Mathematical recreations are often assumed to consist of toys, games, and puzzles, such as the Chinese tangram, Rubik’s Cube, or magic squares. However, this field also includes other delightful activities that include mathematical components but typically fall within realms such as magic, visual art, or handcrafts. Indeed, the MAA’s Special Interest Group on Recreational Mathematics defines its charge broadly. Similarly, paper folding is both an enjoyable pastime—as millions of origami aficionados can attest—and a longstanding technique for teaching geometry to audiences from small children to adults in various locations around the world.

This article traces the history of paper folding in mathematics education, from its origins in the German kindergarten movement through its recognition as “a new and very simple method of effecting certain [geometric] constructions” by Felix Klein [1897, p. 42]. I focus particularly on the 1893 text *Geometrical Exercises in Paper Folding*, written by Sundara Rao (1853–after 1923; his name was often spelled “Row” on his publications) of Madras, India, while that country was a colony of Great Britain. Inspired in part by efforts to reform the teaching of geometry in the British Empire, the book then gained readers in the United States as that nation was rising to global prominence. The story thus not only sheds light on an episode in the teaching of geometry, but also helps readers make connections between mathematics and world history and geopolitics. Additionally, mathematics instructors will find outlines for a few classroom activities based on Rao’s book that are suitable for use in precalculus, geometry, and history of mathematics courses, as well as capstone seminars and in-service workshops aimed at the preparation and professional development of secondary mathematics teachers.

**Figure 1.** Two of the surviving copies of Rao’s book were owned by significant 19th-century American mathematicians: Artemas Martin (1835–1918) and Florian Cajori (1859–1930). W. W. Beman’s interest in the work is explored later in the article. Note also the alternative spelling of Rao’s name (Row). American University and GoogleBooks.

Folding bits of paper has a long informal history that stretches around the world; a small part of that story relates to mathematical topics. From at least the 16th century, a few printed geometry textbooks included folded solids [e.g., Billingsley 1570]. Some cabinets of curiosity boasted folded paper models of Platonic solids, and authors such as Albrecht Dürer included patterns for such models in their books [Cromwell 1997, esp. pp. 126–128]. By the 1780s, mathematicians such as A.G. Kästner of the University of Göttingen were using relatively complicated models of polyhedra in their teaching. Some of the surfaces that Kästner used are still held by the Göttingen Collection of Mathematical Models and Instruments.

In the 19th century, as paper became cheap and as the early education of children attracted wider attention,^{[1]} paper folding became part of the learning of much younger people. In particular, the German Friedrich Froebel (1782–1852) and his disciples in the kindergarten movement developed special paper and patterns for paper folding by their students. Froebel, born in a small town in central Germany, opened a school in Thuringia (Germany) in 1816. During the 1830s, while charged with administering a Swiss orphanage, he began to consider the teaching of very young children. Froebel started a school for them in 1837 and soon dubbed such places for the cultivation of children “kindergartens.” In the course of the 1840s, he vigorously promoted kindergartens through publications, training schools, travel, and sales of apparatus. Germany had seven kindergartens in 1847, and another forty-four opened in the revolutionary year of 1848. Then, in 1851, Prussian authorities banned kindergartens as socialistic and atheistic.^{[2]} Froebel died the following year, and his disciples scattered across Europe as well as to Great Britain, the United States, and India, where they concentrated in the Madras Presidency (Province).

**Figure 2.** Lithograph of Friedrich Froebel by C. W. Bardeen, ca 1897. Library of Congress.

To occupy the hands and improve the minds of kindergarten students, Froebel had introduced a series of some twenty “gifts” or “occupations.” These ranged from a simple set of cloth balls; to a variety of wooden blocks, sticks, slats, and jointed slats; and to devices for drawing, sewing, weaving, interlacing, making skeleton models, and modelling clay. Square sheets of paper could be folded and cut into attractive patterns (the thirteenth gift) or folded into interesting forms (the eighteenth gift). Froebel himself emphasized folding squares and rectangles in order to deduce relationships between the areas of squares, rectangles, and triangles, introducing basic principles of geometry [Froebel 1903, pp. 96–117]. Other kindergartners, as kindergarten teachers were then known, emphasized somewhat different geometric relationships, including creating an equilateral triangle [Ronge and Ronge 1858, pp. 47–48]. During the same period, children were encouraged to fold attractive patterns, sometimes in the form of a square and at other times in more complex patterns intended to resemble organic forms. Examples of such paper folding survive in scrapbooks prepared by teachers in training.^{[3]}

**Figure 3.** A presentation of Froebel’s process for folding from Hermann Goldammer’s

*Fröbels Beschäftigungen für das vorschulplichtige Alter* (Berlin: Habel, 1874), plate 40/p. 617.

Reprinted in [Friedman 2018] and available via a CC BY-NC-ND 4.0 license.

[1] On the development of techniques for manufacturing paper less expensively—which encouraged the manufacture and diffusion not only of paper folding but also silhouettes, playing cards, textbooks, board games and a host of other products, see Mueller [2014].

[2] An extensive literature on Friedrich Froebel and the dissemination of the kindergarten in Europe and the United States exists. See Beatty [1995]; Brosterman [1997]; Cosgrove [1989]; and Allen [1988].

