the painting.

*Conic Curve*, an oil or acrylic painting on masonite, is #11 in the series. It was completed in 1966 and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) CONIC CURVE (APOLLONIUS). It has a wooden frame.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Apollonius of Perga

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.06

- catalog number
- 1979.1093.06

- accession number
- 1979.1093

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

## 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, Kenneth E. Behring Center

## Painting -

*One Surface and One Edge (Möbius)*- Description
- Most geometric surfaces have a distinct inside and outside. This painting shows one that doesn’t. Take a strip of material, give it a half-twist, and attach the ends together. The result is a band with only one surface and one edge. Mathematicians began to explore such surfaces in the nineteenth century. In 1858 German astronomer and mathematician August Ferdinand Möbius (1790–1868), who had studied theoretical astronomy under Carl Friedrich Gauss at the University of Goettingen, discovered the one-sided surface shown in the painting. It has come to be known by his name. As often happens in the history of mathematics, another scholar, Johann Benedict Listing, had found the same result a few months earlier. Listing did not publish his work until 1861.

- If one attaches the ends of a strip of paper without a half twist, the resulting figure is a cylinder. The cylinder has two sides such that one can paint the outside surface red and the inside surface green. If you try to paint the outside surface of a Möbius band red you will paint the entire band red without crossing an edge. Similarly, if you try to paint the inside surface of a Möbius band green you will paint the entire surface green. A cylinder has an upper edge and a lower edge. However, if you start at a point on the edge of a Möbius band you will trace out its entire edge and return to the point at which you began. Since Möbius's time, mathematicians have discovered and explored many other one-sided surfaces.

- This painting, #34 in the series, was executed in oil on masonite and is signed: CJ65. The strip is shown in three shades of gray based on the figure’s position. The shades of gray, especially the lightest shade, are striking against the rose-colored background, and this contrast allows the viewer to focus on the properties of the Möbius band. The painting has a wooden frame.

- Crockett Johnson's painting is similar to illustrations in James R. Newman's
*The World of Mathematics*(1956), p. 596. However, the figures are not annotated in the artist's copy of the book.

- Location
- Currently not on view

- date made
- 1965

- referenced
- Moebius, August Ferdinand

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.23

- catalog number
- 1979.1093.23

- accession number
- 1979.1093

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

## Painting -

*Simple Equation (Descartes)*- Description
- In a pathbreaking book
*La Géométrie*, René Descartes (1596–1650) described how to perform algebraic operations using geometric methods. One such explanation is the subject of this Crockett Johnson painting. More specifically, Descartes described geometrical methods for finding the roots of simple polynomials. He wrote (as translated from the original French): "Finally, if I have z² = az -b², I make NL equal to (1/2)a and LM equal to b as before: then, instead of joining the points M and N, I draw MQR parallel to LN, and with N as center describe a circle through L cutting MQR in the points Q and R; then z, the line sought, is either MQ or MR, for in this way it can be expressed in two ways, namely: z = (1/2)a + √((1/4)a² - b²) and z = (1/2)a - √((1/4)a² - b²)."

- To verify that z = MR is a solution to the equation z²= az - b², note that the square of the length of the tangent ML equals the product of the two line segments MQ and MR. As ML is defined to equal b, its square is b squared. The length of MR is z, and the length of MQ is the difference between the diameter of the circle (length a) and the segment MR, that is to say (a – z) . Hence b squared equals z (a – z) which, on rearrangement of terms, gives the result desired.

- Crockett Johnson's painting directly imitates Descartes's figure found in Book I of
*La Géométrie*. A translation of part of Book I is found in the artist’s copy of James R. Newman's*The World of Mathematics*. The figure on page 250 is annotated.

- This oil or acrylic painting on masonite is #36 in the series. It was completed in 1966 and is signed: CJ66. It has a wooden frame.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Descartes, Rene

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.24

- catalog number
- 1979.1093.24

- accession number
- 1979.1093

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

## Painting -

*Multiplication through Imaginary Numbers (Gauss)*- 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

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

## Painting -

*Squared Circle*- 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) = &#*&#);(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=(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 1.772435. 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

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

## Painting -

*Golden Rectangle*- 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

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

## Painting -

*Square Roots of One, Two and Three*- 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

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