Ancient Indian Rope Geometry in the Classroom - Mathematics in Ancient India

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[h=1]Ancient Indian Rope Geometry in the Classroom -[/h]Ancient Indian Rope Geometry in the Classroom Ancient Indian Rope Geometry in the Classroom - Mathematics in Ancient India ›

Author(s):
Cynthia J. Huffman (Pittsburg State University) and Scott V. Thuong (Pittsburg State University)



[h=3]Introduction[/h]

Whether intentional or not, mathematics permeated many aspects of life for various ancient cultures, including religious aspects. For example, the Pythagoreans, a semi-religious secretive ancient Greek community, believed All is number. In order to build the religious temples and pyramids in ancient Egypt, the engineers and architects needed a working knowledge of basic geometry. In ancient China, mathematics was used in calendar development for knowing when to celebrate religious events. And in ancient India, geometry was used in constructing various religious altars.​

It is the latter example that we focus on in this article. More specifically, we take a look at ancient Indian rope geometry used in the construction of altars for different fire sacrifices. GeoGebra applets are included to illustrate the ancient Indian rope geometry, as well as to allow the reader to explore. The article concludes with a collection of related activities that can be used in the classroom.​

Figure 1. An Agnicayana fire sacrifice ritual in 2011 in Panjal, Kerala. This ritual, taking 12 days to perform, calls for a bird-shaped altar constructed out of 1005 bricks in homage to the god Agni. (Photo courtesy of Professor Michio Yano.)​

Cynthia J. Huffman (Pittsburg State University) and Scott V. Thuong (Pittsburg State University), "Ancient Indian Rope Geometry in the Classroom," Convergence (October 2015)

There are difficulties encountered when one studies the history of mathematics in ancient India. Some of this is due to gaps in original source material. There are also discrepancies among secondary and tertiary sources. Additionally, at one point in the West, the history of Indian mathematics was often overlooked in favor of a more Eurocentric view. More recently, there have been efforts to correct this neglect with the publication of popular books for mainstream audiences like The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph [Joseph], and more reliable specialized research by scholars such as Kim Plofker, author of Mathematics in India [Plofker2] and a chapter on Indian mathematics in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, edited by Victor J. Katz [Plofker1]. For more information on the debate about Eurocentrism in the study of Indian mathematics, see Clemency Montelle’s review [Montelle] of The Crest of the Peacock.
​

The earliest evidence of ancient Indian mathematics dates back to 3000 BCE with the Harappan culture. [Joseph, p. 217] The earliest texts, dated during the second millennium BCE, are the Vedas, written in a form of Sanskrit. The word “Veda” translates as “knowledge.” In addition to hymns, these texts contain procedures for conducting religious rituals. [Plofker2, p. 5 and Plofker1, p. 385] Elements such as fire and water, as well as gods such as Agni (of fire) and Indra (of rain and thunderstorms) were worshiped in the religion of the Vedic period. The religious rituals involved the recitation of hymns and sacrifices. Priests performed these rituals for both nobles and wealthy commoners with the goal of enhancing fertility, wealth, and an afterlife shared with their ancestors. This style of worship is also seen today in the Hindu religion. There are a vast number of Vedic rituals, we mention only several. See the recent book by Shrikant Prasoon [Prasoon] for further discussion.​

• The Agnistoma ritual involving the ceremonial beverage soma. The beverage soma was made by juicing the stalks of a certain plant, although the identity of this plant remains unclear, and is frequently speculated about. Soma imbued the drinker with immortality. For example, the gods Agni and Indra were said to drink soma.
• Rituals involving the construction of sacrificial fire altars. Examples include the Agnihotra, an oblation to Agni, as well as the Agnicayana.
• Healing rituals found in the Atharvaveda, a Veda on medicine.
• The Ashvamedha ritual for the prosperity of a nation.
• The Rajasuya, a ritual performed by a powerful king after a successful military campaign.

