Painting - Bouquet of Triangle Theorems (Euclid)

The mathematician Euclid lived around 300 BC, probably in Alexandria in what is now Egypt. Like most western scholars of his day, he wrote in Greek. Euclid prepared an introduction to mathematics known as The Elements. It was an eminently successful text, to the extent that most of the works he drew on are now lost. Translations of parts of The Elements were used in geometry teaching well into the nineteenth century in both Europe and the United States.

Euclid and other Greek geometers sought to prove theorems from basic definitions, postulates, and previously proven theorems. The book examined properties of triangles, circles, and more complex geometric figures. Euclid's emphasis on axiomatic structure became characteristic of much later mathematics, even though some of his postulates and proofs proved inadequate.

To honor Euclid's work, Crockett Johnson presented not a single mathematical result, but what he called a bouquet of triangular theorems. He did not state precisely which theorems relating to triangles he intended to illustrate in his painting, and preliminary drawings apparently have not survived. At the time, he was studying and carefully annotating Nathan A. Court's book College Geometry (1964). Court presents several theorems relating to lines through the midpoints of the side of a triangle that are suggested in the painting. The midpoints of the sides of the large triangle in the painting are joined to form a smaller one. According to Euclid, a line through two midpoints of sides of a triangle is parallel to the third side. Thus the construction creates a triangle similar to the initial triangle, with one fourth the area (both the height and the base of the initial triangle are halved). In the painting, triangles of this smaller size tile the plane. All three of the lines joining midpoints create triangles of this small size, and the large triangle at the center has an area four times as great.

The painting also suggests properties of the medians of the large triangle, that is to say, the lines joining each midpoint to the opposite vertex. The three medians meet in a point (point G in the figure from Court). It is not difficult to show that point G divides each median into two line segments, one twice as long as the other.

To focus attention on the large triangle, Crockett Johnson executed it in shades of white against a background of smaller dark black and gray triangles.

Bouquet of Triangle Theorems apparently is the artist's own construction. It was painted in oil or acrylic and is #26 in the series. It was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) BOUQUET OF TRIANGLE THEOREMS (/) (EUCLID).

Reference: Nathan A. Court, College Geometry, (1964 printing), p. 65. The figure on this page is not annotated.

Currently not on view