1

Śrī Chakram:

The Geometric Duet in Praise of

Tripura-Sundari

K. Chandra Hari

Abstract

Present study seeks to bring out the sacred rationale Srī Chakram based on the classical precepts on

construction. Modern studies by Kulaichev, Rao CS etc had depicted the Siddha precepts as inadequate

to derive the optimal configuration guided by the erroneous interpretations. True rationale of the

classical construction method on the other hand leads to the identification of the tāntrik

characterization of the sacred object of worship. The conventional method of deriving the chords is

shown to be defective and the illustration has been given of the right method as:

������� ���� � ��� � ���� � 2 � ε � ���48 � → ���� �����

������� ���� � ��� � ���� � 2 � ε� → ��� ������� With the right bases derived, the traditional elements lead to precise angles which are integer or half-

integer multiples of Kundāmśa i.e. 360/81. The Jupiter inner triangle of the 5th chord with the

naturally obtained classical angles as 49.5:49.5:81 along with the Moon and Rahu equilateral triangles

lead to a remarkably concurrent and concentric mode of 9 interlocking trinagles. The uppermost Sun

triangle for such a configuration is found to have a full chord with the angle structure 54:54:72 and the

prime vertical over which the Sri Chakram manifests can be interpreted as oriented eastwards to the

Krttikā (Alcyone) nakshatra. Result as above of the 360 bisector of the solar triangle finds credence with

the legends on Kārtikeya vis-a-vis Trikkārtika celebrations. Solutions from the Sriyantraexplorer and

the drawings made on Autocad are presented to illustrate the truth of the classical precepts and the

Siddha wisdom. Artistic or aesthetic features may be added subject to the fixtures provided by the

Siddha wisdom.

The notion of an under-determined problem yielding numerous solutions and the modern quest for an

optimal mathematical solution arose as a result of the missing import of the Siddha precepts in fixing

the bases correctly. Work on Autocad has shown that the drawing turns easier when the solar apex

angle is chosen as 720 so that the bisector or the prime vertical is oriented towards the sidereal longitude

of Kr �ttikās. Present work got ably supported by Kodathu Suresh Kesava Pillai with all the needed

drawings and analysis of dimensions on Autocad. Present work also marks the end of a phase of 19-

year luni-solar period which began with the Sivarātri of 1994 and set to conclude on the next Sivarātri

of Kumbham, 10 March 2013, coinciding solar transit of 325:25.

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1. Introduction

Srichakram as it has become popularly known evokes a feeling of awe, a mystery in respect

of its construction, the underlying mathematics and its purpose as a tool giving access to the

divine. Most of the modern discussions have been beating around the bush without knowing

even the very meaning of the Siddha geometrical construction and ascribing attributes and

descriptions in a superficial manner. Some have gone to the extent of striking parallels with

the triangular faces of one of the many Pyramids at Cairo to claim glory for the Yantra of

Tripurasundari. There have been few modern scientific studies as well by Kulaichev1, CS

Rao2 and SR Tiwari3. Tiwari’s work is unique by its reliance on Jyotihsastra notions in

explaining the traditional geometrical construction method. Given the present author’s

nearly two decades of work in integrating Jyotihsastra and Tantra over a mathematical

framework, the most famous of all tantrik gadgets viz., the Sri Chakra comes as a natural

object of study for seeking further evidence in the matter of the one and only force,

Mahākāli, as the medium underlying both Jyotisha and Tantra. In fact not only

Jyotihśastra and Tantra but integration of all the Vidyas enunciated by the triad of eyes

must be realizable when the precepts are followed without distortion created by the

vikalpam. Present study of Sri Chakram is being presented in two parts – the first part

presenting a discussion of the theory and the second part discusses the practical application

in deriving the Sri Yantra as prescribed by the Siddhas.

2. Śrī Chakram – Classical Descriptions

Among the references I could lay hands upon, the complete description of the construction

methods as known in Sanskrit, Tamil and Malayalam are found only in the Sri Chakra-

pūjakalpam of Chattampi Swami. His verses explain the Kāśmīra tradition of drawing

method. The verses give a complete description of the method and leave no scope for any

confusion as is being alleged at many websites. Sri Chakra of course has its art content and

therefore practical exercises in drawing the same may have to be learned from the

traditional scholars.

¸ÉÒSÉGòÊ´ÉÊvÉ& ¹ÉhhÉ´ÉiªÉRÂóMÉÖ±ÉɪÉɨÉÆ ºÉÚjÉÆ |ÉÉM|ÉiªÉMÉɪÉiÉÆ* SÉiÉÖ̦ɮÆúMÉÖ±Éèζ¶É¹]èõººÉÆ ÉÞiÉÉÊxÉ SÉ ¦ÉÚ{ÉÖ®Æú*1*

Take a section of the prime vertical 96 units long and in the 4 units at both the rising and

setting points, the earth-abode (Bhūpuram) is created.

1 Kulaichev, AP., Sri Yantra and its Mathematical Properties, IJHS, 19, 1984, pp. 279-292 2 Rao, CS, Sri Yantra-A Study of Spherical and Planar Forms, IJHS, 33(3), 1998 3 Sudarshan Raj Tiwari, Sri-Chakra: Rediscovering the Rules of its Construction from First Principles, personal communication.

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+xiÉzÉÇ ÉÉÆMÉÖ±ÉÆ YÉäªÉÆ ¨ÉvªÉä {ÉjÉxiÉÖ ¹ÉÉäb÷¶ÉÆ* BEòÉnù¶ÉÆMÉÖ±ÉÆ YÉäªÉ¨É¹]õ{ÉjÉÆ ºÉ¨ÉÉʱÉJÉäiÉÂ**

Inner 9 units are used for the 16-petalled flower and the following 11 units for the 8-petalled flower.

näù´ÉÒºiÉÖiÉÉä ¨Éä MÉÆMÉɴɱ±ÉÒºiÉÖiÉäÊiÉ |ÉSÉIÉiÉä* iÉjÉ iÉä <¹]õ B´É¨ÉÉt¨ÉÆMÉÖ±ÉÒ¨ÉÉxÉÉxiÉ®äú * xÉ´É®äúJÉÉ Ê´É±ÉäJÉxÉÒªÉÉ ´ÉÞkɨÉvªÉä <iªÉlÉÇ&**

In the 48 unit diameter of the inner circle formed by the base of 8-petalled flower, draw the

9-chords at intervals of 6, 6, 5, 3, 3, 4, 3, 6 and 6 units.

+Étä ÊuùiÉÒªÉä +¹]õ¨ÉEäò xɴɨÉä SÉ ªÉlÉÉGò¨ÉÆ* ¨ÉÉVVÉǪÉänÂù MÉÖhɦÉÉMÉÉƶÉÉxÉ ´ÉÞkÉÉnäùEòjÉ SÉÉxªÉiÉ&** SÉiÉÖlÉǹɹ`öªÉÉä& {ÉÉ·Éæ iÉÉäªÉÉƶÉÆ {ÉÊ®ú¨ÉÉVVÉǪÉä±ÉÂ* {É\SɨɺªÉÊvɪÉÉƶÉxiÉÖ ¨ÉÉVVÉǪÉäSSÉÉxªÉ{ÉÉ·ÉÇiÉ&**

The chords have to be reduced on both sides equally by 3, 5, 4, 3 units respectively for the 1st, 2nd, 8th

and 9th. For the 4th and 6th chords, 16 units and 19 units are to be reduced for the 5th chord.

iÉÞiÉÒªÉÉxiÉÉÎnÂùuùiÉÒªÉÉxiÉÉSSÉiÉÖlÉÉÇSSÉ {É\SɨÉɱÉÂ* |ÉlɨÉÉxiÉɱÉ ºÉÚjɪÉÖMÉÆ ´ÉÞkÉiÉÉä xɴɨÉÉÊnùEÆò* ºÉ{iÉɹ]õ¨ÉªÉÉä®úxiÉÉkÉlÉÉ ¹É¹]õxÉ´ªÉxiɪÉÉä&* ´ÉÞkÉÉÊnùºÉÚjɪÉÖMɳýGò¨ÉɱÉ ºÉÚjÉuùªÉÆ ÊuùvÉÉ**

For the reduced chords or bases of triangles 1 to 5 and 6 to 9, the triangular line-doublets are

of two types. In the former, the 3rd base meets the circle and in the latter it is the 7th base that

gets connected to the circle and the rest in order stated i.e. 3rd, 2nd, 4th, 5th and 1st base lines

meeting the intersections of prime vertical and the chords successively moving up in order

from the circle. 7th, 8th, 6th and 9th are in the opposite direction likewise.

