Aiding the Teaching of Geometry and Affording Mathematical Recreation: Paper Folding in the Spirit of Sundara Rao of Madras – Pedagogical and Popular Influence

As with many proposals for improving mathematics education, it is difficult to trace the influence of Rao’s proposals on classroom practice. British schoolteacher E. M. Langley mentioned the book in a 1904 article on unconventional lessons in mathematics, using a modified form of Rao’s argument for employing paper folding to find the sum of a geometric progression [Langley 1904]. In 1905, two British mathematicians working in Göttingen, Grace Chisolm Young (1868–1944) and William Henry Young (1863–1942), published The First Book of Geometry [Bradley 2006, pp. 15–24, esp. p. 18]. In this textbook, they recommended having children make diagrams and models using paper. Their presentation attempted considerably more axiomatic structure than that given by Rao, and used a wider range of constructions (e.g. a coin for drawing circles, folded models of polyhedra—with provision for flaps and cuts to hold edges together). The Youngs explained in their preface that they found these methods better suited than those of Rao to actual practice. As they wrote:

There have been efforts to introduce paper folding as a means of teaching, but they have not been of a satisfactory nature. The book of Sundara Row (Geometrical Exercises in Paper Folding) has little to recommend it. It is too difficult for a child, and too infantile for a grown person [Young and Young 1905, p. vii].

The Youngs’ book would be translated into German, Yiddish, and Italian, but it too did not receive a wide audience. In summary, there is no evidence that paper folding became common in the plane geometry classroom. Rao’s book did remain in print, with another edition in 1905 as well as reprints in 1917 and in 1966. Rao himself published further geometrical constructions in The Educational Review in the 1920s [Rao 1923–1924; 1925a].^[20] A few British authors also sought to extend his discussion of using paper folding to construct conic sections, although this too did not become common [Hardcastle 1909–1910; Lotka 1907].

Rao, Klein, Beman, and Smith had perceived paper folding within the traditions of mathematics in Germany and India. Their work, developed in response to British calls for reform of geometry teaching, was a contribution to an international educational and mathematical community.^[21] Throughout the first half of the 20th century, paper folding was discussed in books on mathematical recreations and in popular mathematical lectures. Teachers such as A. Harry Wheeler (1873–1950) developed a passion for folding polyhedra and other surfaces, although they did not confine themselves to starting from square sheets of paper. The Mathematical Gazette and other journals for schoolteachers continued to occasionally publish short papers on mathematical paper folding, including four articles reprinted in Pritchard [2003]: [Markowsky 1991; Brunton 1973; Nevill 1996; and Gibb 1990]. Meanwhile, at mid-century, through the efforts of dedicated hobbyists, including a few mathematically-minded folk, paper folding achieved considerable popularity as both an artistic endeavor and child’s play. By this time, it was closely associated with Japanese origami.

In fact, Japanese discussion of folding square sheets of paper as a recreation for both children and adults, and indeed as a form of art, reshaped discourse on the subject. The Japanese word for paper folding, origami, was widely used in English, both in Britain and the United States. David Lister [2009] and others have described the events surrounding the popularization of Japanese origami, so I shall note them only briefly here.

From the 1920s, aspects of paper folding appealed to magicians, to those recovering from medical procedures, to those seeking peaceful entertainment, and to admirers of Japanese culture. In 1958, dedicated paper folder Lillian Oppenheimer (1898–1992) of New York City spearheaded the formation of the Origami Center [Lister 2009; Berger 1958]. An exhibit on paper folding, curated by Edward Kallop (1926–2016), soon followed at the Cooper Union. The title of this exhibition, Plane Geometry and Fancy Figures, suggests that mathematical aspects of paper folding had not been entirely forgotten. At the same time, the figures shown were not generally those intended for instruction in plane geometry such as those of Rao. To be sure, Jack S. Berger, a student in chemical engineering at the Cooper Union Engineering School, showed paper folding as it related to conic sections such as the hyperbola, parabola, and ellipse. However, the exhibit placed far more emphasis on how paper was folded into three-dimensional shapes. Allan Sass, another chemical engineering student at Cooper Union, provided models of the truncated dodecahedron and an Archimedean solid called the rhombicuboctahedron. He also showed what he called a pentakis dodecahedron (a dodecahedron with a five-sided pyramid on each face) and a trapezoidal icositetrahedron (a polyhedron with twenty-four identical kite-shaped faces). The Engineering School at Cooper Union further supplied a string, wire and wood model of a three-dimensional figure called a hyperbolic paraboloid, made by one Stephen Forman. Mary F. Blade, a faculty member at the Cooper Union who taught drawing and mathematics, is acknowledged in the catalog, and it seems likely that the mathematical models were made under her direction. Three other geometric models were made by Jack J. Skillman of Chicago, who is remembered as a painter and origami artist [Kallop 1959].

Figure 18. The title page for the 1959 exhibition at the Cooper Union. Smithsonian Institution Libraries.

Most of Plane Geometry and Fancy Figures was devoted to paper forms of animals, plants, boats, clothing, and the like. Books on origami showed a similar emphasis. However, three of Lillian Oppenheimer’s sons were mathematicians and at least one of them, Martin Kruskal (1925–2006), took an interest in mathematical origami [Miura et al. 2007]. Books on recreations, some aimed at high school mathematics students, also mentioned paper folding [Johnson 1957]. Hence it is not entirely surprising that the tradition of paper folding to illustrate mathematical principles lived on, and indeed endures to this day.