[3] Brosterman [1997, p. 79] shows a paper folding album from Philadelphia created around 1875. The Margaret MacLachan Papers in the National Louis University Archives and Special Collections, Chicago, contain samples of her paper folding from when she trained as a kindergarten teacher in the 1890s. The Museum of Modern Art in New York has a scrapbook of paper folding designs compiled by Fannie E. Kachline in the 1890s.

Also in the 19th century, mathematics educators were questioning then-standard approaches to teaching geometry as the subject moved from colleges into secondary schools while enrollment in secondary education expanded. From at least the time of the introduction of the printing press, university students in Britain and continental Europe had routinely studied geometry by memorizing translations and condensations of Euclid’s *Elements*. By 1850, many European countries had abandoned Euclid for use in introductory geometry, preferring alternative organizations of propositions and more practical treatments. As Joan Richards [1988] and Michael Price [1994] have shown, British mathematicians and mathematics teachers disagreed sharply about what texts were most appropriate for students.

The London firm of Macmillan published two textbooks that well illustrate this debate. The first was by Isaac Todhunter (1820–1884) of St. John’s College, Cambridge University, who had graduated from there as Senior Wrangler in 1848. In the preface to his *The Elements of Euclid for the Use of Schools and Colleges*, which first appeared in 1862, Todhunter wrote:

In England the text-book of Geometry consists of the Elements of Euclid; for nearly every official progamme of instruction of examination explicitly includes some portion of this work. Numerous attempts have been made to find an appropriate substitute for the Elements of Euclid; but such attempts, fortunately, have hitherto been made in vain. The advantages attending a common standard of reference in such an important subject, can hardly be overestimated; and it is extremely improbable, if Euclid were once abandoned, that any agreement would exist as to the author who should replace him. It cannot be denied that defects and difficulties occur in the Elements of Euclid, and that these become more obvious as we examine the work more closely; but probably during such examination the conviction will grow deeper that these defects and difficulties are due in a great measure to the nature of the subject itself, and to the place which it occupies in a course of study [Todhunter 1862, pp. vii–viii].

Todhunter went on to present Euclid’s definitions, postulates, propositions, theorems, and corollaries.

**Figure 4.** Isaac Todhunter as an undergraduate at Cambridge, as drawn by T. C. Wageman,

ca 1848, and James Maurice Wilson as photographed by Marcus Guttenberg in the 1870s.

MacTutor and National Portrait Gallery, available via a CC BY-NC-ND 3.0 license.

Another graduate of St. John’s College of Cambridge University, James Maurice Wilson (1836–1931)—who was Senior Wrangler in 1859 and then taught mathematics at Rugby School—took a different view of Euclid. In the preface to his *Elementary Geometry*, first published in 1868, Wilson wrote:

Euclid’s recognised and acknowledged faults as a system of Geometry, and as a specimen of analysed reasoning, are of slight importance compared with others of greater magnitude. The real objections to Euclid as a text-book are his artificiality, the invariably syllogistic form of his reasoning, the length of his demonstrations, and his unsuggestiveness [Wilson 1868, vol. 1, p. v].

Wilson’s book, while including many of the results of Euclid’s *Elements*, did not follow the order of the classic text.

In 1871, a few British secondary school teachers who were critical of the use of Euclid as a geometry text banded together with two university faculty members to form the Association for the Reform of Geometry Teaching, an organization promptly renamed the Association for the Improvement of Geometry Teaching (AIGT). They strove to develop a standard curriculum for geometry teaching that could be incorporated into national and university matriculation examinations, but the organization accomplished its goal of replacing Euclid as the standard introduction in geometry for students in Britain and British colonies only slowly [Price 1994]. It did welcome and publicize alternative approaches to the discipline, including Sundara Rao’s *Geometrical Exercises in Paper Folding*. The goals of the AIGT broadened over the years, and it officially changed its name to the Mathematical Association in 1897. Rao joined the society in 1904 [Anonymous 1911, p. 32].

**Figure 5.** Thomas Archer Hirst (1830–1892) was the first president of AIGT. MacTutor.

At the same time some sought to replace Euclid as a textbook, others, both in Britain and in the United States, sought further reforms. These instructors hoped to provide budding scientists and engineers with a thorough knowledge of the practical mathematics needed in their work. Handling protractors and drawing instruments, studying mathematical models, using slide rules and mathematical tables, and exploring functions using graph paper were all deemed valuable activities. Some reformers stressed the advantages of combining the study of geometry and algebra, the better to understand functional relationships. Many of these reforms were associated with the name of the British educator John Perry (1850–1920), whose influence on British mathematics teaching has been discussed by Brock and Price [1980]. David L. Roberts [2012] has considered the reception of his ideas in the US.^{[4]} In Massachusetts, Englishman Walter Smith developed a curriculum for teaching industrial design that included a set of wooden models. Other drawing courses, such as that of Louis Prang, encouraged use of models made from paper and even supplied templates for them [Kidwell 1996]. William T. Campbell [1899] would develop this approach in detail at the end of the century, while William W. Speer [1888] of Chicago extended the idea of folded paper models to children.^{[5]}

**Figure 6.** Campbell’s 1899 *Observational Geometry* showed how to

draw, cut out, fold, and paste geometrical solids such as the cube. Internet Archive.

