Painting - Heptagon from Its Seven Sides

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.

Date Made: 1973

Painter: Johnson, Crockett

Location: Currently not on view

See more items in: Medicine and Science: Mathematics, Science & Mathematics, Crockett Johnson, Art

Exhibition:

Exhibition Location:

Credit Line: Ruth Krauss in memory of Crockett Johnson

Data Source: National Museum of American History

Id Number: 1979.1093.74Catalog Number: 1979.1093.74Accession Number: 1979.1093

Object Name: painting

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

Guid: http://n2t.net/ark:/65665/ng49ca746a5-1654-704b-e053-15f76fa0b4fa

Record Id: nmah_694698

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.