More Classroom Activities Based on Ancient Indian Rope Geometry

Introduction

One source of mathematics from ancient India is a group of texts known as the Śulba-sūtras, which were written between 800 and 350 BCE (a timeline can be found at the MacTutor History of Mathematics Archive). These works contained instructions for building a variety of altars for fire sacrifices. The name of the texts, Śulba-sūtra, translates as “rules of the cord.” The four best known Śulba-sūtras, named after their authors and listed in order of importance, are Baudhāyana-śulba-sūtra, Āpastamba-śulba-sūtra, Kātyāyana-śulba-sūtra, and Mānava-śulba-sūtra (for more information on the history of the Śulba-sūtras and Vedic rituals, see [Plofker1], [Plofker2], and [Prasoon]). These fire altar manuals, with instructions for laying out altars of fixed area in particular shapes and then filling them in with a fixed number of bricks of various shapes, necessarily contained much mathematics. Mathematical ideas included measurement, constructing right angles, transforming from one geometric shape to another such as square to rectangle or trapezoid, approximating irrational numbers like \(\sqrt 2\) and \(\pi,\) working with fractions, and an equivalent of the Pythagorean theorem prior to Pythagoras.

Several GeoGebra applets and other classroom activities based on the Śulba-sūtras can be found in a previous article by the authors, "Ancient Indian Rope Geometry in the Classroom" [Huffman and Thuong], here in Convergence. The current article is a continuation with additional GeoGebra applets and activities based on the translation found in [Sen and Bag] of the Baudhāyana-śulba-sūtra, which we will abbreviate BSS. We follow the labelling and notation of the Sen and Bag translation. For example, BSS 1.6 will refer to Stanza 1, Verse 6. Words in parentheses in the translations throughout the article and applets have been added by the translator to aid the reader. It should be noted that the original text is very terse with no diagrams or verification of the correctness of the mathematics.

The Baudhāyana-śulba-sūtra is the most complete of the Śulba-sūtras, and is also considered by many to be the most important. It begins with a brief description of some units of measurement before providing two different methods for constructing a square. The square is the basis for the fire altar constructions. The first square construction method in the BSS incorporates constructing a perpendicular bisector, while the second method uses the converse of the Pythagorean Theorem to form right angles (see Figure 1). In their previous article, the authors have provided an activity for elementary and middle school students that involves measurement using three units of length from ancient India, as well as both indoor and outdoor student activities (see "Ancient Indian Rope Geometry in the Classroom – Student Activities") based on Baudhāyana’s first method of constructing a square. Indoor and outdoor student activities using the second method in BSS are included at the end of this article.

Figure 1. A 3-4-5 triangle is used to form a right angle in the authors' second method of constructing a square.

After the methods for constructing squares, the Baudhāyana-śulba-sūtra (BSS) provides methods for constructing rectangles, and for transforming squares into various shapes. GeoGebra applets for adding and subtracting squares, transforming a rectangle into a square of equal area, and approximating the square root of two are provided in the authors' "Ancient Indian Rope Geometry in the Classroom" [Huffman and Thuong] here in Convergence. The present article contains GeoGebra applets for two constructions and four more transformations from BSS. The two constructions are for a rectangle (BSS 1.6), and for a square (BSS 1.8). The four transformations are square to rectangle (BSS 2.3–2.4), rectangle to trapezoid (BSS 2.6), rectangle to triangle (BSS 2.7), and square to circle (BSS 2.9). Proofs are provided for each applet, but students could be shown the applets only and asked to prove themselves that the construction or transformation actually yields the desired shape.