Born in 1853, Sundara Rao received an introduction to geometry as an undergraduate at Kumbakonam College, a school in the town of Kumbakonam, about 160 miles south of Madras. He obtained his B.A. in mathematics there in 1874. As Alex D. D. Craik has noted [2007, esp. pp. 259–269], it seems almost certain that Rao would have studied from Todhunter’s textbooks as an undergraduate. His instructor would have been William Archer Porter (ca 1825–1890), who had graduated as a Third Wrangler in mathematics at Cambridge in 1849, the year after Todhunter. Porter taught at Kumbakonam from 1863 until 1872 and again from 1874 to 1878. Rao’s father, T. Gopala Rao (or Row), also was on the staff of the college, assuming a variety of posts over the period between 1857 and 1882. Despite these connections to education, upon graduation Rao became a civil servant, working for the British government in Madras.

**Figure 7.** Undated photograph of Sundara Rao and Kumbakonam College in 2011. mAnasa-taraMgiNI,

“Paper folding, Sundara Rao and geometrical constructions” (26 November 2016), and *Wikimedia Commons*.

Although he was apparently not directly involved in any of them, the trends in geometry teaching that were mentioned previously had an impact on India during the years of Rao’s career. For example, by the 1870s, kindergarten had come to India, particularly Madras.^{[6]} One of those most interested in promoting the practice was Isabel Bain Brander. Bain, born in about 1846, was first educated at home by her mother and eldest sister, and afterwards in schools in Brentford and Oxfordshire, at Queen’s College in London, and in Paris. She received teacher training at a normal school in Salem, Massachusetts, and at Home and Colonial College, London [Anonymous 1907].^{[7]} In 1870 she was appointed by the Duke of Argyll, Secretary of State for India, to open a Government Normal School for women in Madras. The school grew under her direction. Attracted by Froebel’s work, and repelled by the rote teaching given to young children in Madras, in 1875 Bain introduced kindergarten training into lower classes of a practice school associated with the teacher’s school. She also devoted an hour a week after school hours to the instruction of normal school students in kindergarten occupations [Cotton 1898, p. 254]. The following year, Bain married James Brander, and she took an early retirement. He died two years later, in June 1878. That September, Isabel Brander published a brief account of her early experience with kindergarten in India, noting the importance of introducing materials and using songs and stories that reflected the experiences of Indian children [Brander 1878]. From 1880, she was able to put these ideas into practice once again, as the inspector for schools for girls and women for the entire Madras Presidency. By 1885, knowledge of a few of Froebel’s gifts was required of young children sitting for examinations at schools for both Europeans and Asians that received grants from the British government [Cotton 1898, pp. 254–255]. Brander also continued to publish on kindergartens [Brander 1887; 1896; 1899–1900].^{[8]} After she retired a second time and left India in 1903, both British and Indian teachers continued to run kindergartens, sometimes using a somewhat larger range of Froebel’s gifts [May, Kaur, and Prochner 2014, esp. pp. 139–145].

**Figure 8.** Sketch of Isabel Bain Brander. Mary Frances Billington,

*Woman in India* (London: Chapman & Hall, 1895), p. 36.

Attempts to improve geometry teaching at the primary and secondary levels also took place in India. One reformer was B. Hanumantha Rao, who had graduated from Kumbakonam College the same year as Sundara Rao (he was first in the class, Sundara Rao second; the two men were not related) [Row 1875, pp. 110–111]. Hanumantha Rao stayed on at Kumbakonam as an assistant master and then taught at the Government Normal School in Madras. In 1885, he published *First Lessons in Geometry*. The book was reviewed in *Nature*, where the anonymous reviewer’s comments suggest the spirit of the times:

A modern mathematical movement has taken hold of able mathematical teachers of the mild Hindoo. Mr. Rau [*sic*] candidly repudiates all claim to originality for his matter, as in its compilation he has consulted the best English and French text-books for pure as well as for practical geometry. “If ‘Euclid’s Elements’ is unsuited for beginners who study it in their own native tongue, how much more so should it be in this country, where it is taught in classes consisting generally of lads between ten and twelve before they have had time to master the difficulties of a foreign language, and before too, I may add, they can benefit by its rigorous logic” [Anonymous 1887].^{[9]}

A second edition of Hanumantha Rao’s book, published in 1888, was commended in *The India Magazine* and in *Nature* [Rau 1888; Pope 1888; Anonymous 1888]. Hanumantha Rao would go on to become Professor of Mathematics at the College of Engineering in Madras. He was a founding member of what became the Indian Mathematical Society and served as the organization’s first president from 1907 to 1912.

**Figure 9.** Hanumantha Rao and title page from the second edition of *First Lessons in Geometry*.

Indian Mathematical Society and American University Digital Research Archive.

[6] For a general introduction to infant schools in British India, see May, Kaur, and Prochner [2014, pp. 111–148]. Nineteenth-century kindergarten is discussed on pp. 141–145.

[9] I have not found any evidence that copies of this edition exist; the citation according to the review is: B. Hanumanta Rau [*sic*], *First Lessons in Geometry. For the Use of Technical, Middle, and High Schools*, Madras: Addison & Company, 1885.

