ĹšrÄŤ Chakram: The Geometric Duet in Praise of Tripura-Sundari K. Chandra Hari Abstract Present study seeks to bring out the sacred rationale SrÄŤ 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 tÄ ntrik 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 Îľ 2 Îľ
48
→
→
With the right bases derived, the traditional elements lead to precise angles which are integer or halfinteger multiples of KundÄ mĹ›a 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 KrttikÄ (Alcyone) nakshatra. Result as above of the 360 bisector of the solar triangle finds credence with the legends on KÄ rtikeya vis-a-vis TrikkÄ rtika 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 ttikÄ s. 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 19year luni-solar period which began with the SivarÄ tri of 1994 and set to conclude on the next SivarÄ tri of Kumbham, 10 March 2013, coinciding solar transit of 325:25. 1
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 Chakrapūjakalpam 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 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. 2
2
+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 devīkrte... etc presents the correct classical method. 3
´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 4
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 5
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. 6
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. Mandala 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 7
{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 Mandalas 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 Kundapushpam 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: 360
4.4444& 81
80
0.987654321& 81 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 KundaLagna completes 1 revolution. In other words, 1 nakshatra of Kunda-Chakra is equivalent to 9.87654321 minutes of the Jyotischakra. 800
9.87654321 800 81 9.87654321& 81 i.e. 800 minutes of Kundachakra or 1 Nakshatra of Kundachakra is 9.87654321’ of the Jyotischakra observed with the naked eye. 800 * 27 9.87654321 * 27 266.66667+ 4.4444& 81 360 360
1.23456789 81 100 8
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 Mandalas 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: 9
Table-1: Chord Lengths and Bases Derived Position from
Triangle Radius 1 2 3 4 5 6 7 8 9
24 24 24 24 24 24 24 24 24
Top
Centre
6 12 17 20 23 27 30 36 42
18 12 7 4 1 3 6 12 18
Chord
Base
Planets
31.75 41.57 45.91 47.33 47.96 47.62 46.48 41.57 31.75
27.780 34.641 45.913 15.776 11.990 15.875 46.476 34.641 27.780
Sun Moon Mars Mercury Jupiter Venus Saturn Rahu 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 ε Triangle
Base
Height
1D 2D 3D 4D 5D 6U 7U 8U 9U
27.780 34.641 45.913 15.776 11.990 15.875 46.476 34.641 27.780
21 30 31 16 7 15 30 30 25
12345 78
Equal Sides Angles 25.178 56.52 34.641 60.00 38.575 53.48 17.839 63.76 9.216 49.42 16.971 62.11 37.947 52.24 34.641 60.00 28.600 60.94
9 and Triangle Elements Apex Angle 66.96 60.00 73.04 52.49 81.15 55.77 75.52 60.00 58.11
Sum of Circumcircle Angles Radius 180.0 15.09 180.0 20 180.0 24 180.0 9.94 180.0 6.07 180.0 9.60 180.0 24 180.0 20 180.0 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 10
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 Mūlaprakris 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 1 1D 2D
