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Sri Yantra


Sri yantra, also known as Sri Chakra, is called the mother of all yantras because all other yantras derive from it. In its three dimensional forms Sri Yantra is said to represent Mount Meru, the cosmic mountain at the center of the universe.

The Sri Yantra is conceived as a place of spiritual pilgrimage. It is a representation of the cosmos at the macrocosmic level and of the human body at the microcosmic level (each of the circuits correspond to a chakra of the body).

Sri Yantra is first referred to in an Indonesian inscription dating to the seventh century C.E. It may have existed in India, its country of origin, long before the time of its introduction to Indonesia.

The Sri Yantra is a configuration of nine interlocking triangles, surrounded by two circles of lotus petals with the whole encased within a gated frame, called the "earth citadel". The nine interlocking triangles centered around the bindu (the central point of the yantra) are drawn by the superimposition of five downward pointing triangles, representing Shakti ; the female principle and four upright triangles, representing Shiva ; the male principle. The nine interlocking triangles form forty three small triangles each housing a presiding deity associated with particular aspects of existence.

Man's spiritual journey from the stage of material existence to ultimate enlightenment is mapped on the Sri Yantra. The spiritual journey is taken as a pilgrimage in which every step is an ascent to the center, a movement beyond one's limited existence, and every level is nearer to the goal. Such a journey is mapped in stages, and each of these stages corresponds with one of the circuits of which the Sri Yantra is composed from the outer plane to the bindu in the center.

The Sri Yantra is a tool to give a vision of the totality of existence, so that the adept may internalize its symbols for the ultimate realization of his unity with the cosmos.

The goal of contemplating the Sri Yantra is that the adept can rediscover his primordial sources. The circuits symbolically indicate the successive phases in the process of becoming.

   

Nine Triangles in a Circle

Four triangles pointing up.
Five triangles pointing down
Central figure
Complete Sri Yantra
Figure 1

The Sri Yantra is composed of a central figure that is surrounded by two circular rows of petals and then by a rectangular enclosure called the bhupura. In this study we will be focusing mainly on the central figure which is composed of nine overlapping triangles and a bindu point. Four of the triangles point up, the other five point down. In the most popular configuration the two biggest triangles (green triangles in figure 1) touch the outer circle on all three points. In some other versions there are either one or two more triangles that touch the outer circle (See figure 5).

Interconnections

sri yantra marmas
Figure 2
Triple intersections.

When looking at the figure we notice that there is a high degree of interconnectedness between the nine triangles. This the main reason why it is so difficult to draw. This means that every triangle is connected to one or more of the other triangles via common points. Changing the location of one of the triangle usually requires changing the size and position of many other triangles.

Figure 2 shows where the triple intersection points are located. These are the points that lock together the triangles. You can't move one without also moving the others.

Notice also that the two biggest triangles are touching the outside circle on three points and that the apex of every triangle is connected to the base of another triangle.

False Sri Yantra
Figure 3 
False Sri Yantra

As with everything else there is a tendency to simplify and/or distort things , so that over time knowledge gets eroded. In the case of the Sri Yantra this has led to what some call the "false Sri Yantra". It's a version that is so far from the original figure that it is missing some of the most basic characteristic of a Sri Yantra. An example of such a false Sri Yantra is shown in figure 3. Here we see that the apex of most triangle is not connected with the base of another triangle as indicated by the red arrows. This reduces greatly the difficulty of drawing the figure and leads to something that looks like a Sri Yantra but isn't.

 

The First Key: Concurrency

Figure 4 
Errors in the drawing will lead to extraneous secondary triangles.

The obvious challenge when drawing a Sri Yantra is to achieve near perfect concurrency. Meaning that all the triple intersection meet at the same point rather than crisscrossing.

Figure 4 shows a detail of a Sri Yantra with the error circled in red. The lines should intersect at the same point but instead they crisscross each other and form an extraneous triangle. Using the right sequence to draw the Sri Yantra will ensure that there will be errors only in two of the triple intersections.

