Preamble:

This blog is a "popular" article and maybe of general interest, because it is about some pretty math in an ancient Chennai temple's pillars, kind of math that is current and

used in physics and science in general. This does not mean that the people who made

these pillars had these in mind. In fact they must have had some entirely different truth

to convey, but found the same symbolism useful. If we keep our eyes, ears, and importantly minds, open India can be truly fascinaing. Please keep in mind that the pictures of pillars in this blog are from a temple taken with permission of the temple authorities. Although I dislike any form of copyright on things I did not create, I ask you to use these if you must with care.

This blog is a "popular" article and maybe of general interest, because it is about some pretty math in an ancient Chennai temple's pillars, kind of math that is current and

used in physics and science in general. This does not mean that the people who made

these pillars had these in mind. In fact they must have had some entirely different truth

to convey, but found the same symbolism useful. If we keep our eyes, ears, and importantly minds, open India can be truly fascinaing. