IN THE SULBA SÛTRAS
Name: John Price, Mathematics and Finance (b. Melbourne, Australia., 1943).
Address: Department of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia.
Fields of interest: Symmetry principles in finance, symmetry principles in physics (Vedic literature, Transcendental Meditation, teaching investment courses).
*Applied Geometry of the Sulba Sutras (in Geometry at Work, C.A. Gorini, Ed., MAA Notes, Washington, 2000, pp.46-55).
*Foreign Exchange Option Symmetry (with V.A. Kholodnyi) World Scientific, New Jersey, 1998
*Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl
inequality, (with M.G. Cowling) SIAM Journal of Mathematical Analysis 15
The Sulba Sûtras, part of the Vedic literature of India, describe
many geometrical properties and constructions such as the classical “Pythagorean”
between the sides of a right-angle triangle and arithmetical
formulas such as calculating the square root of two accurate to five decimal
places. Beyond these constructions, there are deep and elegant symmetries
such as a pair of formulas for converting a square to a circle and vice
versa. Also there are beautiful blueprints for constructing citis or ceremonial
platforms in images such as falcons, wheels and tortoises. At this level
the Sutras seem to be a finely crafted manual for expert artisans. Certain
key words, however, suggest that the applications go far beyond.
1. SULBA SÛTRAS
The Sulba Sûtras are part of the Vedic literature, an enormous body of work consisting of thousands of books covering hundreds of thousands of pages. Maharishi Mahesh Yogi explains that the Vedic literature is in forty parts consisting of the four Vedas plus six sections of six parts each. These sections are the Vedas, the Vedângas, the Upângas, the Upa-Vedas, the Brâhmanas, and the Prâtishâkhyas. Each of these parts has been described as by Maharishi as "expressing a specific quality of consciousness". This means that often we have to look beyond the surface meanings of many of the texts to find their deeper significance.
The Sulba Sûtras form part of the Kalpa Sûtras which in turn are a part of the Vedângas. There are four main Sulba Sûtras, the Baudhayâna, the Âpastamba, the Mânava, and the Kâtyâyana, and a number of smaller ones. The general formats of the main Sulba Sûtras are the same; each starts with sections on geometrical and arithmetical constructions and ends with details of how to build citis which, for the moment, we interpret as ceremonial platforms or altars. One of the meanings of Sulba is “string, cord or rope.” This could be a reference to the fact that measurements for the geometrical constructions are performed by drawing arcs with different radii and centres using a cord or sulba.
Three outstanding features are the wholeness and symmetry
of their geometrical results and constructions, the elegance and beauty
of the citis, and the indication that the Sutras have a much deeper purpose.
1.1 Wholeness and Symmetry
When each of the main Sulba Sûtras is viewed as a whole, instead of a collection of parts, then a striking level of symmetry and efficiency becomes apparent. There are exactly the right geometrical constructions to the precise degree of accuracy necessary for the artisans to build the citis. Nothing is redundant. For example, the units of measurement used could supply accuracy for the diagonal of one of the main bricks of "roughly one-thousandth of the thickness of a sesame seed."
There is also a remarkable degree of internal symmetry
such as the way that the ‘square to circle,’ ‘circle to square’ and ‘square
root of two’ constructions fit together with an accuracy of 0.0003%.
1.2 Beauty of the Citis
Each of the citis are low platforms consisting of layers
of carefully shaped and arranged bricks. Some are quite simple shapes such
as a square or a rhombus while others are much more involved such as a
falcon in flight with curved wings, a chariot wheel complete with spokes,
or a tortoise with extended head and legs. These latter designs are particularly
beautiful and elegant depictions of powerful and archetypal symbols, the
falcon as the great bird that can soar to heaven, the wheel as the ‘wheel
of life,’ and the tortoise as the representative of stability and perseverance.
1.3 Deeper Significance
Sanskrit is a rich language full of subtle nuances. Words can have quite different meanings because of their context and, in any case, frequently there is no reasonable English equivalent. We shall examine examples such as citi, vedi and purusa.
When these and other examples are combined with the general
direction of all the Vedic literature towards describing "qualities of
consciousness," we are led to the conclusion
that the Sulba Sûtras are describing something far beyond procedures
for building brick platforms, no matter how far-reaching their purpose.
2. Geometrical Symmetry
Most of the geometric procedures
described in the Sulba Sûtras start with the laying out of a prâcî
which is a line in the east-west direction. This line is then incorporated
into the final geometric objects or constructions, generally as a centre
line or line of symmetry. Some of the constructions include methods for
determining the length of the diagonal of a right-angled triangle (referred
to in the west as Pythagoras’s theorem), constructing a square equal to
the sum of two other squares and constructing a square equal to the difference
of two squares. We outline the procedure in
the Baudhâyana Sulba BSS for converting a square into a circle and
a circle into a square.
2.1 Converting a square into a circle
Verse I, 58 of the BSS describes the procedure for constructing a circle with area approximately equal to that of a given square. A pictorial outline of the procedure is given in Figure 1. In modern mathematical terms, the sûtra asserts that the radius r of a circle with area equal to the area of a square with side b is
r = b ´ (2 + Ö 2) / 3.
2.2 Converting a circle into a square
Verse I, 59 of the BSS states:
If you wish to turn a circle into a square, divide the diameter into eight parts and one of these eight parts into twenty-nine parts; of these twenty-nine parts remove twenty-eight and moreover the sixth part (of the one part left) less the eighth part (of the sixth part).
d - d/8 + d / (8 ´ 29) - d / (8 ´ 29) ((1 / 6 - 1/ (6 ´ 8)) = 9785 ´ d / 11,136.
2.3 Symmetry of the square-circle constructions
Even though each of the previous constructions have a considerable degree of accuracy (around 1.7 %), certainly sufficient for a construction of bricks and mortar, it when the two constructions are combined that we see something quite remarkable.
Suppose we start with a square of side 2 and hence area
4. By using the above procedure we would convert this to a circle with
area approximately equal to 4.069. Now converting this back to a square
using the square-to-circle construction, we arrive at a square with side
2.00006, giving an accuracy of 0.0003%.
3. Applications to the construction of citis
In the second part I will describe the applications of the geometrical and arithmetical constructions of the Sulba Sûtras to the construction of citis. Each of the citis is constructed from five layers of bricks, the first, third and fifth layers being of the same design, as are the second and fourth.
A lengthy sequence of units is used; the two that are referred to the most are the angula and the purusa. They are approximately 2 cm and 240 cm with 120 angulas equal to a purusa.
The heights of each layer are 6.4 angulas which is about 13 cm and the successive layers are built so that no joins lie along each other. This last requirement is sometimes difficult to achieve and adds to the aesthetics of the finished product as much as to its strength. Generally each layer has 200 bricks.
The following are displays
of the ûuyena or falcon citi and the rathacakra or wheel citi.
I will explain in more detail the geometrical and construction features
of these and other citis in my presentation.
In a final discussion I shall argue that the Sulba Sûtras were written by people with a high level of mathematical knowledge and that they are part of a description, or perhaps a map, of the structure and qualities of the field of consciousness.
(Picture is available by request.)