Painting - Construction of the Heptagon

Three very similar paintings in the Crockett Johnson collection are closely related to the construction of a side of an inscribed regular a heptagon which he published in The Mathematical Gazette in 1975. The paper presents a way of producing an isosceles triangle with angles in the ratio 3:3:1, so that the smallest angle in the triangle is π/7. This angle is then inscribed in a large circle, and intercepts an arc length of π/7. A central angle of the same circle intercepts twice the angle, that is to say 2π/7, and the corresponding chord the side of an inscribed heptagon.

Crockett Johnson described the construction of his isosceles triangle in the diagram shown in the image. The horizontal line segment below the circle on the painting corresponds to unit length BF in the figure, and the triangle is ABF. The light colors of the painting highlight important points in the construction - marking off an arc of radius equal to the square root of 2 with center F, measuring the unit length AO along a marked straight edge that passes through B and ends at point A on the perpendicular bisector, and finding the side of the regular inscribed heptagon.

This version of the construction of a heptagon is #108 in the series. The oil painting on masonite with chrome frame was completed in 1975 and is unsigned. It is marked on the back: Construction of the Heptagon (/) Crockett Johnson 1975. See also paintings #115 (1979.1093.77) and #117 (1979.1093.79) in the series.

Reference: Crockett Johnson, "A Construction for a Regular Heptagon," Mathematical Gazette, 1975, vol. 59, pp.17–21.

Currently not on view