As noted above, Sundara Rao was a clerk in the treasury department, not a teacher. Yet, in 1893 his book *Geometrical Exercises in Paper Folding* was published by Addison & Company in Madras. Even though Rao was too old to have attended a kindergarten himself and presumably had no experience working in one, he cited Froebel’s ideas as an inspiration. In the book’s introduction, Rao wrote: “The idea of this book was suggested to me by Kindergarten Gift No. VIII. – Paper-folding. The gift consists of 200 variously coloured squares of paper, a folder, and diagrams and instructions for folding. The paper is coloured and glazed on one side. The paper may, however, be of self-colour, alike on both sides” [Row 1893, p. 1]. He particularly recommended the kindergarten gift sold by Messrs. Higginbotham and Co., and he noted that a packet of 100 sheets of paper accompanied his book.

Rao did not say how he learned about kindergarten gifts. His specific mention of Higginbotham’s suggests that he investigated kindergarten material at that well-known bookstore, perhaps hoping to purchase supplies for his own family. He might also have learned of Brander’s ideas from other sources. His reference to “Gift No. VIII” suggests that he was not deeply immersed in Froebel’s theory—kindergarten gift VIII usually refers to drawing, while gift XVIII is paper folding. Of course, this may be a typographical error.

**Figure 10.** Advertisement for Higginbotham and Co., Madras, ca 1890.

Sriramy, “Higginbotham’s – a home for literature,” *[Madras] Hidden Histories* (17 January 2015).

Additionally, Sundara Rao specifically mentioned Hanumantha Rao’s *First Lessons* in the preface to his *Geometrical Exercises in Paper Folding*. He noted that his classmate had made frequent allusions to paper folding in the discussion of plane figures in his *First Lessons*, “but the hint has not been generally taken by teachers” [Row 1893, pp. iii–iv].

Rao had not sought to write an entire textbook on geometry; rather, his aim was to show how regular polygons, circles, and other plane curves could be folded or pricked on paper. This allowed him, as he wrote in his preface:

To introduce to the reader some well known problems of ancient and modern Geometry, and to show how Algebra and Trigonometry may be advantageously applied to Geometry, so as to elucidate each of the subjects which are usually kept in separate pigeonholes [Row 1893, p. iv].

**Figure 11.** Table of contents for *Geometrical Exercises in Paper Folding*. Internet Archive.

Rao’s book cut a wide swath. In the first nine chapters, he presented methods for folding paper to produce nine regular polygons. Some of the material he presented built toward such constructions. For example, he showed how to divide a line segment “in medial section” (i.e., so that the length of one section was the mean proportional between the length of the other section and the entire length). He would use this construction to fold an angle of π/5 radians, which proved most useful in constructing a pentagon.

At other times, Rao used his constructions to suggest theorems relating to both plane geometry and other mathematical topics. For example, his discussion of the square not only included an illustration of the Pythagorean theorem, but also presented the binomial theorem (algebraically as well as geometrically), and described a construction for finding the sum of the geometric series \(\sum_{n=1}^{\infty} (\frac{1}{2})^n\) [Row 1893, pp. 4–5].^{[10]}

**Figure 12.** Pages 4–5 from *Geometrical Exercises in Paper Folding*. GoogleBooks.

Not all of Rao’s constructions involved paper folding alone. In a chapter on conic sections, he considered a parabola as the set of points equidistant from a point (the focus) and a line (the directrix). He assumed that the axis of the parabola was on a horizontal line that bisected a square, the focus was a point on that line, and the left edge of the square was part of the directrix. He folded a line parallel to the axis, and noted where it intersected the directrix. Then, folding down this parallel line, he found the point on it that was equidistant from the focus and the directrix. By pricking a series of such points, one obtained the outline of a parabola. From this relatively straightforward beginning, Rao went on to find such conic sections as the ellipse and the hyperbola. In a subsequent chapter, he considered more elaborate curves, including the cissoid, the conchoid, the ovals of Cassini, the logarithmic curve, the catenary, the cardioid, and the cycloid. I mention the pricking of points on a curve not only to illustrate the range of Rao’s interests but also because paper pricking was another of Froebel’s occupations for kindergarten students.

Rao’s comments about who might read his book focused on British India. In the preface, he pointed out that paper folding was not entirely foreign to those who wrote Sanskrit and Mahrati, as paper was folded vertically or horizontally to keep the lines and columns straight. Paper folding also had been associated with correspondence, with letters folded so that part of a sheet formed an envelope. Further, paper folding had been tied to making models of polyhedra in order to illustrate Euclid’s Book XI. And then, of course, there was the example noted earlier of Hanumantha Rao and his textbook. However, Sundara Rao had a wider audience in mind. As he wrote:

I have sought not only to aid the teaching of Geometry in schools and colleges, but also to afford mathematical recreation to young and old, in an attractive and cheap form. “Old boys” like myself may find the book useful to revive their old lessons, and to have a peep into modern developments which, although very interesting and instructive, have been ignored by the Madras University [Row 1893, p. vi].

As it happened, the faculty of Madras University showed relatively little interest in the book. However, as we shall see, *Geometrical Exercises* became a classic in the literature of mathematical recreations.

Rao sent a copy of his book to the AIGT, which had recently began publishing a journal entitled *The Mathematical Gazette*.^{[11]} The December 1894 issue of that serial noted receipt of the book with the comment:

The author has sought, in our opinion with success, “not only to aid the teaching of geometry in schools and colleges, but also to afford mathematical recreation to young and old.” The book should be in the hands of all those who have to introduce geometrical ideas to young pupils. We commend it especially to kindergarten training colleges for teachers [Anonymous 1894].