Very few Sri Yantras achieve perfect concurrency. Mathematically speaking it is not possible. But practically speaking a satisfactory level of precision can be achieved. It is difficult to achieve this when doing the drawing by hand but not impossible. Often the lines are made thicker to hide the errors at the intersections. A good level of accuracy can be achieved with a pencil and ruler and a lot of patience. A better accuracy can be achieved with a drawing program such as AutoCAD or Visio. The greatest amount of accuracy will be achieved by using a mathematical program such as Mathematica to compute the figure.

The One and the Many

type I type II type III
Figure 5 
Three different configurations of the Sri Yantra.

It would seem at this point that all one needs to do is to make sure that the lines match precisely at the triple intersections (concurrency) and our job is done. Not so!

Why are there so many different versions of the Sri Yantra out there? Figure 5 shows a few examples of Sri Yantras. In these examples the differences are obvious. The differences are usually more subtle and require closer examination. Like snow flakes there seems to be an infinite number of different Sri Yantras. Why is that? How can that be? Isn't there a precise and complete method that would tell us how this famous sacred figure should be drawn? If there is one we haven't found it yet.

The reason is simple. The criteria of concurrency (precise intersections) is not enough to fully define the Sri Yantra. Over time people have assumed that being able to produce a figure where the lines meet precisely at the intersections will produce a unique figure. This has lead to the current multiplicity of figures available.

Let us take the simple example of drawing a triangle. If the only criteria required is that the figure must have three sides then you can draw a infinite number of different triangles with three sides. If on the other hand you are asked to draw a triangle where the sides are of equal length then there is only one way to draw such a triangle (not taking size into account).

The Sri Yantra is a geometry with five degrees of freedom, which means that up to five different criterion can be used to define it. This is why we have to decide on the location of five lines when drawing the figure. Five degrees of freedom is not a lot considering that there is a total of nine triangles. This is because of the high degree of interconnectedness between the triangles. This effectively limits the possibilities and variations that can be achieved.

The Second Key: Concentricity

Figure 6 
Concentricity: the center of the innermost triangle coincide with the center of the outer circle.

Lets now take a look at the bindu point; the small point located in the central triangle. It should be located in the center of the innermost triangle. This can be achieved precisely by placing the bindu at the center of a circle that fits inside this triangle (see figure 9). This is known in mathematics as the incenter of a triangle.

To achieve a perfectly centered figure however, the bindu should also be located at the center of the outer circle. This is illustrated in figure 6. The red cross shows where the center of the outer circle is located. The small red circle shows where the center of the innermost triangle is. As we can see in this figure they coincide. This is not the case for most Sri Yantras.

Sri Yantra
Figure 7 
Sri Yantra that doesn't meet the concentricity criteria.

Figure 7 shows an example of a Sri Yantra where the center of the innermost triangle doesn't match perfectly with the center of the outer circle. The green dot (center of the innermost triangle) is not aligned with the center of the red cross (center of the outer circle).

The Third Key: Equilateral Inner Triangle

geodesic dome
Figure 8 
R. Buckminster Fuller stands in front of his geodesic dome.

The equilateral triangle is a perfect and minimal structure. It is the simplest, strongest and most fundamental structure in geometry and computer graphics. It has the highest degree of tensegrity for a minimum amount of structural elements. That is why it is so prevalent in the structural designs created by Buckminster Fuller. This is also why the geodesic dome, a spherical structure composed of small triangles is the only man-made structure that becomes proportionally stronger as it increases in size.

Figure 9 
Equilateral triangle as the expression of Rishi, Devata, Chanda.

The Sri Yantra symbolizes, among other things the unfoldment of creation. The bindu represents the unmanifest, the silent state. The next level in the expression of the Universe is represented by the innermost triangle. This level represents the trinity of rishi, devata, chanda, or the observer, the process of observation and the object being observed. At this point the symmetry of creation is still intact and will be broken when it reaches the next level which represent the grosser aspects of the relative.