½þ¨ªÉÇ{É´ÉǺɨÉÉä{ÉäiÉÆ ¦ÉpùºÉÎxvɺɨÉÎx´ÉiÉÆ* ªÉSSÉGÆò ±ÉʳýiÉɪÉÉ& iÉx¨ÉÆMɳÆý xÉ <iÉ®únÂù ¦É´ÉäiÉÂ**

The chakra of 18 parvas and 24 sandhis is the auspicious abode of Lalita and any distortion is inauspicious.

Swami has quoted other verses and the corresponding Malayalam verses of same import.

When we look for alternate sources of information, the most popular account today known

in the Kerala tradition is the appendix given by K. Krishnan Nair to his translation of the

work Sri Chakra by S.Sankara Narayanan. It presents a very concise description but the

reference for the same is not given. Nair has also quoted the method from Nityotsavam, an

ancillary text of the Parasuramakalpasutram. Given such different descriptions and the

approaches seen quoted in some of the websites, it is easy to realize that the verse Vyase

devīkr te... etc presents the correct classical method.

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´ªÉɺÉà näù´ÉÒEÞòiÉä iÉkɨɱɱÉÊ´ÉMÉiÉä SÉɹɦÉÉMÉä ZɺÉÚjÉÉxÉÂ* EÞòi´ÉÉ{ÉÚ®úÒnù³ýÉxiÉè MɴɴɱɨÉ֦ɪÉÉä&´ÉÉÇiɪÉÉäºiÉÉäªÉ¦ÉÉMÉÉxÉÂ** vÉxªÉƶÉÆ ¨Éä |ɨÉÉVªÉÇ |ÉlɪÉiÉÖ ´É¯ûhÉÉOÉÉÊhÉ ºÉÚjÉuùªÉÉÊxÉ* |ÉÉEÂò{ÉÉiÉÆ ®úÉäÊvÉMÉÉxÉÆ Éʽþ¨É漃 iÉÖ ÊvÉMÉÆ Ênù´ªÉºÉÚxÉÖÆ ÊSÉ®Æú |ÉÉEÂò**

These lines complete the subtle description of the nine interlocking triangles which in turn

produce the 43 triangles and the following rough translation can be attempted:

Draw a circle with east west diameter of fourty eight units Divide the dia with chords at units 6-6-5-3-3-4-3-6-6 and 6 Reduce the chords in order from top as 1 to 9 on both sides 3 and 4 units on the 1st and 2nd, 4 and 3 units on 8th and 9th 16 units from the 4th and 6th and 19 units from the 5th chord Make the triangles east-west in order with the reduced lines The 1st is to become base and meets the 6th middle as vertex Likewise the 2nd puts its vertex on 9th and 3rd onto the circle 4th is to meet the vertex on 8th and 5th onto 7th, 9th onto 3rd 8th to 1st and 6th to 2nd while the 7th to the circle at east point

The Kaivalyashrama version quoted by Tiwari differs a bit and gives the reduced parts of

the nine chords as 3, 5, 0, 16, 18, 16, 0, 4 and 3. Devistuto...and the Malayalam verses given

by Chattampi Swami in his Sri Chakra Puja-kalpam also give the same erasers as in the

above verses. This difference will be examined in details in the next section.

Rao CS who had applied the most complicated mathematics to the Sri Yantra had

concluded that the original tantra figures are erroneous owing to the deviation from the

optimal plane model he had derived. His results are4:

The values in the two lower rows of the table may be contrasted to have a feel of the errors

spotted by the modern mathematical exercise. 3 is 3.6618 and 6 is 6.2 and 5.8 etc and the

emerging conclusion is that the siddhas were incompetent to construct the Sri Chakram

correctly.

Kulaichev also had similar conclusions earlier as may be noted from his 1983 paper:

4 Rao, CS, Sri Yantra-A Study of Spherical and Planar Forms, IJHS, 33(3), 1998, p.224

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In contrast to the above observations by Rao and Kulaichev, the Huet’s quest for Sri Yantra

had led to experiences and observations like:

• Our first 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...

(1.2 A more rigorous geometric analysis)

The difficulty of the above experiements had left undecided whether Sri Yantra was indeed uniquely

defined in the real plane, under-specified or even impossible... This investigation solved our query:

Theorem: Sri Yantra is an under-determined Euclidean plane geometry problem with 4 real

parameters, admitting an infinity of solutions around the classical Sri Yantra.

Further, Gerard Huet5 has summed up the outcome of his bibilographic search in the

following words:

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 aesthetic criterion.

However, even though the first part of the conclusion was reached, witness 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 fashion.

Doubts became thus to enter the mind of the author as to the precise definition of Sri Yantra. Even a

serious study such as [19] contained inconsistencies. It defines descriptions of it, culminating in its Figure 10, which are clearly different from its final colour rendition presented in the frontispiece. The

frontispiece figure conforms to the mathematical analysis given above, and thuswe may ascertain that it is a precise graphical rendition of Classical Sri Yantra. But the awkward sloping of the innermost

shakti triangle of the latter 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 Bhaāaskararāaya's Nityas�od �asikārn�ava, 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 an approximate description

of Sri Yantra, and by no means as precise instructions for its geometrical construction.

It was not clear at this point 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 to be preferred....

Huet is a Sanskrit scholar working in the field of Sanskrit linguistics related programming

studies and familiar with Indology. He goes on to describe a number of references where in

the pitcographs of the Sri Chakra are given and concludes as:

5 Gérard Huet. Sri Yantra Geometry. Theoretical Computer Science 281 (2002) pp. 609-628

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We finally mention that numerous books on symbolism mention Sri Yantra, but they usually show

incorrect representations of it, either reproducing the False Sri Yantra from [20] (e.g. Campbell), or it's

upside-down inverse (e.g. Jung).

Another geometric study of the diagram has come recently to our attention [4]. But this study mentions

only approximate constructions and dubious angular relationships with the Great Pyramid of Cheops.

Kulaichev and Rao CS have given detailed discussions on the mathematical aspects of ‘9

triangles, 5 down and 4 up inside the circum-circle’ without caring to look at the rationale of

the classical method of construction. Most of the websites and later authors except Tiwari

SR got swayed by the above works of magic by mathematics devoid of siddha experience

and have patented their own drawing methods of what they consider as the optimum

geometrical features needed. Paul Delisle has a wonderful website sriyantraresearch.com

and presents software tools like ‘Sriyantra Explorer’ with which any number of ‘9 triangles

& circumcircle’ pictographs can be created6.

All those who have ventured to study Sri Chakra have found fault with the classical method

and chose to give a new method based on ‘their aesthetic’ considerations. Hence the million

$ question arises:

• Can the ancient wisdom be wrong as is made out by the modern mathematical

derivations?

Sri Chakra is a siddha creation of immense spiritual wisdom and tantrik application and

unless the under lying rationale and the artistic freedom involved are understood and

reconciled, one may not be able to reach the right conclusions. Huet’s theorem brings out

clearly the fact that the mathematical solution is not unique and the choice of a solution as

acceptable depends on the tantrik characterization contained in a particular solution.

3. The Classical Śrī Chakram

Truth of the classical method can be examined using simple geometrical analysis involving

the chords and angles. Sri Chakra was not created by any super-science as is happening with

the modern computers nowadays. The siddha wisdom encapsuled in the drawing methods

can be understood only if we employ insights that can be gained from the related Sakta

works. Familiarity with the Sakta tradition is inevitable if we are to understand the

geometrical properties of Sri Chakra.

• What exactly is the object of the classical description quoted in the verses Devi

vyasekrte?

Modern interpreters have taken the description to mean the whole gamut of 43 triangles, the

marmas and the sandhis etc and in that process have forgotten the very crux of the Sri

6 Both Sudarsan Raj Tiwari and Paul Delisle had been very positive and helpful in their interactions and spared their valuable time to clarify my doubts.

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Chakra. The crux of Sri Chakra in fact is the nine triangles pointing outwards which make

up the nine Mūlaprakrtis and the Charana-konas of Bhagavati and the verse quoted above

defines the 9 angles or ‘footings’ of the Tripurasundari. This fact has been clearly brought

out by Sastri and Ayyangar in their commentary to the verse 11 of Saundaryalahari7. But

the tragedy is that none of the modern investigators including Kulaichev, CS Rao, Patrick

Flanagan, Gerard Huet, Mcleod & Bolton, Russian works quoted by Kulaichev in his later

notes seen on the web or resourceful websites like sriyantraresearch.com give any

cognizance to the special characterization given for the nine prime triangles. It is obvious

that the right interpretation of the classical verses shall bring out such facts which shall in

turn validate the approach and if the approach is wrong, inconsistencies shall creep in as to

smear the whole exercise.

