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 5 th 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 36 0 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 72 0 so that the bisector or the prime vertical is oriented towards the sidereal longitude of Krttikās. 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
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� → ��� ������� 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.
2
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.
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
5
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
6
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.
7
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