Tantric tradition suggests that there are two ways of using the sriyantra for meditation. In the 'outward approach', one begins by contemplating the bindu and proceeds outwards by stages to take in the smallest triangle in which it is enclosed, then the next two triangles, and so on, slowly expanding outwards through a sequence of shapes to the outer shapes in which the whole object is contained. This outward contemplation is associated with an evolutionary view of the development of the universe where, starting with primordial matter represented by the dot, the meditator concentrates on increasingly complex organisms, as indicated by increasingly complex shapes, until reaching the very boundaries of the universe from where escape is possible only through one of the four doors into chaos. The 'inward' approach to meditation, which starts from a circle and then moves inwards, is known in tantric literature as the process of destruction.
The mathematical interest in the sriyantra lies in the construction of the central nine triangles, which is a more difficult problem than might first appear. A line here may have three, four, five or six intersections with other lines. The problem is to construct a sriyantra in which all the intersections are correct and the vertices of the largest triangles fall on the circumference of the enclosing circle. We shall not go into the details of how the Indians may have achieved accurate constructions of increasingly complex versions of the sriyantra, including spherical ones with spherical triangles. Bolton and Macleod (1977) offer a simple overview of the subject; Kulaichev (1984) goes into the 'higher' mathematics implicit in constructing different types of sriyantra.
There is, however, a curious fact about all the correctly constructed sriyantras, whether enclosed in circles or in squares. In all such cases the base angle of the largest triangles is about 51°. The monument that comes to mind when this angle is mentioned is the Great Pyramid at Gizeh in Egypt, built around 2600 bc. It is without doubt the most massive building ever to have been erected, having at least twice the volume and thirty times the mass of the Empire State Building in New York, and built from individual stones weighing up to 70 tonnes each. The slope of the face to the base (or the angle of inclination) of the Great Pyramid is 51°50'35.
It is possible from the dimensions of the Great Pyramid to derive probably the two most famous inational numbers in mathematics. One is pi, and the other is phi the 'golden ratio' or 'divine proportion', given by (1 + sqr-rt 5)/2 (its value to five decimal places is 1.61803). The golden ratio has figured prominently in the history of mathematics, both as a semi-mystical quantity (Kepler suggested that it should be named the 'divine proportion') and for its practical applications in art and arAhitecture, including the Parthenon at Athens and a number of other buildings of Classical Greece. In the Great Pyramid, the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle theta to the ground, to half the length of the side of the square base, equivalent to the secant of the angle theta. The original dimensions of the Great Pyramid are not known exactly, because later generations removed the outer limestone casing for building material, but as far as we can tell the above two lengths were about 186.4 and 115.2 metres respectively. The ratio of these lengths is, to five decimal places, l.618 06, in very close agreement with phi. The number phi has some remarkable mathematical properties. Its square is equal to itself plus one, while its reciprocal is itself minus one. But the most intriguing feature is its link with what are called the Fibonacci numbers.
The Fibonacci numbers are the sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ...
where each number equals the sum of its two predecessors. This sequence pops up in a variety of natural phenomena - in pattems of plant growth and in the laws of Mendelian heredity, for example. It is easily shown that the ratio between successive Fibonacci numbers gets closer to phi the hurther up the sequence one goes. In the Fibonacci sequence given above, the ratio of 233 to 144 gives the value of phi calculated from the dimensions of the Great Pyramid.
The quantity pi can also be found in the dimensions of the Great Pyramid. If its height (1466 metres) is taken to be the radius of a circle, the perimeter of its base (4 x 230.4 = 921.6 metres) is almost equal to the circumference of that circle (2pir = 921.6 metres). The product of pi and the square root of phi is close to 4.
The largest isosceles triangle of the sriyantra design is one of the face triangles of the Great Pyramid in miniature, showing almost exactly the same relationship between pi and phi as in its larger counterpart. It would be idle to indulge in any further speculation.
Many of the accurate constructions of sriyantras in India are very old. Some are even more complicated than the one shown. There are those that consist of spherical triangles for which the constructor, to adlieve perfect intersections and vertices falling on the circumference of the circle enclosing the triangles, would require knowledge of 'higher mathematics whidh the medieval and ancient Indian mathematicians did not possess' (Kulaichev, 1984, p. 292). Kulaidhev goes on to suggest that the achievement of such geometrical constructs in Indian mathematics may indicate'the existence of unknown cultural and historical altematives to mathematical knowledge, e.g. the highly developed tradition of special imagination'.