Vedic Geometry

            Arya Samaj

            Arya Samaj was founded in India in 1875 by Swami Dayanand Saraswati as a movement to spread the Vedic knowledge.  Towards this goal, it started a number of educational institutions called gurukuls.  No good deed goes unpunished.  The founder was assassinated in 1983 (https://en.wikipedia.org/wiki/Arya_Samaj).  However, Arya Samaj spread voraciously, not just in India but throughout the world. Educations institutions were established – with names often starting with DAV (Dayanand Arya Vir). Initially as an author of this Math story, I just created a fictitious DAV Sunday School in an imaginary town in USA.  My research subsequently showed how widespread  Arya Samaj was in USA and other western countries.  However, the story still remains imaginary as written originally.

            Raju Sahu

            Raju Sahu was a 12 year old boy living in the Texas town of Timbell.  He came here with his parents when he was 2 year old because his father and mother got decent jobs  here.  Raju excelled in his studies and was liked by everyone in his school and in the neighborhood.  He  also played baseball with his friends.

            Raju Attended  A Sunday  School

            Raju’s parents wanted to make sure that he did not forget his Indian culture – not the Bollywood kind but the ancient Vedic culture.  The town had a small Sunday school called DAV Sunday School run by the local Arya Samaj members, and Raju was sent there every week.  Typically the Sunday school session lasted about two hours.  There would be a half an hour lecture by a some one, an informal session and a yagya which was called havan.   Raju was impressed that the speakers would have a question-answer session at the end.  It was called Shankasamadhan (reconciliation of doubts).  This was different from the sermons at the Hindu temple or those at the church where you were not allowed to question the preacher.

            Sulbha Sutras – the rope principles 

            This Sunday, it was a presentation by Miss Gyan Devi on the practice of yagya in Vedic brahmin households.  A typical household would have a Tretha agni yagya equipment well designed using rope principles (Sulbha sutras).  It consisted of three vessels which were joined together. These were an ancestral vessel (Garhapatya), a vessel called Ahavaniya and another one called Daksina.   One of the vessels was circular, one square shaped and the third one formed a semicircle. A constraint was that all the vessels must have the same areas.

            Experts would use altars of different shapes in yagyas for diverse purposes.  Some   examples are shown in this table.

Altar TypeShapePurpose
SheyanaFalconObtain prosperity
KantaCraneObtain honor
AlajaAlaja birdObtain Authority
With PraugaTriangleDestroy enemies

            These altars were made of square, right angle triangles and circular bricks to make intricate designs.  This meant that the geometry had to be well developed even though there were no compasses.   The tools available were sticks, ropes and sesame seeds for length measurements: 34 of them in a row formed the unit angula.

            Shankasamadhan

            Raju was almost mesmerized with the presentation but had several questions.

            Raju: Miss, approximately what time period in the history was this ?

            Miss Gyan Devi: Vedic period was from 1500 – 500 BCE but the texts containing Sulbha Sutra were part of Baudhayana Sutras in 800-600 BCE.

            Raju:  How could they do all this without knowing the Pythagoras theorem or the value of the constant pi ?

            Miss Gyan Devi: Pythagoras gave this theorem around 500 BCE but it was known to the Aryans well before that.  Here is the mantra described it

दीर्घचतुरश्रस्याक्ष्णया रज्जु: पार्श्र्वमानी तिर्यग् मानी च यत् पृथग् भूते कुरूतस्तदुभयं करोति ॥ dīrghachatursrasyākṣaṇayā rajjuḥ pārśvamānī, tiryagmānī, cha yatpṛthagbhūte kurutastadubhayāṅ karoti.  (A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together.)

            You and I will discuss how they would determine a circle with the same area as a square and then you can present it to the class next Sunday.

              Raju was happy with the answer.  Afterwards, he met with Miss Gyan Devi who gave him some references (https://www.youtube.com/watch?v=s723-3hkUjA, https://en.wikipedia.org/wiki/Baudhayana_sutras),  and he prepared the following presentation on Sulbhasutra, squares and circles.

            Raju’s Presentation

            Raju: Make a square ABCD.  The objective is to construct a circle with the same area as this square using Sulbha Sutras.

            Draw the lines AC and BD as diagonals of the square and call their point of intersection E.  Draw a circle with the center E and half the diagonal length as the radius (see Figure).  From E, draw a vertical line that meets the square line CD at X and the circle at Y.  On EY mark the point Z such that YZ = 2 ZX.  Draw a circle with the center E and radius EZ.   This circle would have the same area as the square ABCD.

            Miss Gyan Devi: Raju, now with the help of your knowledge of geometry today, prove that the circle with the radius EZ has the same area as the square ABCD.

            Raju:  Let us say the length of each side of the square is 2a.  Then its area would be 4a2.  The half diagonal of the square would have a length of a√2 or 1.4142a.  Then length EY = 1.4142a and EX = a.  Therefore length of EZ = (1 + 0.41142/3)a = 1.1381a.  The area of a circle of radius 1.1381a = π (1.1381a)2 = 4.069 a2 compared to the area of the square which is 4a2.

            Venu (another student): So the Vedic answer is only an approximation.

            Raju: Yes, but remember so is the value of π. You might know it to hundred thousand decimal places but still it is not exact.   By the way, in these calculations if the area of the circle were to be 4 and radius to be 1.1381, we would say that the  value of pi would be 3.088 as compared to 3.1416 as we know it to be today.  Remember, these calculations from almost 2600 years ago.

            Challenge

            Extend the Vedic geometry Sulbha Sutra method to draw a semicircle with the same area as a square.

            Solution:  From the square ABCD, draw a square ACUV using the diagonal AC as one of its sides.  If the area of ABCD is 4a2, ACUV will have an area of (a2√2)2 or 8a2.  Now using Raju’s method for the square ACUV will give a circle with an area of approximately 8a2 or a semicircle of the area 4a2.