This reflects the unfoldment from unity to trinity as expounded in the Vedic literature. According to the Veda the Universe becomes manifest when unbounded awareness becomes aware of itself. The spark of self awareness ignites creation. At this point Unity divides into the trinity of rishi (the observer), devata (process of knowing) and chanda (the object of perception). The same idea is also found in the bible as the principle of the holy trinity.

The central triangle is the central lens of the Sri Yantra. If as some suggest, this pattern is capable of emitting a significant amount of subtle energy, the importance of having a well balanced and centered figure becomes obvious.

For these reasons we believe that the central triangle should be equilateral in an optimal Sri Yantra configuration. For this to happen the highest down pointing primary triangle must have an angle of 60 degrees (see figure 9 and 11).

 

Center of Mass

center of mass center of mass

Figure 10 
Left: Figure with Concurrency criteria only. Center: Figure with Concurrency and concentricity. Right: Figure with Concurrency, concentricity and equilateral central triangle.

Now lets see if we can find ways to confirm the idea that we are getting closer to a perfectly balanced configuration. Another measure of overall balance of a structure is the center of mass. This is the point in the geometry where it would balance if it was a solid object.

Figure 10 shows a detail view of the central triangle of three different Sri Yantras. The left figure shows a configuration where onlyconcurrency is achieved. In this case the bindu (red dot), the center of the outer circle (green dot) and the center of mass (blue dot) are not aligned.

The central figure shows a Sri Yantra that achieves concurrency and concentricity. As a result the bindu (red) and the center of the outer circle (green) overlap nicely. The center of mass still doesn't overlap however.

On the right we see that for a figure drawn with the three criterion that we have suggested (concurrency, concentricity and equilateral central triangle), the three centers overlap and we have a perfectly centered and balanced figure.

The Sri Yantra from Sringeri Temple

Sri Yantra
Figure 11
Sri Yantra from Sringeri temple
.

The Sringeri temple in India claims to have the oldest Sri Yantra. This temple is one of the four pillars founded by none other thanShankara during the first millennium. Assuming that older Sri Yantras are closer to the original configuration lets see how this Sri Yantra compares to our optimal version. Obviously it is not possible to be certain that it is the oldest Sri Yantra on Earth but it is certainly older than most of the versions available. The shape of the petals and the bhupura are good indicators that it is a old Sri Yantra configuration.

Figure 11 shows an alleged picture of this Sri Yantra and a diagram that was drawn from the picture. Taking into account the distortions caused by the camera and printing we can see that the figure has many of the same characteristics as our optimal figure. The bindu is well centered and more importantly the centermost triangle has an angle very close to 60 degrees.

The Three Flavors of the Sri Yantra

Sri Yantra sri yantra pyramid spherical sri yantra
Plane form
Pyramidal form (Meru)
Spherical form (Kurma)

Since the Sri Yantra is based on triangles it is very appropriate that there are currently three main ways to represent this figure. The first and probably the most common is the plane form, which is what we have been looking at so far. 

The second is the pyramidal form called Meru in India. Mount Meru is a mythical mountain. So named because of the mountain shape of the figure. 

The third and rarest form is the spherical form or Kurma. Kurma was the second incarnation of Vishnu, the turtle incarnation. This refers to the similarity between this form and the shell of a turtle. It is interesting to note that there seems to be some confusion with the use of these two terms. The pyramidal form is often wrongly referred to as Kurma. This form is the rarest because of the extremely high level of difficulty involved in generating it. We have not yet found a correct physical representation of a spherical Sri Yantra. There are many attempts but very few have succeeded. 

