Painting - <I>Squared Rectangle and Euler Line</I>
date: label: date made
content: 1972
name: label: painter
content: Johnson, Crockett
notes: label: Description
content: Crockett Johnson had a longstanding interest in squaring figures, that is to say, constructing squares equal in area to other plane figures. Euclid had shown in his <I>Elements</I> (Book II, Proposition 14) how to construct a square equal in area to a given rectangle. Crockett Johnson developed his own construction, one case of which served as the basis of this painting. The rectangle, the square of equal area, and a circle used in the demonstration are shown in various shades of pink.
label: Description
content: Two drawings from Crockett Johnson’s papers illustrate his ideas. The one that relates most closely to this painting is labeled A in his figure. In it, the given rectangle is ABED. The angles at the corner A and D are bisected, and the bisectors extended to meet at point C. The line from corner B through C meets side DE at point X. Line segments CL and XS are constructed parallel to AD. By this construction, the segment DL is half the length of AD. From center X, one may draw a line segment of length DL that intersects CL at point O. The figure and painting then show a circle of radius OX and center O that intersected side AD at V (where OV equals DL and is perpendicular to AD), and side BE at F. The point Y on the circle is on OV extended. As Crockett Johnson states in his notes, XY squared equals the product of AB and AD.
label: Description
content: The Euler line of a triangle includes three points. These are the intersections of the altitudes, of the perpendicular bisectors (lines perpendicular to the sides at their midpoints), and of the medians (lines drawn from a vertex to the midpoint of the opposite side). For an inscribed right triangle, both the perpendicular bisectors and the medians intersect in the center of the inscribing circle, while the altitudes meet at the right angle of the triangle. In the painting there are three right triangles inscribed in the circle. These are triangles XEF, XYF, and VXY in the diagram. The Euler line for the first two triangles is XOF, the Euler line for the third is VOY. The colors of Crockett Johnson's painting draws special attention to XOF, and it is this line he mentions in his figure for the painting.
label: Description
content: The painting is on masonite, and is #94 in the series. It has a blue-black background and a black wooden frame. It is signed on the back: SQUARED RECTANGLE AND EULER LINE (/) Crockett Johnson 1972.
label: Location
content: Currently not on view
setName: label: See more items in
content: Medicine and Science: Mathematics
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content: Science & Mathematics
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content: Crockett Johnson
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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: 1979.1093.62
label: catalog number
content: 1979.1093.62
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: Measurements
content: overall: 65 cm x 60 cm x 4.5 cm; 25 9/16 in x 23 5/8 in x 1 3/4 in