## Painting - Aligned Triangles (Desargues)

- In the 17th century, the French engineer and architect Girard Desargues (1591–1661) explored interconnections between extensions of the lines within a pencil of three line segments (a pencil of line segments consists of several line segments originating at a common point). His theorems, as published in his own extremely obscure work and also by his contemporary, Abraham Bosse, were extended in the 19th century, and proved of fundamental importance to projective geometry.
- Crockett Johnson's library contains discussions of Desargues' theorem by H. S. M. Coxeter, N. A. Court, Heinrich Dorrie, and William M. Ivins. This painting most resembles a figure from Coxeter, although the diagram is not annotated. Suppose that the vertices of two triangles (PQR and P'Q'R' in Figure 1.5B from Coxeter) lie on a pencil of three line segments emanating from the point O. Suppose that similarly situated sides of the two triangles can be extended to meet in the three points denoted by A, C and B in the figure. According to Desargues' theorem, A, C, and B are collinear.
- In the painting, the two concurrent triangles are shown in shades of gray and black, while the top of the pencil of three lines is in shades of gold. Extensions of the sides and their points of intersection are clearly shown. Both the figure and the background of the painting are divided by the line joining the points of intersection
- The painting is #63 in the series. It is painted in oil or acrylic on masonite, and has a brown wooden frame. The painting is signed: CJ70.
- References:
- Newman, J. R.,
*The World of Mathematics*, p. 133. Figure annotated. - Court, N. A.,
*College Geometry*(1952), pp. 163–5. The figure is not annotated. - Coxeter, H. S. M.,
*The Real Projective Plane*, (1955 edition), p. 7. The figure resembles the painting but is not annotated. - Dorrie, Heinrich,
*100 Great Problems of Elementary Mathematics: Their History and Solution*(1965), p. 267. There is an annotated figure here for another theorem of Desargues, the theorem of involution. - Field, J. V.,
*The Invention of Infinity: Mathematics and Art in the Renaissance*(1997), pp. 190–206. - Ivins, William M. Jr.,
*Art & Geometry: A Study in Space Intuitions*(1946), pp. 87–94. - Location
- Currently not on view

- National Museum of American History

- referenced
- Desargues, Girard
- painter
- Johnson, Crockett

- Credit Line
- Ruth Krauss in memory of Crockett Johnson

- Physical Description
- masonite (substrate material)
- wood (frame material)
- Measurements
- overall: 64 cm x 123.8 cm x 4.5 cm; 25 3/16 in x 48 3/4 in x 1 3/4 in

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

- 1970

- Object Name
- painting

- ID Number
- 1979.1093.38
- accession number
- 1979.1093
- catalog number
- 1979.1093.38