Painting - <I>Equal Areas, Their Triangular Square Root and Pi</I>
date: label: date made
content: 1970-1975
name: label: painter
content: Johnson, Crockett
notes: label: Description
content: 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.
label: Description
content: 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.
label: Description
content: 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.
label: Description
content: 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.
label: Description
content: This painting is unsigned and its date of completion is unknown.
label: Location
content: Currently not on view
setName: label: See more items in
content: Medicine and Science: Mathematics
label: See more items in
content: Science & Mathematics
label: See more items in
content: Crockett Johnson
label: See more items in
content: Art
creditLine: label: Credit Line
content: Ruth Krauss in memory of Crockett Johnson
dataSource: label: Data Source
content: National Museum of American History
identifier: label: ID Number
content: 1979.1093.59
label: catalog number
content: 1979.1093.59
label: accession number
content: 1979.1093
objectType: label: Object Name
content: painting
physicalDescription: label: Physical Description
content: masonite (substrate material)
label: Physical Description
content: wood (frame material)
label: Physical Description
content: metal (frame material)
label: Measurements
content: overall: 122 cm x 120.5 cm x 3.8 cm; 48 1/16 in x 47 7/16 in x 1 1/2 in