# sri yantra

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Sri YantraGerard HuetMarch 1990

IntroductionSri Yantra (also written Shri Yantra) is a sacred diagram of Tantric Hindhuism. Its symbolism is explained in Zimmer 15]. It consists in three concentric parts: An inner gure of interpenetrating triangles. This gure, symmetric in its vertical central axis, contains both upward-pointing triangles (vahni), symbolizing the male element (\Purusha", the Person), and downward-pointing triangles (sakti), symbolizing the female aspect (\Prakriti", i.e. energy) of divinity. There are 4 male triangles, and 5 female ones. Their inter-penetration symbolizes the complementarity of the opposite principles in creating the illusion of duality through ignorance, whereas the general symmetry and balance of the whole gure symbolizes the more profound reality of Unity of God through its various unfoldings. Two concentric circles, wearing a regular lotus (padma) design. The inner pattern has a period of 8, the outer one a period of 16. These patterns symbolize the sanctity of the inner diagram, used in Yoga meditation as a linear yantra 15]. A triple Greek-like key-pattern, the bhupura, or sisirita (shivered) frame. It symbolizes the ground-plan of a temple with 4 square openings, in the 4 directions of space. This sanctuary is the seat (p tha) of one's chosen deity (ista-devata), representing one's Higher Self. . .. This Sri Yantra diagram is one of the most sacred symbols of Tantric Hindhuism, and it is traditionally used in Yoga meditation exercises. The original edition of Zimmer 14] had erroneous descriptions of Sri Yantra, according to the later english translation 15]. Unfortunately, that one too is marred with errors in diagrams, and unclear references to inside-out construction instructions for the drawing of the triangular central pattern. The present paper presents a more systematic approach to the formal de nition of Sri Yantra.

Our rst approach was completely experimental: the author tried to draw Sri Yantra in free hand, and failed. A more systematic attempt with a computer drawing system failed too. The only reliable model available then was the frontispiece gure in 15], all the other diagrams in the book being obviously erroneous, or at least signi cantly di erent from the frontispiece gure. Instructions for drawing the diagram lead to various inexact drawings, such as the False Sri Yantra shown in Figure 1. Using the Macintosh package FreeHand, and measuring precisely Zimmer's frontispiece diagram, a correct Sri Yantra was then produced, shown below as Figure 2, and measured to serve as initial approximate con guration.

1.1 Drawing experiments

1 In search of Sri Yantra

Figure 1: False Sri Yantra The di culty of the above experiments had left undecided whether Sri Yantra was indeed uniquely de ned in the real plane, under-speci ed, or even impossible. Using the notation XA for the horizontal coordinate of point A in Fig. 1, and Y A for its vertical coordinate, and writing A&B ! c for \points A and B determine the line c joining them" and a b ! C for \lines a and b determine their intersection C ", we compute a system of real coordinates constraints as follows. is the inner circle of the diagram. We choose Y Q, XF , Y P , XA and Y J as parameters, and we get:

1.2 A more rigorous mathematical analysis

YQ !Q Q&O ! e YF =YQ YJ !J J &T ! h XA e ! A YJ e !H YQ h!I XU = 0; Y U = Y Q U &A ! f2

T

V G i U E D b C W B c P N a F M

S R d I e h H g A K J Q

Z f L

O

Figure 2: Approximate Sri Yantra

YP f ! C F &H ! g 0T g ! V XW = 0; Y W = Y P F &W ! a a f !D a h!G YJ f !B XZ = 0; Y Z = Y A C &Z ! c c g!M I &B ! d d 0 !YL f YL!L d YP !P a YM ! E P &E ! i a YV !S g YA!K3

i YD !N d YG!RThe parameters are correct if point (0; Y G) is on line i, which gives one extra constraint. We thus obtain a problem with 4 degrees of freedom, which admits of several solutions. However, we have to take into account the further constraints that the various points of the diagrams should be inside the circumbscribing circle ; we thus have a very shallow range for the 4 real parameters, leading to solutions which are esthetically rather close to Figure 2. A standard Newton approximation solving of the constraint on G from various initial solutions leads to the following particular solution, where the diameter of circle is taken as unit length: De nition Classical Sri Yantra is de ned as: YF=0.668 XF=0.126 YP=0.463 XA=0.187 YJ=0.398 YL=0.165 YA=0.265 YG=0.769 YV=0.887 YM=0.603 YD=0.551 This investigation solved our main query: Theorem Sri Yantra is an under-determined Euclidean plane geometry problem with 4 real parameters, admitting an in nity of solutions around the Classical Sri Yantra. The wider variation of the various solutions is on Y L, and this obviously a ects the esthetic rendering of the diagram, and consequently its magic powers. It became clear at this point that a synthetic study of the above solutions was essential in getting a more precise approximation to the traditional diagram. The graphic description language Postscript 3] was chosen for the graphic synthesis of the solutions. The mathematical analysis of the previous section pertains only to the triangular area. The circular, as well as rectangular patterns, were obtained by measurements from the original model from the frontispiece gure in 15]. The lotus patterns were also obtained by a trial-and-error ddling with 4