Apex Base No. of Apex Corrected New Base Correction Kundāmsa correction Angle Kundāmsa Apex Base % ′ 2 3 4 5 6 7 9 10 66.96 15.07 27.780 66.666 15 27.624 0.6 -18′
3D 4D
60.00 73.04 52.49
13.50 16.43 11.81
34.641 45.913 15.776
60 73.333 53.333
13.5 16.5 12
5D 6U
81.15 55.77
7U 8U 9U
75.52 60.00 58.11
18.26 12.55 16.99 13.50 13.08
11.990 15.875 46.476 34.641 27.780
80 55.555 75.555 60 57.777
18 12.5 17 13.5 13
0′ 17′ 51′ -69′ -13′ 2′ 0 -20′
34.641 46.158 16.071
0.0 -0.5 -1.9
11.747 15.802 46.504 34.641 27.589
2.0 0.5 -0.1 0.0 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. 11
(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 Mandalas) 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
1D 2D 3D 4D 5D 6U 7U 8U 9U
27.780 34.641 45.913 15.776 11.990 15.875 46.476 34.641 27.780
21 30 31 16 7 15 30 30 25
Circumcircle Radii R Original New 15.09 15.04 20.00 20.00 24.00 24.09 9.94 10.02 6.07 5.96 9.60 9.58 24.00 24.01 20.00 20.00 16.36 16.31
12
Change R in % 0.3 0.0 -0.4 -0.8 1.8 0.2 0.0 0.0 0.3
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 Mandala 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 1 31.75 41.57 45.91 47.33 47.96 47.62 46.48 41.57 s31.75
Base used 2 27.780 34.641 45.913 15.776 11.990 15.875 46.476 34.641 27.780
Erased Parts 3 3.0 4.0 0.0 16.0 18.0 16.0 0.0 4.0 3.0
Corrected Erasers Base new 4 5 27.62404 3.12 34.64102 4.00 46.1577 -0.13 16.0705 15.85 11.7474 18.12 15.80242 16.04 46.50361 -0.01 34.64102 4.00 27.58897 3.14
% 6 3.9 0.0 0.0 -0.9 0.7 0.2 0.0 0.0 4.8
Corrected Kundāmsa Apex 5 6 66.666 15 60 13.5 73.333 16.5 53.333 12 80 18 55.555 12.5 75.555 17 60 13.5 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. 13
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 Mūlaprakrtis Tri- Classical Apex Erasers Apex Change Kunda Erasers Apex Change Kunda angle erasers angles +2 angle in apex fraction (-1) angle in apex 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 8U
0
75.56
0
75.56
0.00
0
0
75.56
0.00
0.00
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 14
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 Mūlaprakrti consisting of the 4 Srikanthas 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 15
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 1D 2D 3D 4D 5D 6U 7U 8U 9U
1 2 3 4 5 6 7 8 9
Classical Angles Side Apex 56.52 66.96 60 60 53.48 73.04 63.76 52.49 49.42 81.15 62.11 55.77 52.24 75.52 60 60 60.94 58.11
Corrected Apex Side 66.67 56.67 60.00 60.00 73.33 53.33 53.33 63.33 80.00 50.00 55.56 62.22 75.56 52.22 60.00 60.00 57.78 61.11
Planet Sun Moon Mars Mercury Jupiter Venus Saturn Rahu 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 8 < No.
16
1
Position of chords From Angles % Units Top 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
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 â&#x2030;&#x2C6; 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). â&#x20AC;˘
5D = 50, 2D = 8D =60
â&#x20AC;˘ 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: 17
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. 18
Concurrency , Concentricity , User Angle U2: 60° , User Angle D2: 60° , User Angle
(Fig.4)
Seed : 80.48 , 0.26 , 0.29 , 0.47 , 0.12 <
Relevant data is furnished in Table-9 below: Triangle
Base
Height
1D 2D 3D 4D 5D 6U 7U 8U 9U
27.780 34.641 45.913 15.776 11.990 15.875 46.476 34.641 27.780
21 30 31 16 7 15 30 30 25
Equal Software Sides Angles Solution 54.727 25.178 56.52 60 34.641 60.00 53.334 38.575 53.48 60.726 17.839 63.76 50 9.216 49.42 59.880 16.971 62.11 51.589 37.947 52.24 60 34.641 60.00 62.478 28.600 60.94