The remarks quoted are unsigned, but a later postcard indicates they were from the founding editor of the *Gazette*, Edward M. Langley (1851–1933).^{[12]} That same month, Langley sent a letter to the much better-known journal *Nature. *Here, he commended teachers who wished to introduce students to geometrical ideas gradually, before exposing them to the full rigors of Euclidean geometry. Langley recommended kindergarten materials as a first step and then listed eight books that might be helpful for further study. One of them was Rao’s *Geometrical Exercises* [Langley 1894]. Langley’s list inspired another Englishman, Louis C. Miall (1842–1921), to publish his own recommendations on elementary geometry textbooks. This appeared in his 1897 book *Thirty Years of Teaching* [Miall 1897, p. 118]. Miall too recommended Rao’s book.^{[13]}

Copies of Rao’s book also made it outside the British Empire. An example belonging to the American mathematical journalist Artemis Martin, inscribed with the author’s compliments, is in the library of American University. Rao’s compatriot B. H. Rao had posed a problem in Martin’s short-lived journal *The Mathematical Magazine*, which might explain why Martin received a copy of the booklet.^{[14]}

**Figure 13.** Undated photos of Artemas Martin and Felix Klein. *Wikimedia Commons* and *Wikimedia Commons*.

Another reader, of far greater importance both to mathematics generally and to the diffusion of knowledge of Sundara Rao’s work, was the German Felix Klein (1849–1925). By 1893, Klein was professor of mathematics at Göttingen University, where he trained a generation of research mathematicians. The summer of that year, he successfully represented German mathematics at an international congress of mathematicians held in Chicago in conjunction with the World’s Fair. Klein also edited the *Annalen der Mathematik*, a leading German journal. It is possible that Rao sent a copy of his book to Klein directly or to that journal. In the spring of 1895, Klein presented a short series of lectures on famous problems of elementary geometry at the third meeting of the Vereins zur Förderung des Mathematischen und Naturwissenschaftlichen Unterrichts (Association for the Promotion of Mathematics and Science Education), an organization of teachers in German gymnasia (schools). These lectures were based on material he had given previously in short courses at Göttingen, and they were compiled from Klein’s notes by one of the students who had attended a course, F. Tägert.

Klein’s lectures were intended to introduce teachers to the consequences of recent results in geometry for the kind of mathematics they taught. More specifically, he talked about the implications of recent discoveries related to the three problems of classical geometry—the duplication of the cube, the trisection of an angle, and the quadrature of the circle. Ancient geometers strove in vain to solve these problems using only ruler and compass constructions; Klein argued, using algebraic arguments he considered quite elementary, that such solutions were impossible.

Of particular importance to the history of paper folding as a mathematical recreation in the United States is a casual comment at the beginning of the fifth chapter of Klein’s 1895 *Vorträge űber Ausgewählte Fragen den Elementargeometrie* (*Lectures on Selected Questions in Elementary Geometry*). The preceding chapter offered a ruler and compass construction of a polygon with seventeen equal sides. Klein’s next topic was to be algebraic constructions, but he first mentioned “a new and very simple method of effecting certain constructions, *paper folding*” [Klein 1897, p. 42]. He noted that Hermann Wiener, who taught in Darmstadt, had constructed models of the regular polyhedra in this way.^{[15]} Klein also pointed out that, at the same time and in Madras, the Indian mathematician Sundara Row [*sic*] had worked along similar lines (ignoring Rao’s emphasis on plane figures rather than polyhedra). Klein mentioned *Geometrical Exercises in Paper Folding *by title and commented that “the author shows how by paper folding we may construct by points such curves as the ellipse, cissoids, etc.” [Klein 1897, p. 42].

**Figure 14.** Klein’s reference to Row’s *Geometrical Exercises in Paper Folding.* Internet Archive.

Klein’s lectures struck a chord with at least two American readers. The first was David Eugene Smith (1860–1944) of the Michigan State Normal School in Ypsilanti. Smith published a review of Klein’s *Vorträge *in the November 1895 issue of *School Review*, a journal of secondary education associated with the University of Chicago. Smith enthused that the small book “deserves a prominent place in the library of every teacher of elementary geometry.” He went on to comment that:

It is very rare that a man of Professor Klein’s ability in the higher mathematics condescends to take up questions that are quasi-elementary, or, doing so, treats them in a manner at once scholarly and intelligible to those who are not his peers [Smith 1895].

Klein’s short book received a longer and equally enthusiastic review from Charlotte Angas Scott (1858–1931) of Bryn Mawr College, published in the *Bulletin of the American Mathematical Society* in March 1896*.* Like Smith—and like Klein himself—Scott emphasized the logical solution of problems in elementary geometry rather than intuitive proofs. Also like Smith, she made no reference to Sundara Row. After a detailed summary of Klein’s book, she suggested that an English-speaking association “would do a service in publishing a thoroughly good translation of this inspiring work and circulating it as widely as possible” [Scott 1896, p. 164]. The British journal *Nature* gave a summary of Scott’s review in its 9 April 1896 issue. Noting her suggestion that Klein’s pamphlet should be translated, they suggested that she undertake the task [Anonymous 1896].