Step 1: Nine Charanakonas of Bhagavati

The focus that the author of Saundaryalahari gives to the nine angles may be noted from

verse 11 of Saundaryalahari:

SÉiÉÖ̦É& ¸ÉÒEòh èö& ʶɴɪÉÖÊiÉʦÉ& {É\SÉʦɮúÊ{É |ÉʦÉzÉÉʦÉ& ¶ÉƦÉÉäxÉÇ ÉʦɮúÊ{É ¨ÉÚ±É|ÉEÞòÊiÉʦÉ&* jɪɶSÉi´ÉÉË®ú¶ÉiÉ ´ÉºÉÖnù±ÉEò±ÉɸÉÊjÉ´É±ÉªÉ ÊjÉ®äúJÉÉʦÉ& ºÉÉvÉÈ iÉ´É SÉ®úhÉEòÉähÉÉ& {ÉÊ®úhÉiÉÉ&**

This verse puts the 9 angles or Yonis – the apex angles of the 4 + 5 = 9 triangles formed on

the reduced chords which are referrred to as Srikantha and Sivayuvati – to be placed at a

higher level of cognition when one attempts to understand the Sri Chakra. Further, when

the astronomical or the macrocosmic interpretation is considered, a Mandala means 400 or

360/9 and in navāmśa each of the Mandala becomes 40x9 = 3600. Man dala therefore refers

to the Yoni which encompasses the whole wheel of 3600 in dormancy or latent form. Within

a Mandala, there are 3 nakshatras which in navamsa breeds the 27 nakshatras of the

Chakra. It must be noted here that the 3 nakshatras in turn present the 12 padas and the

Chakra is constituted by 108 of the nakshatra-padas over which the Chandra-kala rides. At a

deeper level, each of the nakshatra is divided into 3 parts 13.3333/3 = 3600/81=4.4444 and

nine of such 4.44440 arcs make up a Mandala and thus at the macrocosmic level, each of the

nine Yonis or Pādas of Bhagavati is constituted by arcs of 4.44440. Rightly the arc 3600/81

=4.44440 is called the Pādāmśa where Pāda in katapayādi mens 81. The 81 Pāda-Padmams

makes a triad of 27 each to constitute the 1200 arcs of the Jyotischakra between the 00 Ketu

or Sikhi points and apparently reflects the description of Her Chakra as in Subhagodaya.

ÊjÉJÉhbÆ÷ iÉä SÉGÆò ¶ÉÖÊSÉ®úÊ´É ¶É¶ÉÉRÂóEòÉi¨ÉEòiɪÉÉ ¨ÉªÉÚJÉè& ¹ÉϲjɶÉqù¶ÉªÉÖiÉiɪÉÉ JÉhb÷EòʱÉiÉè&*

7 Sastri SS and Ayyangar TRS, Saundaryalahari, Theosophical Publishing House, Adayar, p. 65

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{ÉÞl´ªÉÉnùÉè iÉi´Éä {ÉÞlÉMÉÖÊnùiÉ´ÉÎi¦É& {ÉÊ®ú´ÉÞiÉÆ ¦É´Éäx¨ÉÚ±ÉÉvÉÉ®ú |ɦÉÞÊiÉ iÉ´É ¹É²SÉGòºÉnùxÉÆ**

Your abode, the wheel, has a 3-fold structure of 360 rays or degrees involving the triad of

fire, sun and moon. The six chakras manifest by the five elements which rise and fall upon

the same.

The 3-fold structure of the Jyotischakra is quite popular in Jyotisha with the Rāśi-Nakshatra

sandhis at 00, 1200 and 2400 i.e. the confluence of the 12-fold 30 degree solar divisions and

the 27-fold lunar houses of 13020’ each. Confluence at these junction points percolate down

in the 1200 arc in terms of the 3 Man dalas of 400 each and each Mandala in turn has a 3-fold

structure of nakshatras and 9-fold structure of Pādāmsas i.e. 120 = 40 x 3 = 4.4444 x 9.

Thus the 3 and 9-fold divisions of the Jyotischakra in concurrence with the Sri Chakra call

for the existence of 4.4440 divisions i.e. a flower of 81 petals or Kunda-pū or Kunda-

pushpam alias Pāda-padmam in cryptic terminology. The Pādapadma or the Lotu Feet of

Bhagavati in fact means the Pā-da or 81 divisions of the Jyotischakra where in She dwells as

Samayā or Mahākāli – the goddess of time, the creatrix of time, manifestation or the

changes personified, the fifth force underlying the manifestation that remains hitherto

unknown to modern Physics. Jyotischakra with its 81 divisions of 4.4440 in fact is the

original Kunda-puram, abode of 81 petals.

Some interesting aspects of the above division and the numbers involved are noteworthy:

36081 � 4.4444&

8081 � 0.987654321&

Jyotisha defines the human birth by the rule that the Kunda-Lagna must be a triangular

longitude of Moon i.e Lagna or Ascending East Point x 81 = Moon ± 1200, 00. This implies

that during the rise of an arc of 4.4444 degree, there can be only 3 destinies as the Kunda-

Lagna completes 1 revolution. In other words, 1 nakshatra of Kunda-Chakra is equivalent

to 9.87654321 minutes of the Jyotischakra.

80081 � 9.87654321 800 � 81 � 9.87654321&

i.e. 800 minutes of Kundachakra or 1 Nakshatra of Kundachakra is 9.87654321’ of the

Jyotischakra observed with the naked eye.

80081 * 27 � 9.87654321 * 27 � 266.66667+ � 4.4444&

36081 � 1.23456789 � 360

100

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Given such importance of the Kundapushpam in Jyotisha, it is natural to expect that the arc

of 4.4440 might have played a significant role in the construction of Sri Chakra as

representation of the one and only one Cosmic Being – the fully integrated existence of

microcosm and macrocosm – the experience of Universe or the Advaita. It is likely that the

nine Mūlaprakrtis represented by the nine triangles may have in their design, the nine-fold

Jyotischakra of 9 Man dalas and the 9-fold design of the Mandalas themselves so that the

macrocosmic charanakonas of Bhagavati making up the Kunda-pushpa of 81 petals is

implicit in the same.

Mathematical Evidence for Kunda-pushpa in Sri Chakra

In the following part, the elements of Sri Chakra as per the classical description are

examined to adduce evidence in support of the above thesis. A rought schematic of the

triangles is given in fig.1.

(a) Base of 9 Triangles

Classical description calls for the 9 horizontal chords on an east-west line (up-down) which

are numbered as 1 to 9 on the circumference. The nine bases derived from the chords to

shape the triangles have been named as 1 to 9 inside the circum-circle seen aside. In deriving

the bases the following reduction in parts have been applied to the 1-9 chords successively:

3, 4, 0, 16, 18, 16, 0, 4 and 3

Results obtained for the chord lengths at units 6-6-5-3-3-4-3-6-6 and 6 from the top of the

diameter of 48 units are presented in Table-1:

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Table-1: Chord Lengths and Bases Derived

Triangle Radius Position from

Chord Base Planets Top Centre

1 24 6 18 31.75 27.780 Sun

2 24 12 12 41.57 34.641 Moon

3 24 17 7 45.91 45.913 Mars

4 24 20 4 47.33 15.776 Mercury

5 24 23 1 47.96 11.990 Jupiter

6 24 27 3 47.62 15.875 Venus

7 24 30 6 46.48 46.476 Saturn

8 24 36 12 41.57 34.641 Rahu

9 24 42 18 31.75 27.780 Sikhi

(b) Apex Angles of 9 Triangles = 9 Mūlaprakrtis

Above base lengths have been used to compute the other elements of the nine triangles using

the height information contained in the classical instructions. Formation of the triangles

using the bases finds illustration in fig.1 and the height of each triangle may be derived from

the position of the bases and the apexes. Table-2 presents the data:

Table- 2: ,-./ � 012345 � 6 � εεεε � 1234578 9 and Triangle Elements

Triangle Base Height Equal Apex

Angle Sum of Angles

Circumcircle Radius Sides Angles

1D 27.780 21 25.178 56.52 66.96 180.0 15.09

2D 34.641 30 34.641 60.00 60.00 180.0 20

3D 45.913 31 38.575 53.48 73.04 180.0 24

4D 15.776 16 17.839 63.76 52.49 180.0 9.94

5D 11.990 7 9.216 49.42 81.15 180.0 6.07

6U 15.875 15 16.971 62.11 55.77 180.0 9.60

7U 46.476 30 37.947 52.24 75.52 180.0 24

8U 34.641 30 34.641 60.00 60.00 180.0 20

9U 27.780 25 28.600 60.94 58.11 180.0 16.36

Surya to Guru reduced chords form the bases of the 5 downward (D) triangles of Sakti while

the Sukra to Ketu ones form the bases for the 4 upward (U) triangles of Siva. Among these,

the triangles 3D of Mars and 7U of Saturn are fixed by the rule of having the circumcircle of

radius R. There is absolutely no reason to doubt the correctness of the elements prescribed

for construction in classical texts as the circumcircle computed for the 3D and 7U is

precisely 24 units. Given the verse 11 of Saundaryalahari, the apex angles of the 9 triangles

which make up Srikantha and Sivayuvati represent the 9 Mūlaprakrtis and this is the key to

the construction of a Sri Yantra as per the tantrik precepts. It is apparent from the classical

instructions worked out above that the 2D triangle of Moon and 8U triangle of Rāhu are

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equilateral and such a geometric feature cannot be discarded with superficial accusations

that the elements given by the traditional texts are rounded off integers and approximate.