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 19
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 ε (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 Chord2ε*Chord/48 and not Chord - 2ε. ?G>HIG> :;<=>F DEFG 0:;<=> 6 ε 20
:;<=> 78
9
→ :<==GIJ KGJ;<>
?G>HIG> :;<=>F DEFG :;<=> 6 ε
→ 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 Eraser Positio Base Chord ε1 = ε2 = Parts Sl.No. Radius From From C 2ε*C/48 2ε Tradition True ε Top Centre 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 21
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. 1 2 3 4 5 6 7 8 9
Beginning Starting Planet Point Ketu 0 Chandra 40 Guru 80 Ketu 120 Chandra 160 Guru 200 Ketu 240 Chandra 280 Guru 320
End Point 40 80 120 160 200 240 280 320 360
End Star Krittika Ardra Sarpa Utram Swati Jyeshtha Utradam Satabhishak Revati
End Planet Sun Rahu Budha Sun Rahu Budha Sun Rahu Budha
Triangle Sun Moon Mars Merc Jup Ven Sat Rahu 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 22
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 bisecting 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
23
by taking 49.5 â&#x2030;&#x2C6; 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
1 2 3 4 5 6 7 8 9
3 4 0 16 19 16 0 4 3
24 24 24 24 24 24 24 24 24
Position from Top Centre 6 18 12 12 17 7 20 4 23 1 27 3 30 6 36 12 42 18 24
Chord 31.75 41.57 45.91 47.33 47.96 47.62 46.48 41.57 31.75
Base used Correct Traditional 27.780 25.749 34.641 31.569 45.913 45.913 15.776 15.329 11.990 9.958 15.875 15.624 46.476 46.476 34.641 33.569 27.780 25.749
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 1 2 3 4 5 6 7 8 9
3 4 0 16 19 16 0 4 3
Base
Position
25.749 31.569 45.913 15.329 9.958 15.624 46.476 33.569 25.749
21 30 31 16 7 15 30 30 25
Equal sides angles 24.632 58.49 33.899 62.25 38.575 53.48 17.741 64.40 8.590 54.58 16.912 62.49 37.947 52.24 34.376 60.77 28.120 62.75
Angle in solution 59.435 62.902 53.685 66.027 54.58 62.909 52.738 62.531 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 25
(Fig.10) Relevant data is provided in table-14 below: Sl.No. Eraser 1 2 3 4 5 6 7 8 9
3 4 0 16 18 16 0 4 3
Base
Position
25.749 31.569 45.913 15.329 11.958 15.624 46.476 33.569 25.749
21 30 31 16 7 15 30 30 25
Equal sides angles 24.632 58.49 33.899 62.25 38.575 53.48 17.741 64.40 9.206 49.50 16.912 62.49 37.947 52.24 34.376 60.77 28.120 62.75
Angle in solution Angle = 49.5 Angle = 54.58 53.955 59.435 60.4474 62.902 52.976 53.685 62.135 66.027 49.5 54.58 58.081 62.909 51.996 52.738 59.890 62.531 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 Raoâ&#x20AC;&#x2122;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: 26
Table-15: Crux of CS Rao’s Mathematical Derivation
b 7D to 9D Sani to Sikhi 0.464 11.1 0.456 11.0 0.438 10.5 0.467 11.2 0.469 11.2 0.561 13.5 0.482 11.6 0.500 12
c
d
Bindu to Sani
Bindu to Kuja
0.223 0.237 0.218 0.261 0.257 0.279 0.261 0.250
0.289 0.283 0.269 0.305 0.308 0.279 0.287 0.292
5.4 5.7 5.2 6.3 6.2 6.7 6.3 6
e 6.9 6.8 6.5 7.3 7.4 6.7 6.9 7
1D to 3D Surya-Kuja 0.488 11.7 0.456 11.0 0.440 10.6 0.472 11.3 0.481 11.5 0.514 12.3 0.467 11.2 0.458 11
g Bindu-Guru 0.106 0.105 0.097 0.120 0.122 0.101 0.108 0.042
2.5 2.5 2.3 2.9 2.9 2.4 2.6 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 1D 2D 3D 4D 5D 6U 7U 8U 9U
Diameter Divisions 5.8839 12.1018 17.1011 19.8549 22.5986 26.6031 30.2649 36.1098 41.8422
