On Mathematical Idea Inside The Yantra Art

Mathematical Idea

It is a common place that nobody can expose exactly geometrical structure of the Sri Yantra performed through instruments: ruler, compass, 0.3 mm pencil etc.  I don’t name computer between the tools because it was not a primary implement for the geometrical art of yantra. Those illustrations modern readers can see in one book or another are only a feeble echo of possibly calculated solution and splendid view.  

Thus we cannot go further in our preview of mathematical idea if we shall not name the main details  have been lying in a base of Sri Yantra’s construction. 

Traditionally we begin to move from outside to the centre.  The first sacred geometrical figure is Bhupura, or a square with four gates. It has double (usually) or triple line, colour or colourless.  It can resemble a double svastika, if it will consist of eight prolonged bulges on four sides of the Bhupura.

The second principal figure is Mekhala, or triple concentrical circles (color or colorless) usually had been drawing without a compass.  The area being defined between Bhupura and Mekhala has own name, the first from chakras: Traylokya-mohana chakra, or Enchantress of the Triple World.  It is usually yellow on the whole, or one gate is white (down, East), another is blue (right, North), and two others are yellow (up and left, West and South).  Inside and outside the first chakra there are many pictures of the gods, goddesses, and sacral animals on the original hand-painted Indian copies.


Fig. 1  Traylokya-mohana chakra (yellow area), or Enchantress of the Triple World

The next two following chakras are Sarva-saparipuraka and Sarva-sankshobhana.  Very often researchers these lotus petals don’t carry mathematical gist and can be any size, that is wrong point of view.  Mathematical calculations cover the every line, circle, corner, every dot inside chakra.  These two chakras consist of two rings of sixteen (first, outer, blue) lotus petals and eight (second, inner, red) lotus petals , indicating fulfilment of desire.  Inner part of the petal also can carry images of recurring deities, Fig. 2 and 3.


Fig. 2 Sarva-saparipuraka and Sarva-sankshobhana chakras (blue and red rims of lotus petals)


The first chakra consisting of outer rim of 14 blue triangles on Fig. 3 (left) is Sarva-saubhagyadayaka chakra, or Giver of Auspiciousness, or Provider of All Prosperity.  It is a 14-pointed star polygon from 6 interlocking big triangles with parallel bases inside the mandala.

The second chakra consisting of 10 red triangles (Fig. 3, right) is Sarvartha-sadhaka chakra, or Accomplisher of All Purpose, or Accomplisher of All Objects.  It is a 10-pointed star polygon had been bearing by 6 primary big interlocking triangles.


Fig. 3  Sarva-saubhagyadayaka chakra (left, blue polygon) and Sarvartha-sadhaka chakra (right, red polygon)


The third yantra is Sarva-rakshakara chakra, or Giver of Protection, or Protector of All.  There are two smaller triangles inside commencing six big triangles.  It is again 10-pointed (inner) star polygon (Fig. 4, right, blue polygon).

The fourth yantra consisting of eight triangles is Sarva-rogahara chakra, or Remover of All Deceases, or Remover of All desires and Ills.  This is an 8-pointed star polygon having up-down asymmetry (Fig.4, left, red polygon).

The fifth yantra is Sarva-siddhiprada chakra, or Giver of All Accomplishments, or Bestower of All Attainments.  This is an inverted (usually colorless) triangle in the centre of Sri Yantra with the apex facing East (hanging downward when it is drawn on paper or cloth).  It is a female principle symbolizing the creative generatrix of the cosmos, yoni-mandala (Fig. 4, left, a small red triangle).

The ninth and last chakra in the short preview of Sri Yantra composition is Sarva-anandamaya chakra, or Full of Bliss.  This is a red dot Bindu in the centre of Sarva-siddhiprada chakra (Mula-trikona, primary triangle, female principle, colorless).  Last chakra, the red dot, is a male principle symbolizing the static aspect of the cosmos, male seed of Being, the state of supreme joy (Fig. 4, right, a small red circle).


Fig. 4  Sarva-rakshakara chakra (right, blue polygon), Sarva-rogahara chakra (left, red polygon), Sarva-siddhiprada chakra (left, a small red triangle), and Sarva-anandamaya chakra (right, a small red circle).