Such accusations reflect only our ignorance and incapability to capture the essence of the

siddha wisdom.

(c) Numerous Solutions of an Under-determined Problem

The innumerable versions of the nine interlocking triangles and the circumcircle as seen in

India and also at websites like sriyantraresearch.com are illustrative of the truth of Huet’s

conclusions. Sri Chakra manifests only when the output and especially the 9 triangles

representing the Mūlaprakrtis can be characterized with relevant tantrik rationales. Here the

question arises as to:

1. How the classical output of Table-2 given above fares in the light of tantra?

2. Classical divisions of the diameter and base lengths used have contributed any specific

characteristics to the final tantra product i.e. the Sri Yantra?

A closer look at the 9 Mūlaprakr is or the 9 apex angles of table-2 reveals that the modern

researchers either did not examine them or could not find anything remarkable with the

values and hence they set out in quest of the perfect figure, optimal figure etc. On the other

hand, the classical apex angles derived above are remarkable by their flowering around a

common rationale as may be noted from the Table-3 below:

Table-3: Nine Apex Angles = 9 Mūlaprakrtis

Triangle Apex Angle

No. of Kundāmsa

Base Corrected Apex

Kundāmsa

Apex correction

′

New Base

Base Correction

%

1 2 3 4 5 6 7 9 10

1D 66.96 15.07 27.780 66.666 15 -18′ 27.624 0.6

2D 60.00 13.50 34.641 60 13.5 0′ 34.641 0.0

3D 73.04 16.43 45.913 73.333 16.5 17′ 46.158 -0.5

4D 52.49 11.81 15.776 53.333 12 51′ 16.071 -1.9

5D 81.15 18.26 11.990 80 18 -69′ 11.747 2.0

6U 55.77 12.55 15.875 55.555 12.5 -13′ 15.802 0.5

7U 75.52 16.99 46.476 75.555 17 2′ 46.504 -0.1

8U 60.00 13.50 34.641 60 13.5 0 34.641 0.0

9U 58.11 13.08 27.780 57.777 13 -20′ 27.589 0.7

Columns 6 & 7 are illustrative of the truth of the classical instructions. For the classical apex

angles representing the Mūlaprakrti, with minor corrections, all of them reduce to values

which are integer or half-integer multiples of Kundāmsa = 360/81=4.4440. Column 10

shows the base correction as percentage of the original value in column 4. The classical

values also exhibit the following special characteristics discussed in the next section.

12

(d) Remarkable Ancient Wisdom

1. The equal angles of the isoceles triangles also are integers or half or quarter fractions of

the Kundāmsa 360/81 = 4.444.

2. The Pyramid angle hypothesis falls through as the 3D triangle has a side angle of 53.333

degree which is 12 times the Kundāmsa and the circumcircle for the triangle is precisely

equal to the radius of the initial circle of 48 units diameter.

3. For the 7U triangle, the side angles are equal to 52.222 and less by a quarter of

Kundāmsa to the 3D triangle and both of them share the same circumcircle.

4. The 5D triangle or the “Baindava Griham” – the inner most abode of the Bindu or

Sambhu - has an apex angle of 800 (2 Man dalas) and side angle of 500. The 50 Mayūkhas

or degrees can rightly be interpreted as the 50 alphabets (aksharas or varnas). This is the

axial triangle in which the axis or Aksha i.e. Akarādi Kshakārantham in its most potent

form as the seed of expansion or creation has assumed the form of bindu. The 500 side

angles amply reflect the Aksha-māla contained by the bindu.

5. 800 apex and 50-50 side angles of the 5D innermost triangle presents the correct

geometric picturization of the Yoni as may be noted from the sketch attached as

appendix-1.

6. The lowest side angle is 50 representing the 50 alphabets and all the nine Mūlaprakrtis

contain the same.

7. No apex angle is lower than 53.333 = 12 Kundāmśa and the 9 apexes lie within a range

of 26.6666 = 2 Nakshatras = 6 Kunda-khandas.

8. Comparison of the circumcircle radii of each triangle is also noteworthy:

Table-4: Circumcircle Radii Comparison

Triangle Base Height Circumcircle Radii R

Change R in %

Original New

1D 27.780 21 15.09 15.04 0.3

2D 34.641 30 20.00 20.00 0.0

3D 45.913 31 24.00 24.09 -0.4

4D 15.776 16 9.94 10.02 -0.8

5D 11.990 7 6.07 5.96 1.8

6U 15.875 15 9.60 9.58 0.2

7U 46.476 30 24.00 24.01 0.0

8U 34.641 30 20.00 20.00 0.0

9U 27.780 25 16.36 16.31 0.3

13

The percentage change in the value of the circumcircle radius of 7U and 3D is less than even

0.5% and the maximum change occurred for the innermost down triangle 5 and there also

the circumcircle radii changed very small, less than 2%.

9. Triangles represent the ‘tri-khandas’ of ‘Shodaśī’ as repeatedly described in

Subhagodaya and the siddha astronomical reasons thereof can be explained on the basis

of the nakshatras.

The 27 nakshatras or lunar houses or mansions in fact are the 27-fold Jyotischakra of

Bhagavati which when divided into 3 becomes 81-petalled Kunda-chakra or Kunda-pushpa

of Indu or Moon. Each nakshatra is 1/3rd of the Man dala and contains 800 Kalas which in

turn is the 16-fold configuration of 50 alphabets each.

10. The integer values of erasers given for symmetrical application to the chords have been

declared as approximate by modern scholars without caring to look at the rationale

under which they have been derived. Table-5 below illustrates the truth of ancient

wisdom:

Table-5: Truth of Erased Parts at the Integer Divisions of East-West Diameter

Chord Base used

Erased Parts

Corrected Base

Erasers new

% Corrected Apex

Kundāmsa

1 2 3 4 5 6 5 6

31.75 27.780 3.0 27.62404 3.12 3.9 66.666 15

41.57 34.641 4.0 34.64102 4.00 0.0 60 13.5

45.91 45.913 0.0 46.1577 -0.13 0.0 73.333 16.5

47.33 15.776 16.0 16.0705 15.85 -0.9 53.333 12

47.96 11.990 18.0 11.7474 18.12 0.7 80 18

47.62 15.875 16.0 15.80242 16.04 0.2 55.555 12.5

46.48 46.476 0.0 46.50361 -0.01 0.0 75.555 17

41.57 34.641 4.0 34.64102 4.00 0.0 60 13.5

s31.75 27.780 3.0 27.58897 3.14 4.8 57.777 13

When the Kundāmśa rationale is applied to correct the apex angles to become integer or

half integer multiples of Kunda divisions or the tri-khanda of the nakshatras, the erasers

symmetrically applied to the chords for deriving the base follow automatically with almost

zero errors as seen in column 6. Nearly 4% and 5% errors occur only for the 1st and 9th chord

reduced parts and that too is negligible.

It becomes apparent that the 9-Mūlaprakrtis when derived as per the tantrik rationale leads

to the correct geometrical configuration and helps to re-discover the origin of the ancient

elements prescribed.

If we are to use the differing values of 5 and 19 for erasers, the apex angle corresponding to

5 will be 12.5 Kundāmsa and 16 Kundāmsa for 19.

14

11. The classical instructions lead to a very consistent set of elements for the Sri Yantra and

contain hitherto unknown depths of ancient siddha wisdom. The integers prescribed in

the tradition are the beauty of the Sri Chakram and they can never be approximate as

shown by Rao CS with the modern mathematical analysis. Given the conclusion of

Huet, the analysis of Rao CS may be incomplete and the optimal configuration derived

by him is devoid of the right priors as to choose the right solution as per classical

precepts.

