Painting - One Surface and One Edge (Möbius)

Object Details

referenced

Moebius, August Ferdinand

painter

Johnson, Crockett

Description

Most geometric surfaces have a distinct inside and outside. This painting shows one that doesn’t. Take a strip of material, give it a half-twist, and attach the ends together. The result is a band with only one surface and one edge. Mathematicians began to explore such surfaces in the nineteenth century. In 1858 German astronomer and mathematician August Ferdinand Möbius (1790–1868), who had studied theoretical astronomy under Carl Friedrich Gauss at the University of Goettingen, discovered the one-sided surface shown in the painting. It has come to be known by his name. As often happens in the history of mathematics, another scholar, Johann Benedict Listing, had found the same result a few months earlier. Listing did not publish his work until 1861.

If one attaches the ends of a strip of paper without a half twist, the resulting figure is a cylinder. The cylinder has two sides such that one can paint the outside surface red and the inside surface green. If you try to paint the outside surface of a Möbius band red you will paint the entire band red without crossing an edge. Similarly, if you try to paint the inside surface of a Möbius band green you will paint the entire surface green. A cylinder has an upper edge and a lower edge. However, if you start at a point on the edge of a Möbius band you will trace out its entire edge and return to the point at which you began. Since Möbius's time, mathematicians have discovered and explored many other one-sided surfaces.

This painting, #34 in the series, was executed in oil on masonite and is signed: CJ65. The strip is shown in three shades of gray based on the figure’s position. The shades of gray, especially the lightest shade, are striking against the rose-colored background, and this contrast allows the viewer to focus on the properties of the Möbius band. The painting has a wooden frame.

Crockett Johnson's painting is similar to illustrations in James R. Newman's The World of Mathematics (1956), p. 596. However, the figures are not annotated in the artist's copy of the book.

Location

Currently not on view

Credit Line

Ruth Krauss in memory of Crockett Johnson

1965

ID Number

1979.1093.23

catalog number

1979.1093.23

accession number

1979.1093

Object Name

painting

Physical Description

masonite (substrate material)

wood (frame material)

Measurements

overall: 65 cm x 65 cm x 1.3 cm; 25 9/16 in x 25 9/16 in x 1/2 in

National Museum of American History

Record ID

nmah_694647

Usage of Metadata (Object Detail Text)

CC0

GUID (Link to Original Record)

http://n2t.net/ark:/65665/ng49ca746a5-1b71-704b-e053-15f76fa0b4fa