[11] Edward M. Langley to David Eugene Smith, November 20, 1901, David Eugene Smith Professional Papers (DESPP), Rare Book and Manuscript Library, Columbia University, New York, New York.

[13] Miall claimed he was reproducing Langley’s list, but the books included are not identical.

[14] I thank Susan McElrath of American University for information about the volume in the Artemis Martin collection. For the problem posed to Artemis Martin and his readers by B. H. Rao, see *Mathematical Magazine* 1, no. 12 (October 1884): 220.

As it happened, Smith and Wooster Woodruff Beman (1850–1922) of the University of Michigan beat Scott to the task of translating Klein’s *Vorträge űber Ausgewählte Frägen den Elementargeometrie* into English. In 1895, the pair had coauthored a textbook on plane and solid geometry, following the guidelines suggested by the AIGT [Beman and Smith 1895]. Beman and Smith would go on to publish three further textbooks with Ginn. Their translation of Klein’s book, titled *Famous Problems of Elementary Geometry*, included Klein’s reference to Sundara Rao [Klein 1897].^{[16]}

**Figure 15.** David Eugene Smith and Wooster Woodruff Beman. MacTutor and University of Michigan.

Scott lauded Beman and Smith’s translation in the *Bulletin of the American Mathematical Society* [Scott 1898]. The book also received positive reviews in journals in related disciplines such as astronomy as well as in *Nature* [Anonymous 1898a; 1898b]. As late as 1914, Raymond Clare Archibald (1875–1955) of Brown University used the book as a starting point for a review paper on elementary geometry in *The American Mathematical Monthly* [Archibald 1914].

While these responses undoubtedly encouraged readers of Klein to also find out more about Rao’s book, a slightly earlier review may be more relevant to Beman and Smith’s decision to republish it in the United States. In November 1897, *The Open Court*, a journal edited by Paul Carus (1852–1919) of the Open Court Publishing Company of Chicago, published a short notice of *Geometrical Exercises*. The reviewer (writing under the Greek pseudonym μκρκ) thought that “both translators and publishers deserve the thanks of students for the reproduction of this delightful little book in English” [Anonymous 1897]. This positive review undoubtedly was known to Beman and Smith. More generally, they were aware that Open Court editor Thomas McCormack (1865–1932) actively sought new mathematical titles accessible to a relatively large number of readers.^{[17]} Hence, when they sought to publish an edition of *Geometrical Exercises in Paper Folding*, they turned to Open Court.^{[18]}

Beman and Smith had some difficulty acquiring a copy of Rao’s booklet. According to the preface to their edition of the book, to which they gave the slightly altered title *Geometric Exercises in Paper Folding*, they obtained a copy of the original only “after many vexatious delays.” However, by 1899 Beman had a copy in hand, and the two authors could proceed. Rao had thought his book might interest those intrigued by mathematical recreations and former geometry students, as well as those actively studying and teaching the subject. Beman and Smith envisioned a slightly narrower audience of “American teachers and students of geometry.” They were convinced that the book was “sure to be of interest to every wide-awake teacher of geometry from the graded school to the college. The methods are so novel and the results so easily reached that they cannot fail to awaken enthusiasm” [Row 1901, p. iv].

**Figure 16.** The title page from Beman and Smith’s republication of Row’s book. Internet Archive.

The alterations the American editors made to Rao’s text were slight. The word “geometrical” in the title became “geometric,” “equal” triangles became “congruent,” and “progressions” became “series.” Bazaars became stationery dealers, and a specific allusion to faculty at Madras University became a reference to university professors generally. Citations of specific theorems in Euclid were replaced by citations of the latest edition of Beman and Smith’s 1895 *Plane and Solid Geometry*. The most noticeable change was to the figures. Sundara Rao’s drawings, with black background and white lines, became either black lines on white paper or, more strikingly, halftone photographs of actual folded paper. Halftones were not commonly found in mathematics books of the time; their use here reflects Smith’s more general interest in the role of photography in teaching.

**Figure 17. **Beman and Smith’s presentation of the construction

for finding the sum of the geometric series \(\sum_{n=1}^{\infty} (\frac{1}{2})^n\). Internet Archive.

As one would expect from both the usual advertising practices of Open Court and the pedagogical slant of Rao’s book, teachers—especially geometry teachers—had ample opportunity to learn of the new publication. Benjamin Franklin Finkel (1865–1947), editor of *The American Mathematical Monthly,* described the contents of the volume in detail in the October 1901 issue of that journal. He concluded that “the work is one by which new interest may be awakened in the ever interesting and fascinating subject of geometry” [Finkel 1901]. E. M. Langley, who had reviewed the first edition of Rao’s book in the *Mathematical Gazette*, also noted Beman and Smith’s edition in that journal. He particularly liked the “beautifully executed photographic reproductions of paper folds.” Langley did have doubts about using paper folding as extensively as Rao proposed. As he wrote: “In our opinion the author pushes paper-folding far beyond the stage at which it is really helpful in school teaching, but perhaps an enthusiast for any particular way of doing things does good service in riding his hobby hard and thus attracting attention to its powers” [Langley 1902].