The Sri Chakra as described in the tantra is a mathematical abstraction of the experience of

Jyotischakra by the ancient savants. Its uniqueness arises from the tantrik characterization

of the resultant model. Erasers or the reduced parts prescribed for the nine chords in the

classical precepts in fact represent a characterization given for one of the numerous

configurations emerging within a circle of say radius 24 units. As for example, take a look at

what happens when the erasers are increased by (say) 2 parts and (-) 1 part. Table-6 presents

the relevant data:

Table-6: Prakrtis versus Mūlaprakr tis

Tri- angle

Classical erasers

Apex angles

Erasers +2

Apex angle

Change in apex

Kunda fraction

Erasers (-1)

Apex angle

Change in apex

Kunda fraction

1D 3 66.67 5 62.22 4.44 1 2 68.89 -2.22 -0.5

2D 4 60.00 6 53.33 6.67 1.5 3 62.22 -2.22 -0.5

3D 0 73.33 0 73.33 0.00 0 0 73.33 0.00 0.00

4D 16 53.33 18 40.00 13.33 3 15 57.78 -4.45 -1

5D 18 80.00 20 57.78 22.22 5 17 88.89 -8.89 -2

6U 16 55.56 18 44.44 11.11 2.5 15 62.22 -6.67 -1.5

7U 0 75.56 0 75.56 0.00 0 0 75.56 0.00 0.00

8U 4 60.00 6 53.33 6.67 1.5 3 62.22 -2.22 -0.5

9U 3 57.78 4 55.56 2.22 0.5 2 60.00 -2.22 -0.5

It is evident from the above data that the erasers given by the tradition are intended to give a

specific choice from the many solutions possible. Triangle 5D, is too sensitive to the change

of value of the erasers and change by 1 means a change of 8.8880 – twice the Kundāmśa – in

the apex angle. At the 5D triangle as such the eraser should not have been less than 18 as

the apex had been approaching a right angle. Use of 19 in the tradition for 5D may be an

effort to reduce the apex of 5D to 16 Kundāmśa = 71.1110. Correctness of 18 or 19 can be

ascertained only by a drawing exercise.

For triangles 3D and 7U where there is no reduction of the chord, i.e. erasers are 0, the

impact of introducing the eraser of 1 is to change the apex angle by 2.222 i.e. half of the

Kundāmśa. It is apparent from the above that the classical elements for the drawing of Sri

Chakra are a very consistent set of ‘erasers or reduction to the chords’ and position of

15

chords vis-a-vis height of the 9 triangles that led to a specific angular geometry for the 9

Mūlaprakrtis. When those 9 apex angles are lost, the Sri Yantra is lost in chaos and this is

visible in the innumerable drawings seen since historic times.

As can be expected Prakrtis is a manifold and Mūla-prakrti is unique. The 9 interlocking

triangles within the circle can have many configurations which are representative of the big

and small energy exchanges happening for manifestation. The origin of such a manifold is

the Mūlaprakr ti consisting of the 4 Srikanth as and 5 Sivayuvatis.

4. Kundāmśa and Erasers – Tantrik Characterization of Mūlaprakrti

Above discussion clearly brings out the tuning between the ‘erasers’ (reduction of parts

prescribed for the chords) and the Kundāmśa division of the circle i.e. 360/81 where 81 or

Kunda alias Pāda is a mystic number of the tantras. For the biggest triangles or the longest

chords having circumcircle radius of 24 units i.e 3D has no reduction applied to the chord

and the apex angle is automatically 73.333 =16.5*360/81 and for 7U the apex angle is

17*360/81 =75.55 as per the traditional instructions. Sri Chakra is founded on this cardinal

aspect and those who could not visualize this inner secret of 360/81 failed to recognize the

tantrik characterization applied to the geometric configuration. For the heights of 3D and

7U, a reduction of 2 parts on both sides leads to a reduction of 4.4440 (= Kundāmśa) for the

apex angle. The average change for 4 parts reduction in chord length is 4.620 and hence

given the under-determined nature of the problem, base lengths that meets integer or half

integer multiple differences of 4.4440 to the apex angle can be sought out artistically to make

the Sri Chakra as per the classical method. Triangles 3D and 7U are keys to the problem

and modern researchers have led people astray into eccentric notions like the face angle of

the Pyramid as they missed the insight offered by Kundāmśa for the apex of these triangles.

Axioms for Geometric Construction

1. The innermost sacred triangle 5D has to be 49.50 so that the apex is 810 and visually it

may have been 50-50-80 as transmitted down the generations. Wisdom of 81 may have

been lost in the course of time.

2. The Moon and Rahu triangles are clearly equilateral (60x3=180)

3. Triangles 3D and 7U of Mars and Saturn respectively are on full chords with the

circumcircle of raius equal to 24 units and thus serve as fixtures in the drawing process.

4. Sri Chakra is drawn symmetrical to the Prime Vertical or the East-West line and thus the

apex of 7U must be oriented towards the east point.

5. Symmetry considerations demand that the 7U apex and the 1D apex must be visually

equal.

6. 2D and 8U are equilaterals and are therefore fixtures

16

8 <

7. 4D, 6U and 9U are close to equilateral triangles and therefore must be visually

equilateral

5. Classical Sacred Configuration

Given the above data of the planetary chords (Table-1), angles (Table-2) and axioms, it is

easy to identify the cardinal parameters provided one keeps in mind that the ancient times

when the geometric construction was given shape had no means for precise angular

measures of the kind we have today. Elements of the triangle from the classical chords are

summarized in Table.7 below:

Triangle Chord Classical Angles Corrected

Planet Side Apex Apex Side

1D 1 56.52 66.96 66.67 56.67 Sun

2D 2 60 60 60.00 60.00 Moon

3D 3 53.48 73.04 73.33 53.33 Mars

4D 4 63.76 52.49 53.33 63.33 Mercury

5D 5 49.42 81.15 80.00 50.00 Jupiter

6U 6 62.11 55.77 55.56 62.22 Venus

7U 7 52.24 75.52 75.56 52.22 Saturn

8U 8 60 60 60.00 60.00 Rahu

9U 9 60.94 58.11 57.78 61.11 Sikhi

Based on the above reasoning the Sri Chakra configurations can be sought through the

Sriyantraexplorer assuming that the software gives genuine mathematical solutions and

simultaneously drawing can be attempted on tools like autocad available now for geometric

constructions on computers.

Fig.2 Table-8

No. Position of chords Angles

% Units From Top

1 0.8860 5.5 6 54.04

2 0.7365 12.6 13 60.00

3 0.6422 17.2 17 53.27

4 0.5783 20.2 20 60.25

5 0.5303 22.5 23 49.5

6 0.4482 26.5 27 59.19

7 0.3865 29.4 29 51.56

8 0.2675 35.2 35 60

9 0.1505 40.8 41 61.96

17

Computers give the added advantage that the scatter at the triple intersection points,

junctions, chords, heights and angles can be computed precisely for check with the classical

parameters derived from the verses.

(a) Guru (Jupiter) chord 5 inner triangle as 49.5:49.5:81 and U2=D2=60:60:60 (Moon and

Rahu) as fixtures, the following solution, fig.3 is given by Sri Yantra explorer (line

drawing above along with relevant data)

Fig.3

Guru (Jupiter) 5D by either of the methods have a base of ≈ 12 units when the diameter of

48 units is reduced by 18x2 units and the angle is very close to 50 and is one of the critical

numbers of Tantra as the number of alphabets. The solution as obtained by the Sri Yantra

Explorer is given above (Fig.3).

• 5D = 50, 2D = 8D =60

• Criticism possible

In the fig.3 below, the sun triangle at the top touches the circumcircle as is the case with the

3D and 7U triangles.

The same configuration as drawn out in autocad using the same fixtures is shown in fig.4

below:

18

In this drawing the Sun triangle could be accomodated with the apex angle of 720 along with

the other angles as obtained from the traditional chords derived in the correct manner.

Where as in the optimal solution of the software following the theory outlined by CS Rao,

one obtains only the one of the many numerous solutions of an under-determined problem.

Software solution

In the solution below, it can be seen that for the choice of angle 50 for the inner most

triangle, the solar triangle just leaves the circumcircle and for any value less than 49.5, the

uppermost triangle violates the circumcircle.

