Aiding the Teaching of Geometry and Affording Mathematical Recreation: Paper Folding in the Spirit of Sundara Rao of Madras – Classroom Activities

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].