Ancient Indian Rope Geometry in the Classroom - Fire Altars of Ancient India

Ritual was an extremely important part of the ancient Hindu religion. Fire altars were built for both rituals that occurred at regularly scheduled times throughout the year, as well as for special rituals. The special rituals would involve requests to gain benefits or favors such as wealth, and the shape of the fire altar depended on the favor being requested. The altars were temporary and destroyed once the ritual was completed. For example, the Agnicayana ritual called for a bird-shaped altar constructed out of 1005 bricks in homage to the god Agni. The ritual took 12 days to perform. The purpose of the ritual was to build an immortal body that would transcend suffering and death, both hallmarks of mortal existence [Converse]. The table below shows the various shapes of the fire altars, along with the favor being requested, and the source within the Śulba-sūtras for either information about and/or instructions for building that particular fire altar. The sources are given in the table by using the first letter of each of the four Śulba-sūtras followed by the appropriate verses. The English translation of the Śulba-sūtras by Sukumar N. Sen and Amulya Kumar Bag [Sen & Bag] was used to make the table.

According to Plofker [Plofker2, p. 17],

Many of the altar shapes involved simple symmetrical figures such as squares and rectangles, triangles, trapezia, rhomboids, and circles. Frequently, one such shape was required to be transformed into a different one of the same size. Hence the Śulba-sūtra rules often involve what we would call area–preserving transformations of plane figures, and thus include the earliest known Indian versions of certain geometric formulas and constants.

A 12-day fire altar sacrifice ritual was filmed by scholars in 1975 and made into a documentary called Altar of Fire. For more information on the documentary and to view a 9-minute preview that shows the altar bricks being made and a measuring stick being used, visit the Documentary Educational Resources website. The entire 58-minute documentary is available for purchase at the website.

The first step in any fire altar construction was to lay out the cardinal directions, especially the East-West line. The East-West line had special significance in the construction of the Vedic fire altar. Indeed, on the East-West line are two altars. The altar at the eastern end is square, and contains the Āhavanīya fire, symbolizing the celestial world, heaven. The one at the western end is circular and contains the Gārhapatya fire, symbolizing the terrestrial world. There is a third fire in the southern direction as well, the Dakṣiṇāgni, which symbolizes the air world. See [Kramrisch & Burnier] for further discussion. The GeoGebra applet below moves, one step at a time, through these instructions for laying out the cardinal directions from the Kātyāyana-śulba-sūtra. Click on “Go” to proceed to the next step.

Figure 3. This applet outlines the construction of the East-West line as described in the Kātyāyana-śulba-sūtra, which has special significance in the construction of Vedic fire altars. Note the translation of the original text in quotes. Click "Go" to advance to the next step.

Also, at the beginning of the Baudhāyana-śulba-sūtra, instructions are given for using a measuring cord to construct a square, providing another construction for laying out the cardinal directions. See Activities 2 and 3 on the Student Activities page of this article for indoor and outdoor classroom activities that model this ancient Indian way of constructing a square. The picture below shows the result of Activity 2. Note that there are other methods for constructing a square in the Śulba-sūtras in addition to the two just given, which are not included in this article. For example, one involves Pythagorean triples and is similar to the construction of the Great Altar which follows.

Figure 4. The result of constructing a square with a measuring cord, using an ancient Indian procedure explained in Activity 2, except for the very last step of connecting the four corners (dots) with line segments to form the square.

As mentioned above, a related construction is that of the so-called Great Altar, which is in the shape of an isosceles trapezoid with its altitude parallel to the East-West line, and its longer base facing West. Its bases have lengths 30 paces and 24 paces, and its altitude has length 36 paces. The Great Altar was used in rituals involving the Vedic ceremonial beverage soma [Plofker2, p. 25]. The converse of the Pythagorean Theorem is implicitly used in the construction, in which a mark is made 15 units from an end of a 54 unit rope. If we attach the ends of the 54 unit rope to stakes in the ground 36 units apart, and pull on the mark until the rope is taut, then the resulting triangle has side lengths \(36,\) \(15,\) and \(54-15=39\) units. But then the converse of the Pythagorean Theorem forces the triangle to be a right one. The GeoGebra applet below presents the construction described in the Śulba-sūtra of Āpastamba, and a similar construction also appears in that of Baudhāyana. The applet moves through these instructions one step at a time. Click on “Go” to proceed to the next step.

Figure 5. This applet outlines the construction of the Great Altar as described in the Śulba-sūtra of Āpastamba. Note the translation of the original text in quotes. Click "Go" to advance to the next step.

Āpastamba also described the classical construction of transforming an isosceles trapezoid into a rectangle of equal area, and thus calculated the area enclosed by the Great Altar. That is, cut off a right triangle with leg lengths 3 and 36 units from the northern edge of the trapezoid, and glue it to the southern edge, so as to obtain a rectangle. This yields a rectangle with side lengths 36 and 27 units, and hence area of 972 square units. Below is a translation of the original text from the Śulba-sūtra of Āpastamba, Section 5.7 [Plofker2, p. 26]:

The Great Altar is a thousand [square] paces [or double-paces] less twenty-eight. One should bring [a line] from the south[east] corner twelve units toward the south[west] corner. One should place the cut-off [triangle] upside-down on the other [side]. That is an oblong quadrilateral. In that way one should consider it established.