19

(Fig.4)

Relevant data is furnished in Table-9 below:

Triangle Base Height Equal Software

Solution Sides Angles

1D 27.780 21 25.178 56.52 54.727

2D 34.641 30 34.641 60.00 60

3D 45.913 31 38.575 53.48 53.334

4D 15.776 16 17.839 63.76 60.726

5D 11.990 7 9.216 49.42 50

6U 15.875 15 16.971 62.11 59.880

7U 46.476 30 37.947 52.24 51.589

8U 34.641 30 34.641 60.00 60

9U 27.780 25 28.600 60.94 62.478

As has been discussed by Rao, Kulaichev and others, the iterative drawing of Sri Chakra is a

very complex process and the mathematical precision could not have been achieved in the

drawings. Kulaichev in fact had interpreted the traditional method as approximate and not

involving the co-axial configuration of the circumcircle and the incircle. In his own words: 8

“For this type the method of traditional copying is well known according to which (fig.5) the vertical

diameter is divided into 48 equal parts, after that the horizontal lines of the polygon are drawn on the

levels of subdivisions of 6, 12, 17, 20, 23, 27, 30, 36, 42...However this heuristic method even for so

simple portrayal does not ensure (even for a visual perception) the satisfactory matching of some points

of intersection”

8 Kulaichev (1984), p.285

Seed : 80.48 , 0.26 , 0.29 , 0.47 , 0.12 <

Concurrency , Concentricity , User Angle U2: 60° , User Angle D2: 60° , User Angle

20

Kulaichev had little knowledge of the tantrik tradition in which had produced Sri Chakra as

a part of its ‘magical art’. Prescriptions of the kind he has quoted are only oral transmissions

which have found its way into the later time literature. It is intended as a thumb rule only

and of course the siddha wisdom may have ensured the implicit presence of some rationale

in such memory capsules passed on to the disciples.

It is apparent that the traditional approach had even missed the true rationale of the precept

and had been producing a differing configuration. The truth of the present interpretation

follows from the angles and figures shown above.

5D (Guru) = 49.2 5D (Guru) = 50

Fig.5 Fig.6

The difference between the two configurations is not noticeable by visual logging. For an

angle less than 49.2 as obtained from the precept, the Sun triangle violates the circle while

for angles greater than 50, the Sun triangle withdraws from the 24 unit declination circle.

Kulaichev and others had in fact missed the crux of the problem i.e. creating the bases for

the triangles from the chords. Traditional interpretation sought to erase the given parts say ε

from the chords on both the sides.

:;<=> � 6 � �?6 � @6�^B. C

DEFG � :;<=> � 6 � εεεε (p is the position of the chord in units from the centre and R = 24 units)

In the classical definition, the units are in fact defined as 1U = 1/48 of the whole. This is

true not only for the east-west diameter or Rāhu-sūtra but also for the other lines sought to

be drawn in the process. Reduction of the chords by 2ε parts thus meant Chord-

2ε*Chord/48 and not Chord - 2ε.

?G>HIG> :;<=>F � DEFG � 0:;<=> � 6 � εεεε � :;<=>78 9 →→→→ :<==GIJ KGJ;<>

21

?G>HIG> :;<=>F � DEFG � �:;<=> � 6 � εεεε� →→→→ L=<MN O=EIJPIG

Each chord had to be reduced by its own parts as indicated by the numerals given indicated

as ε. Relevant data is provided in Table-10 below:

Table-10: The True and False Chords

Sl.No. Radius Positio

Chord

C

Eraser Parts

ε

ε1=

2ε*C/48

ε2=

2ε

Base

From Top

From Centre

Tradition True

1 24 6 18 31.75 3 3.97 6.00 25.749 27.780

2 24 12 12 41.57 4 6.93 8.00 31.569 34.641

3 24 17 7 45.91 0 0.00 0.00 45.913 45.913

4 24 20 4 47.33 16 31.55 32.00 15.329 15.776

5 24 23 1 47.96 18 35.97 36.00 11.958 11.990

6 24 27 3 47.62 16 31.75 32.00 15.624 15.875

7 24 30 6 46.48 0 0.00 0.00 46.476 46.476

8 24 36 12 41.57 4 6.93 8.00 33.569 34.641

9 24 42 18 31.75 3 3.97 6.00 25.749 27.780

(a) Astronomical Rationale

The validity of a particular configuration can be adjudged only on the basis of some siddha

rationale that could be attached to the same.

1. The top most down triangle (sakti-1) in fig.4 touches the circumcircle for the 5D apex

angle of 810 and is the maximum possible angle for optimum concurrency in the motif. It

may be noted here that the traditional method takes us straight to this parameter. The

Surya triangle has an equal angle of 54 deg and apex angle of 72 degree and not all

equilateral as it appears in the traditional solution.

Should the first Sakti triangle or the Surya triangle touch the circumcircle?

What is this circumcircle of 24 units radius?

Unless we know about the rationale of the 24 units radius, we cannot answer the

question. It is well known in Indian astronomy that the maximum declination of sun is

24 degree and corresponds to the latitude of Ujjayini. In other words, a declination of 24

degree suggested either of the solstice and for those in the northern hemisphere of earth,

it means the summer solstice.

A 24 degree circle also represents a cone of 24 degree along the circle of which the pole

star (or the celestial north pole) goes round the ecliptic north pole (ENP). ENP in fact is

the ‘Achalesvara’ – a point absolutely stationary for the geocentric view of the sky.

24 thus is a number that represents the earth’s axis or aksham which is also known as the

Meru which the sun is supposed to circumambulate as per the siddha astronomy. The

22

circumcircle as such is a representation of the ecliptic or the apparent path of the sun

around Meru and the centre represents the pole of the ecliptic. At the summer solstice,

the sun achieves the declination of 24 degree and is thus directly above the Meru.

2. Mandalas

The nine Mandalas may be astronomically enumerated as in Table-11 below:

Table-11: The Mandalas of Jyotischakra – Abode of Mahākāli

Sl.No. Beginning Planet

Starting Point

End Point

End Star

End Planet

Triangle

1 Ketu 0 40 Krittika Sun Sun

2 Chandra 40 80 Ardra Rahu Moon

3 Guru 80 120 Sarpa Budha Mars

4 Ketu 120 160 Utram Sun Merc

5 Chandra 160 200 Swati Rahu Jup

6 Guru 200 240 Jyeshtha Budha Ven

7 Ketu 240 280 Utradam Sun Sat

8 Chandra 280 320 Satabhishak Rahu Rahu

9 Guru 320 360 Revati Budha Ketu

These 9 Mandalas populate the orbital plane from the orbit to the centre in triangular

modes:

♣ 4.4444 x 3 = 13.333 ♣13.333 x 3 = 40 ♣ 40.0 x 3 = 120 ♣ 120x 3 = 360

The east-west line differently getting described as the Brahma-sutram and Rahu-sutram etc

in fact is the zodiac diameter 600 – 2400 and so that the six triangles from Sun to Venus get

represented by the 6 Mandalas containing 3 nakshatras each. Six triangles from top to

bottom along a vertical axis to represent 40x 6 = 2400 instead of the 1800 considered on a

Cartesian axis. This is not an imagination and the truth of this interpretation can be verified

from the south Indian style Rāśi chakram (Rāhu-Sikhi) which has 0 and 240 along the same

vertical line (Fig.7).

0--300

Rā-Si

240

The triangular configuration of Sri Chakra from the Surya triangle to the 6th one of Venus

making up the vertical flow of energy from the orbit to the centre is a geometrical replica of

23

the same energy configuration of the Cosmic Siddha depicted in the Zodiac with the 0 –

2400 vertical line. This convergence of the rationales between Sri Chakram and the Rāśi

Chakram is obvious given the common tantrik implications as representations of ‘Universe’

i.e. the micro and macrocosm.

Thus when the circumcircle is the ecliptic of 3600, it transpires that the 9 Mandalas enclosed

by it are the 40 degree divisions (3 nakshatras each). The Mandalas obviously begins at the

zero degree and hence the possibility arises that the ancient savants may have designed the

Sri Chakra with the 1st Mandala of Surya beginning at the zero point where the Surya

triangle had its vertex of 54 degree i.e. 27 x 2 and the apex angle had been 72 degree – the

number again is of astronomical significance – being the number of years that the earth’s

axis takes for 10 precession.

3. Orientation towards the Krttikā

Can there be a connection between the Sri Chakra and the Krttikas? It is well known in the

ancient literature that the Krttikas marked the east in the Indus Valley days. If Sri Chakram

is drawn on the east-west line, it is natural to look for a definition of east in the design.

Considering the Surya triangle in the fig.4 above, its equal angles are 54 degree and

therefore the apex angle turns out to be 72 degree. As discussed above, if the top triangle has

its base intercepting upon the orbit at the east, the apex of triangle 7 (Saturn) will be bi-

secting the apex angle of triangle1 (sun) and thus the east-west line will be at 360 and this

exactly marks the sidereal longitude of Krttikā or Alcyone.

It must be noted that the configuration in the upper and lower halves are different. This had

to be so to have the representation of 2400 across the vertical axis.

4. Moon and Ketu are equilateral triangles while Mercury and Ketu are very close to being

equilateral and thus visually the 4 triangles are equilateral.

5. The 3D triangle has the side angles 12*4.4444 = 53.333 = 4 nakshatras and the deviation

in a complete concurrence is only few minutes of arc. 7U down on the vertical axis

having the same circumcircle as 3D and the side angle of 51.562 i.e. nearly half of

Kundamsa less in contrast to the 3D.