Mathematics of Sri Yantra


A hymn from Atharavaveda is dedicated to an object that closely resembles this. The sriyantra ('great object') belongs to a class of devices used in meditation, mainly by those belonging to the Hindu tantric tradition. The diagram consists of nine interwoven isosceles triangles four point upwards, representing Sakti, the primordial female essence of dynamic energy, and five point downwards, representing Siva, the primordial male essence of static wisdom The triangles are ananged in such a way that they produce 43 subsidiary triangles, at the centre of the smallest of which there is a big dot (known as the bindu). These smaller triangles are supposed to form the abodes of different gods, whose names are sometimes entered in their respective places. In common with many depictions of the sriyantra, the one shown here has outer rings consisting of an eight-petalled lotus, enclosed by a sixteen petalled lotus, girdled in turn by three circles, all enclosed in a square with four doors, one on each side. The square represents the boundaries within which the deities reside, protected from the chaos and disorder of the outside world.

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 asriyantra 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'.

Bolton and Macleod discuss Sri Yantra


The term yantra, which literally means an instrument for holding or restraining, may be used to denote a variety of linear diagrams which play a significant role in the meditative practices of Tantric Hinduism. Yantras may be simple designs such as the cross, triangle, square, circle or lotus pattern, symbolizing basic concepts, or may be more complex combinations of such elements in figures representing in abstract form the particular creative forces in the cosmos which are called divinities. they are closely related to the mandalas used by both Hindu and Buddhist Tantrism, in which geometric design is supplemented by elaborate symbolic images of the deities which by their various forms and attributes indicate different aspects of the hidden order of reality. As Mircea Eliade says (1), the yantra is 'the linear paradigm of the mandala', expressing the same principles in geometric form. Like mandalas, yantras are used in the context of meditation and worship as visual-aids to concentration of the mind leading to realization of abstract principle which is the inner meaning of the visible representation.

The best known and geometrically the most complex yantra is the Sri-yantra, also known as the Sri-yantra, employed by the Sakta school of Tantrism which visualizes the divine primarily in female form. The structure of this yantra is enigmatically described in the Saundarya-lahari (The Wave of Beauty) (2), a lengthy poem praising the great goddess whose dwelling place the Sri-yantra is said to be:

By reason of the four Srikanthas (srikantha is an epithet of Siva) and the five damsels of Siva (which have the nature of Sakti), which are penetrated by Sambhu (i.e. bindu- the dot in the centre) and constitute the nine fundamental natures, the 43 (or 44) angles of your dwelling place are evolved, along with the 8-petalled and 16-petalled lotuses, the circles and the three lines. (stanza 11)

The diagram may be more accurately described as a bilaterally symmetrical figure composed of nine interwoven isosceles triangles, usually depicted with five triangles pointing downwards and four pointing upwards. The former are said to correspond to the yoni representing the dynamic female principle of energy (Sakti), while the latter correspond to the linga representing the static male principle of wisdom (Siva). (The Buddhist Tantrics, incidentally, regard the male principle as dynamic and the female as static.) The central dot called bindu represents the original unity of the male and female principles prior to creation and the paradoxical point female principles prior to creation and the paradoxical point from which the manifestation of the cosmos emerges. The interpenetration of the nine basic triangles gives rise to a number of subsidiary triangles (43 including the central triangle enclosing the bindu) which form the abodes of the deities, representing the particularization of the original creative forces into more concrete manifestations. Sometimes the names of deities and Sanskrit syllables are written into these triangles, or images of the deities are placed in them.

In most versions of the yantra this central design is enclosed by two circular lotus-patterns with eight and sixteen petals, a girdle of three concentric circles, and finally a square arrangement of straight line ('the three lines') with four openings or 'doors' at the cardinal points called 'World House' (bhugra). This square outline, which is common also to mandalas, symbolizes the royal palace in which the deities reside - an area of sacred space protected from the disintegrating forces of chaos. In general, the Sri-yantra is a 'cosmogram' - a graphic representation of the universal processes of emanation and reabsorption reduced to their essential outline. As Eliade puts it, the yantra: 'An expression in terms of linear symbolism of the cosmic manifestations, beginning with the primordial unity.'

