Yantras are diagrams which form part of Hindu Tantric meditation practices, and are used ‘in the context of meditation and worship as visual-aids to concentration of the mind leading to realization of the abstract principle which is the inner meaning of the visible representation‘ (Bolton and Macleod, 1977). The nine isosceles triangles comprise four upwards-pointing triangles (corresponding to linga, representing Siva, the static male principle of wisdom) and five downwards-pointing triangles (corresponding to yoni, representing Sakti, the dynamic female principle of energy). The whole diagram can be considered as a ‘fully-created cosmos arising from the union of male and female principles’ (Khanna, 1979, p73), and each of 43 triangles that are created in the intersection pattern has its presiding deity. These triangles are coloured according to whether there are more ‘male’ (blue) or ‘female’ (yellow) triangles superimposed – the green areas of the stained glass correspond to where an equal number of ‘male’ and ‘female’ triangles are superimposed.
The intersecting pattern of the Sri Yantra is extremely complex and its precise mathematical structure does not appear to have been derived, although a number of approximate construction methods have been developed. In fact it has been suggested that it is impossible to construct such a diagram exactly on the plane and that the Yantra may only be precisely expressible in three-dimensional geometry.
A good reference for the Sri Yantra is Khanna (1979), and the best web page has been provided by Mikel Maron. Mikel describes a variety of construction methods, and I used the Golden Ratio method as the basis for a program in Splus [note in 2022: this was in 2002] to draw the Yantra, although working out the details was quite complex. This construction method gives the largest triangle the proportions of one of the faces of the Great Pyramid of Cheops, in which the ratio of the hypotenuse to half-base (r) is ø, the golden ratio (ø has many properties: it is the solution of the quadratic equation ø^2 = ø + 1, it is the limiting ratio of any Fibonacci sequence, it is (sqrt(5) + 1)/2 = 1.618…). This means the height of the pyramid is r x sqrt(ø^2-1) by Pythagoras, but this is just r x sqrt(ø) by the property of ø . (Nice connection: a circle drawn with its centre at the vertex of the pyramid and radius equal to the pyramid’s height, will have circumference 2 x pi x r x sqrt(ø ) = 7.992. The perimeter of the pyramid’s base is 8 r , a close approximation. )
(Note: I am neither happy with the choice of colours nor glass, and will do another larger version sometime.)
References.
Bolton and Macleod (1977). The Geometry of the Sri Yantra. Religion, 7, 66-85.
Khanna, M (1979). Yantra, the Tantric Symbol of Cosmic Unity. Thames and Hudson, London.
Lawlor, R (1982) Sacred Geometry. Thames and Hudson, London. (although the Sri Yantra on page 9 seems to be upside-down).