1.3 Graphic synthesis of solutions

the Postscript Bezier cubic curves primitives. The relevant Postcript code is available from the author. We show below two examples of gures which are particular solutions of the Sri Yantra equations. In the left gure point V is very high, whereas in the right one point N is very close to line c. The tension between these two pitfalls is the essential di culty in drawing the right diagram.

Figure 3: Two exact Sri Yantras

1.4 Esthetic feedback

An esthetic analysis was made of the resulting gures, leading to more experiments aiming at reaching an optimum feeling of harmony from the contemplation of the diagram. The author must confess his ignorance of Sri Yantra authorities who could obviously help in the search for the most powerful pattern. The main worry was the seemingly mandatory in exion in the slope variation of the upward diagonal lines e, d, c, b, a, and symetrically (but to a lesser extent) of the downward diagonal lines f , i, g, and h. This essentially experimental study converged to the de nition given above of Classical Sri Yantra, and the corresponding rendition is given below in Figure 4. The initial hope of the above mathematical analysis of the yantra was to formally describe a parametric situation admitting multiple solutions which could be optimized according to an esthetic criterion. However, even though the rst part of the conclusion was reached, see the Theorem above, the shallow range of solutions made it absolutely impossible to optimize the diagram to the extent, for instance, that the various triangle slopes vary in a monotonous manner. Doubts became thus to enter the mind of the author as to the precise de nition of Sri Yantra. Even in a serious study such as 15] contained inconsistencies. It de nes descriptions of it, culminating in Figure 10, which are clearly di erent from its colour nal rendition presented in the frontispiece. The frontispiece gure conforms to the mathematical analysis given above, and thus we may ascertain that it is an early precise graphical rendition of Classical Sri Yantra. But its 5

1.5 Bibliographic analysis

Figure 4: Classical Sri Yantra awkward sloping of the innermost shakti triangle makes it less harmonious in some sense than the smoother design in Figure 10 of this work, similar to the False Sri Yantra shown above. The inside-out instructions, attributed to Bhaskararaya's Nityasodasikarnava, are clearly misleading, since there is no hope, except by extraordinary luck, to get points J and Q on the circle determined by its diameter 0T . Actually, this text can be only considered as the description of Sri Yantra, and by no means precise instructions for its geometrical construction. The author then tried to decide which of the two designs was the traditional one. It was not a priori obvious whether the more exact, or the more harmonious drawing, were optimal from the meditation point of view. The frontispiece to 15] is credited to 9], but was added by the translators, and is absent from the original edition 14]. The rst lead was to follow Zimmer himself, who consigned later his observations on Myths and Symbols in Indian Art and Civilization in a volume edited after his death by Joseph Campbell 16]. In there, Fig 36 (also reproduced on the front cover) is a version of the False Sri Yantra. The credits attribute this gure to Sir Woodro e, alias Arthur Avalon. Indeed, this gure may be found on the cover of 1], dated 1914. The next available reference was the work of A. Danielou 4]. The relevant diagram is represented in the French edition on page 537, and we shall call it the Dual Classical Sri Yantra with small gates. The small gates matter in incidental, we mention it just for completeness. What is more important is rst, that it is the mathematically precise yantra (hence the word classical), and second that is is upside down. The diagram is called Shr^ Cakra in this work, i.e. Fortune Wheel, while alluding to its other name of Shr^ Yantra, translated as Diagram of Beauty or Harmony. It is said that the Shr^ Cakra represents the Universal Goddess. At this point it seemed that the Classical Sri Yantra was indeed the correct rendition, but the vertical orientation was in doubt, especially with respect 6

to the remarks on incorrect orientation in footnote q p. 161 of 15]. One of the most confusing sources is Rao, who dis

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