Painting  Rotated Triangle and Reflections
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 Description

Plane figures of the same size and shape can be moved about in several ways and preserve their size and form. Such congruent transformations, as they are called, are combinations of rotations about a point or a line, reflections about a line, or translations in which the figure moves about the plane but the directions of the sides is unchanged.

This painting, which closely follows a diagram from a book by H. S. M. Coxeter, illustrates two properties of congruent transformations. First, a transformation in which only one point remains unchanged is a rotation. In the figure, the triangle PQR passes through a congruent transformation into the triangle PQ'R'. Suppose that the transformation consisted of a reflection. Then triangle PQR could be rotated about the line m to another triangle, PRR[1]. However, these two triangles have a line, and not simply a point, in common. Coxeter went on to argue that any congruent transformation can be constructed as the product of reflections, the number of which can be reduced to three.

In the painting, as in the diagram, there are three congruent triangles. One light blue and gray triangle rotates into another light blue triangle above it to the right (the axis of rotation is perpendicular to the painting). The blue and bluegray triangle is a rotation of the first triangle about the axis m, and a reflection of the other. The background is in two shades of gray, divided by this line of rotation.

The painting is #73 in the series and signed: CJ70. It has a metal frame.

Reference: H. S. M. Coxeter, The Real Projective Plane, p. 153.
 Location

Currently not on view
 date made

1970
 painter

Johnson, Crockett
 Physical Description

masonite (substrate material)

metal (frame material)
 Measurements

overall: 61.5 cm x 77 cm x 2.5 cm; 24 3/16 in x 30 5/16 in x in
 ID Number

1979.1093.47
 catalog number

1979.1093.47
 accession number

1979.1093
 Credit Line

Ruth Krauss in memory of Crockett Johnson
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Medicine and Science: Mathematics

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Art

Crockett Johnson
 Data Source

National Museum of American History