Figure 2. Boys working on a model of the bird-shaped fire altar in an Agnicayana ritual in 2011 in Panjal, Kerala. (Photo courtesy of Professor Michio Yano.)

As mentioned above, the focus of this article is on the construction of sacrificial fire altars in ancient India by means of rope geometry. Cryptic instructions for building these fire altars using measuring cords were given in the Vedic texts called Śulba-sūtras, which translates as “Rules of the cord.” According to Plofker [Plofker2, p. 16-18], there were four main Śulba-sūtras which are known by their authors’ names:​

• Baudhāyana-śulba-sūtra, from the Middle Vedic period, about 800-500 BCE,
• Mānava-śulba-sūtra, also from the Middle Vedic period, about 800-500 BCE,
• Āpastamba-śulba-sūtra, from a time after the previous two but before the following,
• Kātyāyana-śulba-sūtra, possibly from the mid-fourth century BCE.

The dating of these writings is not certain, but is based on comparison of the style and grammar used in these texts with other texts. For more information on the Śulba-sūtras, see Mathematics in India by Kim Plofker [Plofker2] starting on page 16, and also page 387 and following in Chapter 4 of The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, edited by Victor J. Katz [Plofker1].​

Cynthia J. Huffman (Pittsburg State University) and Scott V. Thuong (Pittsburg State University), "Ancient Indian Rope Geometry in the Classroom - Mathematics in Ancient India," Convergence (October 2015)

 

[vgane]

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

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Author(s):
Cynthia J. Huffman (Pittsburg State University) and Scott V. Thuong (Pittsburg State University)


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.

Shape		Request		Source
Falcon		For those desiring heaven		B8, A15-A17, M14
Falcon with curved wings and extended tail		For those desiring heaven		B10-B11, A18-A20
Kite shape				B12
Alaja bird				B13, M14
Rhombus		Destroy existing and future enemies		B15, A12.7-A12.8, K4.4, M15.4
Chariot wheel		To destroy enemies		B16, A12.9-A13.3, K15.14-15.18, K16
Trough		For those desiring food		B17, A13.4-A13.16, K4.2, M15.6
Circle				B18
Pyre		For those desiring prosperity in the abode of the Fathers		B19, A14.7-A14.10, M15.6
Tortoise		Win the world of the Supreme Spirit		B20
Tortoise with rounded limbs		Win the world of the Supreme Spirit		B21
Bird		Wealth		A8-A10
Isosceles triangle		For those with many foes		A12.4-A12.6, K4.3, M15.3

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 [FONT=MathJax_Main]36[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]15[/FONT][FONT=MathJax_Main],[/FONT] and [FONT=MathJax_Main]54[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]15[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]39[/FONT] 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.​

Cynthia J. Huffman (Pittsburg State University) and Scott V. Thuong (Pittsburg State University), "Ancient Indian Rope Geometry in the Classroom - Fire Altars of Ancient India," Convergence (October 2015)

http://www.maa.org/press/periodical...in-the-classroom-fire-altars-of-ancient-india

 

[vgane]

You have have a look at the detailed one by going through the link...This is not to tom-tom about our past but to throw light about how science and mathematics was intertwined in Vedic life

 

[tks]

Informative links.. Thanks
Thumps up

 

[raysundar]

The terms research (or can be interpreted as search again) and discover (or uncover what is already known or hidden from view) are both indicative that whatever science we know today was already known in ancient times. It is no surprise that we are now using Vedic knowledge as the basis for our science or vice versa being amazed that all the science of today is already in one form or the other in ancient Vedic texts or even other Greek texts. Our ancestors were all intelligent people. The Ramayana and Mahabharatha as well other Greek Epics are a testament to great civilizations that flourished in the past. Their DNA still persists in our human race including the Neanderthal DNA which seems to be prevalent among some war mongering groups of people today. All the miracles that are mentioned in ancient scriptures of all religions will be repeated in the future through the constant research and discovery and Ramayana and Mahabharatha will be reenacted once again on this Earth.