6. Why to take 5D angle as 50 instead of 49.5?

5D = 49.5 is the lowest value for which a solution is possible given the traditional method

when it is correctly interpreted. Apex angle will be 810 and thus the configuration can be

explained as having occult significance. (Fig.4). In ancient times, given the popular notions,

it is impossible to conceive methods by which angles may have been getting measured to the

precision of decimal digits. Given the traditional chord section that formed the base, the

angle may have been characterized either by the count of 50 alphabets popular in Tantra or

24

by taking 49.5 ≈ 50 and thus the Sun triangle just touching the solstitial 24 unit declinational

circle. Fig.8 presents the drawing made on Autocad along with angles inscribed.

6. Misinterpretation in the tradition

Sri Chakras in different designs are available today. Most of these are based on a

misinterpretation of the verses given for deriving the bases as given in Table-12 below:

Sl.No. Eraser Radius Position from

Chord Base used

Top Centre Correct Traditional

1 3 24 6 18 31.75 27.780 25.749

2 4 24 12 12 41.57 34.641 31.569

3 0 24 17 7 45.91 45.913 45.913

4 16 24 20 4 47.33 15.776 15.329

5 19 24 23 1 47.96 11.990 9.958

6 16 24 27 3 47.62 15.875 15.624

7 0 24 30 6 46.48 46.476 46.476

8 4 24 36 12 41.57 34.641 33.569

9 3 24 42 18 31.75 27.780 25.749

25

Traditional derivation leads to a Sri Chakra of un-even triangles and distorted quadrangles

as may be noted from the different pictures seen in the web.

(Fig.99)

This is the most common type of Sri Chakra seen and is derived by the wrong method of

derivation of bases from the chords. The angular configuration for the solution subject to

concurrency and concentricity is given in Table-13 below:

Table-13: Conventional & True Chords in Comparison

Sl.No. Eraser Base Position Equal Angle in

solution sides angles

1 3 25.749 21 24.632 58.49 59.435

2 4 31.569 30 33.899 62.25 62.902

3 0 45.913 31 38.575 53.48 53.685

4 16 15.329 16 17.741 64.40 66.027

5 19 9.958 7 8.590 54.58 54.58

6 16 15.624 15 16.912 62.49 62.909

7 0 46.476 30 37.947 52.24 52.738

8 4 33.569 30 34.376 60.77 62.531

9 3 25.749 25 28.120 62.75 66.629

Base 5 and the corresponding angle are critical in deciding the configuration. If one is to

employ the eraser of 18 at base 5, then the angle will be 49.5 degree and the configuration

obtained is:

9 Drawings are based on the Sri Yantra Explorer software

26

(Fig.10)

Relevant data is provided in table-14 below:

Sl.No. Eraser Base Position Equal Angle in solution

sides angles Angle = 49.5 Angle = 54.58

1 3 25.749 21 24.632 58.49 53.955 59.435

2 4 31.569 30 33.899 62.25 60.4474 62.902

3 0 45.913 31 38.575 53.48 52.976 53.685

4 16 15.329 16 17.741 64.40 62.135 66.027

5 18 11.958 7 9.206 49.50 49.5 54.58

6 16 15.624 15 16.912 62.49 58.081 62.909

7 0 46.476 30 37.947 52.24 51.996 52.738

8 4 33.569 30 34.376 60.77 59.890 62.531

9 3 25.749 25 28.120 62.75 62.088 66.629

As concurrency and concentricity are achieved in the solutions, other aspects like evenness

of the triangles etc have not been given sufficient attention. In fact, no yardstick could be

thought of in giving a tantrik or siddha characterization to the drawing.

7. Rao CS Optimal Configuration

Rao’s paper presents the different solutions possible of the 2-dimensional configuration in

terms of the different chord positions b, c, d, e and g as shown in table-15 below:

27

Table-15: Crux of CS Rao’s Mathematical Derivation

b c d e g 7D to 9D Sani to Sikhi

Bindu to Sani Bindu to Kuja 1D to 3D Surya-Kuja

Bindu-Guru

0.464 11.1 0.223 5.4 0.289 6.9 0.488 11.7 0.106 2.5

0.456 11.0 0.237 5.7 0.283 6.8 0.456 11.0 0.105 2.5

0.438 10.5 0.218 5.2 0.269 6.5 0.440 10.6 0.097 2.3

0.467 11.2 0.261 6.3 0.305 7.3 0.472 11.3 0.120 2.9

0.469 11.2 0.257 6.2 0.308 7.4 0.481 11.5 0.122 2.9

0.561 13.5 0.279 6.7 0.279 6.7 0.514 12.3 0.101 2.4

0.482 11.6 0.261 6.3 0.287 6.9 0.467 11.2 0.108 2.6

0.500 12 0.250 6 0.292 7 0.458 11 0.042 1

Here the last row represents the classical precepts. Rao’s solutions differ drastically from the

classical ones in the matter of the elements of Jupiter’s triangle which apparently has the

height increased upon the chord of ≈ 12 units. As a result, the side angle opens up to

eliminate or distort the quadrangle and the 8-triangle enclosure also gets noticeably uneven

triangles at the centre. Rao’s solutions depicted on p.225 also suggest that the Surya-Kuja

height decides the position of Surya relative to the solstitial circle.

The optimal divisions of the diameter derived by Rao are used below to have a look at their

performance:

Table-16: CS Rao Elements for Plane Form

Triangle Diameter Divisions

Base Base

CorrectedHeight

Apex Angle

No. of Kundamsa

C-ircle Radius

Planet

1D 5.8839 27.548 27.254 20.7192 66.67 15.00 14.84 Sun

2D 12.1018 34.738 34.341 29.7404 60.00 13.50 19.83 Moon

3D 17.1011 45.974 46.007 30.8989 73.33 16.50 24.01 Mars

4D 19.8549 15.760 15.545 16.2549 51.11 11.50 9.99 Merc

5D 22.5986 11.980 11.884 7.6663 75.56 17.00 6.14 Guru

6U 26.6031 15.906 16.003 14.5013 57.78 13.00 9.46 Sukra

7U 30.2649 46.336 46.914 30.2649 75.55 17.00 24.22 Sani

8U 36.1098 34.535 34.902 30.2259 60.00 13.50 20.15 Rahu

9U 41.8422 28.090 28.568 24.7411 60.00 13.50 16.49 Ketu

Data has been worked out using the classical instructions i.e. 3, 4, 0, 16, 18, 16, 0, 4 and 3

parts reduction of the chords, for fixing the bases of the 9 triangles. It is apparent that the

optimal configuration found by CS Rao is only one of the numerable configurations and the

Kundāmśa rule can be applied here as well. Corrections to the base on an average are only

1-2% and the method had yielded the 9U triangle also as equilateral like the 2D and 8U.

It is possible that the erased parts used here may not be applicable to the Rao division of the

east-west diameter. Computation is given just to complete the discussion and to convey that

28

any such novel efforts can also be understood in contrast to the traditional method illustratd

earlier.

Fig.11 Table-17

Solution obtained from the

Sriyantraexplorer for CS Rao elements have no tantrik characterization at all as may be

noted from the data given above. As may be noted the Jupiter triangle which is the crux of

Sri Chakra has the angle configuration 52:52:76 and hence the 8 triangle enclosure is

distorted with uneven triangle sizes. Jupiter angle greater than 50 means uneven ashtāram

even though concurrence and concentricity are achieved. Equilaterality also has no meaning

if the 8-triangle enclosure misses its aesthetic character seen in fig.1 with the angle 49.5 for

Jupiter. The distortion of ashtaram in the perfect mathematical solution of Rao CS is similar

to the conventional approach.

Unless the fixtures like 49.5 or 50 for Jupiter, equilaterals for Moon and Rahu and 72 degree

apex for the Sun, innumerable configurations can be derived which attempts to mimic the

sacred abode of Bhagavati.

8. The Coverage of the Circle by Triangles

The coverage of the circle by the 9 triangles is an aesthetic issue which demands tāntrik

characterization. If we are to follow the prevaling conventions, Sri Chakra means the 9

interlocking triangles placed somewhere, somehow in the 24 unit circle. If that is the case,

what need existed for defining a circle of 24 units?

