Cubes, Conic Sections, and Crockett Johnson - Johnson’s Mathematical Paintings

In December 1965, about 20 years after he used mathematical symbols in “Barnaby,” Crockett Johnson started producing artwork based on diagrams used in the proofs of geometric theorems. On February 27, 1966, Johnson wrote to an artist friend noting that his paintings [Reinhardt Papers, Crockett Johnson Correspondence]

have an irresistible appeal to people that like coldly intellectual abstractions and warmly emotional realistic art.  They comprise a series of romantic tributes to the great geometric mathematicians from Pythagoras on up.

Many of the diagrams Johnson used came from James Newman’s The World of Mathematics, a four-volume collection of essays about mathematics and mathematicians published in 1956.  In addition to The World of Mathematics, Crockett Johnson acquired a collection of mathematics books, many of which were geometry books, in which he found visually interesting diagrams.  He also corresponded with a number of professional and amateur mathematicians.  For several years around 1970, Johnson sent diagrams accompanied by short notes about the mathematics in his paintings to a friend, Mickey Rosenau, who, unlike Johnson, had taken many mathematics courses in college.  It was in such a note that Johnson posed the question that appears in the title of this paper.

In 1980, the Division of Mathematics of the National Museum of History and Technology (now the National Museum of American History) of the Smithsonian Institution obtained Johnson’s papers and 80 of his mathematical paintings, all of which appear with descriptions on the NMAH website [see Mathematical Paintings of Crockett Johnson].  Thirty-two of these paintings were included in a 1980 exhibit, “Theorems in Color,” at the museum.  Some of the paintings not in the Smithsonian collection can be seen at Crockett Johnson Homepage: Paintings.

Toward the end of his life, Johnson created a number of paintings based on original constructions.  These included some approximations of \(\pi\) and a construction of a regular heptagon using methods other than straightedge and compass [see Cawthorne and Green 2009].  The construction of square roots of positive integers, discussed in the "Classic Greek Constructions" section of this article, was also considered in a 1967 painting, Square Roots to Sixteen (Theodorus of Cyrene) (Figure 6), which was shown on the poster for the “Theorems in Color” exhibit.  If the legs of the smallest triangle, the beginning of the “spiral” in the middle left, have length \(1,\) then the hypotenuse has length \(\sqrt{2}.\)  Given that all the small legs are of length \(1,\) the hypotenuse of the adjoining triangle has length \(\sqrt{3},\) the next hypotenuse length \(\sqrt{4},\) and so on, until the hypotenuse of the last triangle is \(\sqrt{16}.\)

Figure 6.  Square Roots to Sixteen (Theodorus of Cyrene) by Crockett Johnson (1967).  (Courtesy of the Smithsonian Institution, image number 2008-2509.)

In 1970 and 1975 Crockett Johnson published articles about these constructions in The Mathematical Gazette, a British journal for mathematics educators.  Johnson’s original constructions clearly show that he understated his mathematical abilities as he described them in his 1972 article in Leonardo, the “International Journal of the Contemporary Artist” [Johnson 1972, 99]:

I know little algebra and I avoid using it because algebra, or my ineptness with it, tends to make me lose a graphic grasp of a picture.

Johnson made this comment when referring to Problem of Delos, II (1970), a painting using one of his own constructions to solve the Delian problem.  He continued as follows [Johnson 1972, 99]:

Instead, as I did in approaching the problem of the squared circle, I played with what I knew in advance to be the elements of the problem, imagining them as a construction in motion, an animated film sequence with an infinite number of frames running back and forth between plus and minus limits across the point of solution.

In the next section we return to the question posed by Crockett Johnson in the title of this paper.