Construction by Order of Destruction

sambara-krama

From commentary on Saundarya-laharl written by Kaivalyasrmam.

Attribute to the "Left Hand Path" of Tantrism, the inward approach to meditation, starting from a circle and moving towards the center.

This technique is rather involved, resulting in slight errors at the intersections (marma-sthana) and non-congruent large triangles. This method probably accounts for most of the examples in the literature.


Animation of the Construction Requires Netscape 2+


1. Draw a circle of the required size with a vertical line through the centre and divide this line into 48 equal units.

2. On this line make nine marks at a distance of 6, 12, 17, 20, 23, 27, 30, 36, and 42 units from the top, and draw nine horizontal lines (numbered 1-9) through these marks to meet with the circle.

3. At both ends of the 1st, 2nd, 4th, 5th, 6th, 8th, and 9th lines rub off 3, 5, 16, 18, 16, 4 and 3 units respectively.

4. Join the ends of the 1st line to the centre of the 6th, the ends of the 2nd to the centre of the 9th, the ends of the 3rd to the circle at the bottom of the axis, the ends of the 4th to the centre of the 8th, the ends of the 5th to the centre of the 7th, the ends of the 6th to the centre of the 2nd, the ends of the 7th to the circle at the top of the vertical axis, the ends of the 8th to the centre of the 1st, and the ends of the 9th to the centre of the 3rd.

Construction by Order of Creation

Srsti-krama

From commentary on Sudarya-lahari by Laksmidhara.

Attribute to the Tantric 'Right Hand Path'. The instructions are complicated and somewhat obscure, and it is not obvious that a yantra can be constructed in this fashion without a lot of luck or an example to work from.

This method potentially results in perfect intersections (marma-sthana)



Inner eight-pointed figure

  • A. Draw isosceles triangle A with apex pointing down and vertical line through the apex and centre of base. Place the bindu on this line near the apex of A.

  • B. Above the bindu, draw hiorizontal through the sides of A to form the base of triangle B also drawn with apex pointing down.

  • C. Draw horizontal through apex of A to form base of triangle C with apex pointing up and sides passing through the intersections of the sides of A and the base of B.

Inner ten-pointed figure (antar-dasara):

  • D. Produce base of A both ways to form base of triangle D with apex pointing down and sides passing through the ends of base of C.

  • E. Produce base of C both ways to form base of triangle E with apex pointing up and sides passing through the ends of base of A and B.

  • A1. Produce sides of A to meet horizontal through apex of C.

  • C1. Produce sides of C to meet horizontal through apex of B.

Outer ten-pointed figure (bahir-dasara):

  • F. Produce base of A1 both ways to form base of triangle F with apex pointing down and sides passing through the ends of base of E and C1.

  • G. Produce base of C1 both ways to form base of triangle G with apex pointing up and sides passing through the ends of base of D and A1

  • A2. Produce sides of A to meet horizontal through the apex of E.

  • C2. Produce sides of C to meet horizontal through the apex of D.

Fourteen-pointed figure (caturdasara)

  • H. Produce base of A1/F both ways to form base of triangle H with apex pointing down and sides passing through the ends of base of G and C2. (The circle is not mentioned.)

  • J. Produce base of C1/G both ways to form base of triangle J with apex pointing up and sides passing through the ends of base of F and A2.

  • F1. Produce sides of F to meet horizontal through apex of E.

  • G1. Produce sides of G to meet horizonal though apex of D.

  • A3. Produce sides of A to meet horizontal through apex of G.

  • C3. Produce sides of C to meet horizontal through apex of F.

The Infinite Power of Yantras 

http://www.scribd.com/doc/94247101/tantric-yantras

Solving the Sri Yantra

http://www.scribd.com/doc/36990295/Sri-Yantra

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