# Painting - Heptagon from Its Seven Sides

Usage conditions apply
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
Object Name
painting
1973
painter
Johnson, Crockett
Physical Description
masonite (substrate material)
aluminum (frame material)
Measurements
overall: 82 cm x 41.2 cm x 2.5 cm; 32 5/16 in x 16 1/4 in x in
ID Number
1979.1093.74
catalog number
1979.1093.74
accession number
1979.1093
Credit Line
Ruth Krauss in memory of Crockett Johnson
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Art
Crockett Johnson
Data Source
National Museum of American History
Nominate this object for photography.

Our collection database is a work in progress. We may update this record based on further research and review. Learn more about our approach to sharing our collection online.

If you would like to know how you can use content on this page, see the Smithsonian's Terms of Use. If you need to request an image for publication or other use, please visit Rights and Reproductions.