Tradition had missed the import that the 24 unit circle is a reference circle – the zodiac or

luni-solar ecliptic itself and the geometric representation of her effulgence in action to create

and sustain the process as the Mother of the evolving Universe cannot be a floating motif

within the circle. Her effulgence manifests in the terrestrial realm from Her macrocosmic

No. Position of chords

Angles % Units

From Top

1 0.8566 6.88 5.884 57.22

2 0.7477 12.11 12.102 60.98

3 0.6397 17.29 17.101 53.11

4 0.5876 19.80 19.855 65.75

5 0.5285 22.63 22.599 52

6 0.4473 26.53 26.603 60.57

7 0.3702 30.23 30.265 52.52

8 0.2464 36.17 36.110 60.26

9 0.1418 41.19 41.842 64.82

29

presence as the 9-planets (nava-grahas or nava-yonis or nava-lingas or nava-nagas) and

hence it is quite appropriate to have a configuration where in the Sun triangle leads down

the pattern from the circumference and the triangle motif occupies maximum space in the

24 unit circle. There is no void space for her presence and so the connection between the

circle and the triangle at 00 is essential to have minimum void space. Mars & Saturn

triangles have their vertices on the circle and given the Sun-Mars-Saturn classification as the

major malefics in contrast to Moon-Mercury-Jupiter-Venus group of benefics, it is natural to

expect the Sun triangle to touch the declinational circumcircle and Ketu to be at a point as

far down as possible towards the west on the East-West line.

Given the correct configuration explained earlier with Jupiter 49.5:49.5:81, Moon & Rahu

equilateral and Sun of apex 720, the area covered by the 9 Mūlaprakrtis 1192 units2 in 1810

unit2 of the circle. So the 9 Mūlaprakrtis occupy an area of 65.88% ≈ 66.66 i.e. 2/3rd of the

terrestrial space below the 24 unit declinational circle (Krāntivrttam). Any change in Jupiter

from the classical elements brought out in the study apparently causes only distortions as we

cannot replace the wholesome siddha wisdom that has gone into the conception of this

geometric representation of the Universe.

9. Extending to Larger Scales

The 9-Mūlaprakrtis are invariant against any scale transformations. Table-8 below presents

the relevant data.

Table-18: Circumcircle of 500 mm Radius

Tri- angle

Position from Top mm

Chord Length mm

Base used mm

Base Corrected

Apex angle

Kundāmsa 4.4440 Units

Radius Circumcircle

Reduced Parts for chords

Actual versus Classical

1 2 4 5 6 7 8 10 11 12

1D 125 661.44 578.75 575.428 66.6 15.0 313.35 3.121 3

2D 250 866.03 721.68 721.601 59.9 13.5 416.64 4.002 4

3D 354.167 956.52 956.52 961.490 73.3 16.5 501.85 -0.125 0

4D 416.667 986.01 328.67 334.763 53.3 12.0 208.69 15.852 16

5D 479.167 999.13 249.78 244.703 80 18.0 124.24 18.122 18

6U 562.5 992.16 330.71 329.178 55.5 12.5 199.59 16.037 16

7U 625 968.25 968.24 968.693 75.5 17.0 500.17 -0.011 0

8U 750 866.03 721.68 721.601 59.9 13.5 416.64 4.002 4

9U 875 661.44 578.75 574.702 57.7 13.0 339.68 3.147 3

Columns 11& 12 are illustrative that the reduced parts of the chords known in the tradition

has been derived using the concept of Kundāmśa to characterize the 9 Mūlaprakrtis.

Similarly elements for any other circumcircle can be determined.

30

10. Feasibility and Errors in Drawing

When the first discussion on the mathematical aspects and the circumstances detailing the

origin of this work got released, there was no significant experience at all as to whether the

mathematics can lead to a construction of the Sri Chakram. In the days that followed,

Suresh took up the challenge and real expertize evolved during the last two weeks ending

today with Moon on Pusya nakshatra. Mars and Moon excel in their miracles with the

display of red in unfailing manner and the following data on the errors in Marma are

noteworthy: Fig.12 and the associated data are presented in table-19:

Table-19: Concurrence achieved in Autocad

Triple Jn: 48 U 1080 U

Triple Jn: 48 U 1080 U

a 0.004644 0.1045 i 0.0 0

b 0.004644 0.1045 j 0.0 0

c 0.000032 0.000718 k 0.000028 0.000634

d 0.000032 0.000718 l 0.000030 0.000694

e 0.002373 0.0534 m 0.000030 0.000694

f 0.002373 0.0534 n 0.000028 0.000634

g 0.001581 0.035581 o 0.000014 0.000314

h 0.001581 0.035581 p 0.000014 0.000314

It is apparent from the above data that the mathematical derivation given above is realizable

through drawing manually as well as using software tools like Autocad with the required

perfection. Few drawings from Suresh Kesvapillai are reproduced below:

31

Fig.13

Fig.14

32

11. Conclusions

The study was presented as it progressed on returning from the abode of Tripurasundari at

Kanchi in the document released on 17 December 2012. As a physicist a dream carrying her

instructions to complete the intended work can be set aside but given the outcome of the

present study, she has presented herself with another miracle. Her graceful presence at

Kanchi, Chengannur and Kollur has substantiated her presence between 17 December 2012

and today 30 December 2012 by bringing out the truth of the mathematical derivation of the

chords.

Not having enough time to draw out a conclusion carrying the sum up, I am leaving the

discussion as it is for the discerning reader to draw the right conclusion. The traditional

instructions when rightly interpreted and applied leads to the right configuration of Sri

Chakram which can be qualified in terms of the tantrik rationales. The crux of the matter is:

Q/5RS/5 12345. � ,-./ � 012345 � 6 � εεεε � 1234578 9 →→→→ 1344/ST U/T235

Q/5RS/5 12345. � ,-./ � �12345 � 6 � εεεε� →→→→ V43WX Y4-STZS/

There is a lot more to be discussed in respect of the application of the Sri Chakram and the

astronomical aspects of its application. Kārttikā related legends on Skana-Kārttikeya and

Kumāri Kārtyayani, suggests great antiquity for Sri Chakram, traceable to the Indus Valley

days of Tantra. I have already circulated a note dated 15.12.2012 on the spiritual relevance

of Sri Chakram and few ideas discussed by way of few verses as well.

It is well evident to all those who have been associated with me and familiar with the works

that the present work on Sri Chakram is a sequel to the work on Jyotischakram and the

esoteric evidences produced for the same in works like the ‘Greatest Mahavidya Ritual at

Chitor’. The unearthing of the great sacrifice that matched the rare astronomical

configuration of new moon and Kumbha-samkrama and the astronomical aspects involving

444.444 years finds their mysterious presence in this Sri Chakra study also. Mārgśīrsha

Pournami on Friday that preceded also had been significant in the context of the great

effulgence at Chengannur (known since the times of Ilango Adikal at least as

Chenkamalavalli) and Kollur.

Acknowledgments

I take this opportunity to express my gratitude to Sri Sudarsan Raj Tiwari who undertook

the laborious task of giving a detailed discussion on the draft work and the same reached me

last night 29.12.2012. His views add to my confidence that others will be able to recognize

the sacred Siddha wisdom of the classical precepts. I would like to quote the following

remarkable words of Tiwari –

33

(6) For me the most interesting and refreshing inference you make is that the rub offs are correct integer

values as 24th part of the chord itself or Base=chord - 2ɛ x chord/48. Apparently, you take that the

given divisions of the brahmasutra to locate the chords are also as accurate as it is. I think it is

important to note that integer approximations given in ancient documents do not mean that they were

wrong or that the ancients did not know the accurate solutions, only that the style of communication

(through round figures and ‘iyad mahad rahasyam’ type of phrases made the exact figures an esoteric

knowledge that was transmitted only to the initiated.

Thanks are also due to Devipuram for remaining as a source of expression of the glory of Srī

Vidyā. Kamākshi had given the task of arranging darsanam & abhishekam at Kanchi with

Prasanna P Nair and a search for Sri Chakram these days cannot miss the Devipuram

structure and its exponent Dr N.Prahlada Sastri. Some people may have looked upon my

notes as a criticism of the conventional approach but to me the quest for truth is a means to

the manifestation of her role as the revealer. I acknowledge my pathway through

Devipuram – the first source of study materials outside my own library and the reference I

had of Mr Krishnan Nair’s work.

Q/5RS/5 12345. � ,-./ � �12345 � 6 � εεεε � 1234578 �

Derivation as above of the 9 bases of Mūlaprakrtis and realization in drawing substantiates

her footprints cited in the note of 17 December 2012. In my works I am a Physicist and

speaking about miracles to the public is not to good taste as such. Whatever I have spoken is

only to express the gratitude and highligh the truth of the Pithas for those who regard Sri

Chakram as a sacred object.

Last but not the least, I would like to express my profound regards for Sri Shyam V Rao whose words

have been a source of inspiration to my intellectual pursuits. She has manifested herself as the Universe

and in Her eyes all are placed rightly and a true Sakta won’t be aggrieved of any predicament.