Some reviewers were less skeptical. University of Indiana mathematics professor and educator Robert Judson Aley (1863–1935), writing as editor of the Indianapolis publication *The Educator-Journal*, reported that “to the lover of mathematics the book is a never-ending source of pleasure and surprise; pleasure to be able to fold paper so as to prove many well-known propositions, surprise that he himself had not thought of doing it.” Aley thought the book would be of great value in leading students to the formal study of geometry and “of greater value of keeping them interested in the subject.” He commended the book to “every teacher of elementary mathematics” [Aley 1901].^{[19]} Similarly, *Dominicana: A Magazine of Catholic Literature*, a journal conducted by Dominican priests in San Francisco, thought that “this novel method of teaching geometry will recommend itself to progressive teachers” [Anonymous 1902a]. Mindful of Rao’s mention of kindergarten materials, *Kindergarten Magazine* also reviewed the book. The anonymous commentator thought that the book would not only appeal to teachers of geometry but “will naturally attract the attention of the kindergartener [e.g. the kindergarten teacher] who has already folded the simple geometric forms, and who will readily perceive how naturally the child of kindergarten training would follow this method of proving geometrical theorems” [Anonymous 1902c].

A few more general scientific and engineering journals also mentioned *Geometric **Exercises in Paper Folding*. *Scientific American *listed it among new books on 8 February 1902, echoing Smith and Beman’s comment on the value of paper folding to teachers and students of geometry [Anonymous 1902b]. The *Journal of the Western Society of Engineers* in Chicago commented:

It is surprising to one who has studied the geometry of our schools of (say) the last generation, to find what can be accomplished by such simple means as scraps of paper and a knife as a supplement and practical ocular demonstration of the relations between many geometric problems as laid down in the standard textbooks [W. 1901].

The reviewer not only thought that the book would be of “great assistance” to mathematics teachers, but that it discussed a remarkable range of problems.

Yet another engineer, Princeton University professor of graphics and engineering drawing Frederick Newton Willson (1855–1939), reviewed *Geometric Exercises in Paper Folding* for *Science* magazine. Willson summarized the contents of the book in considerable detail, finding it “a serviceable condensation of mathematical properties, theorems, puzzles, and problems.” He did have doubts that a student familiar with logarithms and positional geometry, as presented in later chapters of the book, would “often stop to obtain his actual results by folding.” As a professor of drawing, he also thought that use of a beveled, nickel-plated steel rule would improve the accuracy of folding. Noting the number of references to works by Beman and Smith, Willson felt led to ask “how far permission to edit carries with it advertising privileges.” Such quibbles aside, Willson found that “Where the claim of the author is so modest and his aim in so high degree attained, the task of criticism is a light one” [Willson 1902].

Finally, mathematician Virgil Snyder (1869–1950) of Cornell University also weighed in on Rao’s book. His review in the *Journal of Physical Chemistry* gave a concise summary of the contents and suggested that geometry teachers would find it of value “as it shows in a forcible and tangible way how properties vaguely known to us by experience are logical and necessary consequences of a few definitions” [Snyder 1902].

[16] For a more extensive listing of Smith’s publications, which mentions his textbooks in passing, see Frick [1936]. A listing that includes textbooks is Ginsburg [1926]. For Klein’s reaction to Smith’s suggestion that he publish a translation of Klein’s book, see the postcard from Felix Klein to David Eugene Smith, November 28, 1895, DESPP.

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.

Are you interested in sharing Rao’s exercises with your students? Here are a few activity starters drawn from his book.

One activity of interest for a college-level geometry course is the side-by-side comparison of Rao’s approach to certain geometric constructions with those found in Euclid’s *Elements. *For example:

- In Chapter III, paragraphs 8–17 (pp. 11–14) (pdf), Rao described a paper-folding construction for dividing a segment in mean and extreme ratio, and some interesting algebraic consequences using the golden ratio. This construction is somewhat different from what appears in Proposition 11 of Book II of Euclid’s
*Elements*. - In Chapter IV, paragraphs 1–9 (pp. 20–23) (pdf), Rao described a paper-folding version of constructing a regular pentagon, along with some follow-up algebra computing the angles and lengths of sides and diagonals thereof. This time, there are a lot more parallels with Euclid’s construction in Book IV, Proposition 11. The main difference is that Rao inscribed a pentagon in a square, whereas Euclid constructed one inscribed in a circle.

Parts of Rao’s book suggest other interesting connections to Greek mathematics that might be explored in a hands-on way within a history of mathematics course. For instance, in Chapter X, paragraphs 1–14 (pp. 39–42) (pdf), Rao dealt with arithmetic and geometric progressions, and he gave some intuitive derivations of sums, at least for the arithmetic progression. He also went into extended proportions, including cube roots, and gave a brief discussion of a (non-Euclidean) measuring device for finding double mean proportions in connection with doubling the cube. The discussion, although brief, is very much along the lines of what appears in Wilbur Knorr’s *The Ancient Tradition of Geometric Problems* [1993, pp. 57–59].