Base 27.548 34.738 45.974 15.760 11.980 15.906 46.336 34.535 28.090
Base Corrected 27.254 34.341 46.007 15.545 11.884 16.003 46.914 34.902 28.568
Height 20.7192 29.7404 30.8989 16.2549 7.6663 14.5013 30.2649 30.2259 24.7411
Apex No. of C-ircle Angle Kundamsa Radius 66.67 15.00 14.84 60.00 13.50 19.83 73.33 16.50 24.01 51.11 11.50 9.99 75.56 17.00 6.14 57.78 13.00 9.46 75.55 17.00 24.22 60.00 13.50 20.15 60.00 13.50 16.49
Planet Sun Moon Mars Merc Guru Sukra Sani Rahu 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 27
any such novel efforts can also be understood in contrast to the traditional method illustratd earlier. Fig.11
Table-17 No. Position of chords From Angles % Units Top 1 0.8566 6.88 5.884 57.22 2 3 4 5 6 7 8
0.7477 12.11 12.102 60.98 0.6397 17.29 17.101 53.11 0.5876 19.80 19.855 65.75 0.5285 22.63 22.599 52 0.4473 26.53 26.603 60.57 0.3702 30.23 30.265 52.52 0.2464 36.17 36.110 60.26 0.1418 41.19 41.842 64.82
9 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 ashtÄ ram 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 tÄ ntrik 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 â&#x20AC;&#x201C; 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 28
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 Triangle 1 1D 2D 3D 4D 5D 6U 7U 8U 9U
Position from Top mm 2 125 250 354.167 416.667 479.167 562.5 625 750 875
Chord Length mm
Base used mm
4 661.44 866.03 956.52 986.01 999.13 992.16 968.25 866.03 661.44
5 578.75 721.68 956.52 328.67 249.78 330.71 968.24 721.68 578.75
Kundāmsa Base Apex Radius 4.4440 Corrected angle Circumcircle Units 6 575.428 721.601 961.490 334.763 244.703 329.178 968.693 721.601 574.702
7 66.6 59.9 73.3 53.3 80 55.5 75.5 59.9 57.7
8 15.0 13.5 16.5 12.0 18.0 12.5 17.0 13.5 13.0
10 313.35 416.64 501.85 208.69 124.24 199.59 500.17 416.64 339.68
Reduced Parts for chords Actual versus Classical 11 12 3.121 3 4.002 4 -0.125 0 15.852 16 18.122 18 16.037 16 -0.011 0 4.002 4 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. 29
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 Pusď&#x20AC;Şya 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: a b c d e f g h
48 U 0.004644 0.004644 0.000032 0.000032 0.002373 0.002373 0.001581 0.001581
1080 U 0.1045 0.1045 0.000718 0.000718 0.0534 0.0534 0.035581 0.035581
Triple Jn: i j k l m n o p
48 U 0.0 0.0 0.000028 0.000030 0.000030 0.000028 0.000014 0.000014
1080 U 0 0 0.000634 0.000694 0.000694 0.000634 0.000314 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: 30
Fig.13
Fig.14 31
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 ε
12345
Q/5RS/5 12345. ,-./ 12345 6 ε
78
9
→ 1344/ST U/T235
→ 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 –
32
(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 - 2É&#x203A; 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 â&#x20AC;&#x2DC;iyad mahad rahasyamâ&#x20AC;&#x2122; 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 SrÄŤ VidyÄ . KamÄ kshi 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 â&#x20AC;&#x201C; the first source of study materials outside my own library and the reference I had of Mr Krishnan Nairâ&#x20AC;&#x2122;s work. Q/5RS/5 12345. ,-./ 12345 6 Îľ
12345 78
Derivation as above of the 9 bases of MĹŤlaprakrtis 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 wonâ&#x20AC;&#x2122;t be aggrieved of any predicament.
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