Thus we had been familiarized with inner structure and the whole composition of ancient Indian diagram and we can easy designate what means mathematical idea concerning Sri Yantra geometry.   Despite the method the nine basic triangles would have been placed inside the circle their location must be led to mathematical strictness of following description:

 All lines and circles must be executed with drawing instruments (a ruler, a compass, a technical pen, or Reißfeder, and templates) and width of the lines must be standard (0.25, 0.30, 0.50 mm and so on) as far as it possible and breath and hands allow it

 Nine principal triangles will produce 43 triangles in all chakras in general

 The ratio between diagonal of defense square (Bhupura, ground plan) and the inner diameter of circle (consisting of nine triangles) must be mathematically determined 

 Six or ten touches inside the kernel (first rim of fourteen color triangles — Sarva-saubhagyadayaka)  have to be right and symmetrical

 Nine main triangles’ bases have to be strictly parallel

 Nine main triangles’ apexes must lie on the bases of each other triangles and on the line of vertical symmetry exactly

 Six ‘cones’ (from left to right) must lie on the central horizontal line passing through the upper point of the central dot, Bindu

 And the last most difficult part — eighteen intersections of three lines must be without blank spaces inside them in spite of  any width of line preferred by author for any particular example

 The Sri Yantra art is geometrical art and its images should have aesthetic impression on adepts and spectators


It is a problem now that the key algorithm for this art and results of its implementation have been still undiscovered for public eyes by the time when the main requirements had been obviously formulated (1994-2011).

I would like to underline the fact that I don’t consider computers’ printouts as art of Sri Yantra, and I bother the less (men’she vsego?) about the zooming existing hand-drawn art works to find inner defects in composition.  A trained programer or a mathematician with a special computer program aren’t Sri Yantra masters.  They just mathematicians and programers exactly in the quantity statistics of the USA or other countries can show in annual reports.  It isn’t a subject for discussion of present review at all.

The path of master is the master himself as he is.  The master can get a virus and die.  The Sri Yantra drawing can be lost or broken.  The mathematical principle of Sri Yantra art is beautiful as is and absolutely unbroken by politicians, presidents, business visionaries, bankers, warlords, gangs, terrorists, religious fanatics, maniacs, and all sort of freaks. 

Probably, to close this theme about wise man and all others the best way is to cite T.S. Eliot: 

“They all go into the dark, The vacant interstellar spaces, the vacant into the vacant, The captains, merchant bankers, eminent men of letters, The generous patrons of art, the statesmen and the rulers, Distinguished civil servants, chairmen of many committees, Industrial lords and petty contractors, all go into the dark…”

It is not a Rig Veda level but after that relief we can definitely come up to discovery of algorithm directly.


Introduction as short as possible

Probably it would be extremely optimistic and naive to believe that the deepest knowledge about Sri Chakra has a root in oral period of human civilization covering 20,000 -1,500 BC.  More realistic point of view is closer to our usual imagination of ancient times.  The best search of its origin approximately could be set in Asia in period 1,500 BC – 8 AD.  As far as I can see, 5th century BC and later.

Two millennia had been elapsed and hand-painted Sri Yantras were accepted by Asian and Western museums.  In the second part of XX century some American scientists tried to find mathematical formulas but without discovery the main principles of construction of star polygons.  As a branch of modern mathematics 14 and18-pointed star polygons is still a blank page in reference books and encyclopedias.

Recently in the paper edition only one exact graphical scheme of Sri Chakra can be named, as it seems to me: published by Indian scholar who had been led his study in Mukambika shrine in Kollar near Bangalore, Karnataka, South India, and it gives us probably the best historic and philosophical explanation in religious framework.  Unfortunately, it will be very hard to accept this partially geometrical picture as a piece of art and a tool for meditation.

Collection of Sri Yantra and More Difficult Star Polygons performed by me in 1994-95 consists of two variations 14-pointed and nine (!!!) variations of 18-pointed star polygons previously unknown for public and specialists.  I would like to suggest results of my meditation with paper and ink for people whose knowledge of the world is deeper than it usually seems passing through everyday routine in the streets, avenues, buildings, offices and so called private life.