Several of Rao’s constructions could also be used as the basis for hands-on explorations of specific mathematical topics at the precalculus level. For example,

- Rao’s “proof by folding” of the infinite geometric sum \(\sum_{n=1}^{\infty} (\frac{1}{2})^n = 1\), beginning with a paper-folding construction of a square, in Beman and Smith’s Chapter I, paragraphs 16–19 (pp. 6–8) (pdf). For this construction, students could shade the regions of the square that represent the various powers of \(\frac{1}{2}\) in different colors in order to make the underlying scheme of the proof more visible.
- Rao’s “folding and pricking” construction of the parabola and its tangent lines, beginning with the analytic definition of a parabola as the locus of points that are equidistant from a given straight line and a given fixed point in Beman and Smith’s Chapter XIII, paragraphs 235–237 (pp. 116–117) (pdf). For this construction, instructors might consider using wax paper, since this renders the tangent lines at each point sufficiently well-pronounced that the shape of the curve can emerge without the need to actually prick the paper in order to obtain the points on the curve.
- Rao’s “folding and pricking” construction of the sine curve, beginning with the construction of a (finite) sequence of right triangles followed by transfer (again, by folding) of the heights of those triangles to obtain the ordinates of points on the curve in Beman and Smith’s Chapter XIV, paragraph 271 (pp. 137–139) (pdf). Rao also called this curve “the harmonic curve” and associated it with the amplitude of a sound wave.

For instructors who work with secondary mathematics teachers (either in a preservice course or an in-service workshop), these and similar activities could be used to launch a discussion about the mathematics involved in the constructions, as well as the use of paper folding as a learning aid more generally. Discussion prompts specific to the particular constructions listed above might include:

- How does Rao’s paper-folding proof for the geometric series \(\sum_{n=1}^{\infty} (\frac{1}{2})^n\) compare with an algebraic approach to that proof? Which do you think most students would find easier to follow? Which do you think most students would find more convincing?
- How does the paper-folding construction of a parabola compare with its construction via point-plotting on graph paper beginning from the equation \(y = ax^2\)? What do students need to know (about parabolas or more generally) in order to carry out each of these constructions? What can they learn (about parabolas or more generally) from either of these constructions? What, if anything, can they learn (about parabolas or more generally) from one of these constructions that would be difficult to learn from the other?
- How does the paper-folding construction of the sine curve compare to your usual method of introducing that curve to students? Do you think the use of right triangles in Rao’s construction of the sine curve would help students to make the connection between right triangle trigonometry and the curve itself? Is it possible to modify the construction in order to construct the cosine curve?

The pedagogical discussion of the advantages and disadvantages of bringing paper folding in the classroom could then culminate by having the participants consider the following quotation from Rao’s book and determine the extent to which they agree or disagree with it.

It would be perfectly legitimate to require pupils to fold the diagrams with paper. This would give them neat and accurate figures, and impress the truth of the propositions on their minds. It would not be necessary to take any statement on trust. But what is now realised by the imagination and idealization of clumsy figures can be seen in the concrete [Rao 1893, p. ii].

A number of disparate trends in the history of mathematics education came together in the small book by Sundara Rao. Inspired by Froebel’s kindergarten occupations, he considered how paper folding might be used to teach geometry. *Geometrical Exercises in Paper Folding*, published by Addison & Company in Madras in 1893, was noted in the *Mathematical Gazette* in 1894 and commended in an 1894 letter to *Nature *by mathematics teacher and founding editor of the *Gazette* Edward M. Langley. Perhaps through these references, the work came to the attention of the distinguished German mathematician Felix Klein, who mentioned it in one of his works. This reference in turn inspired two American mathematicians, David Eugene Smith and Wooster Woodruff Beman, to publish an American edition of Rao’s book, enhancing the presentation by adding photographic illustrations. Thus, Rao himself responded to various components of 19th-century efforts to reform the teaching of geometry, and the reception of his book threaded through a number of the organizations, publications, and individuals associated with those reform efforts.

Furthermore, the apparent influences on Rao and the subsequent discussions of *Geometrical Exercises* all affirmed that paper folding was an appropriate mathematical activity for a broad age range of students. Recall that Rao wrote in his preface:

I have sought not only to aid the teaching of Geometry in schools and colleges, but also to afford mathematical recreation to young and old, in an attractive and cheap form [Row 1893, p. vi].

The story of recreational paper folding additionally involved an expansive geographical area— spanning Germany, Great Britain, India, and the United States—and was shaped by the shifting geopolitical relationships between these nations in the late 19th century, most notably the British colonial administration of India that employed Rao, exposed him to British approaches to geometry teaching, and opened up markets for his book . . . but presumably also constrained his career in ways that are impossible to measure more than a century later. Nonetheless, and even though its direct pedagogical impact was limited, *Geometrical Exercises* found an enthusiastic audience among those engaged in mathematical recreations. It remains in print to this day.

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The author thanks library, museum, and mathematical colleagues across the globe for their help in assembling disparate sources. She is especially grateful to *Convergence *reviewers and editors for useful improvements to the historical account and for adding suggestions of possible classroom activities. Both the author and the editors thank Chuck Lindsey (Florida Gulf Coast University) for identifying specific examples in Rao's book that could serve as the basis of classroom activities drawing connections between Rao's paper folding constructions and ancient Greek mathematics.

Peggy Aldrich Kidwell is Curator of Mathematics at the Smithsonian’s National Museum of American History (NMAH). She became aware of the role of paper folding in geometry teaching as a result of examining hundreds of models in the NMAH collections, and she has also enjoyed seeing models folded by interns and museum staff.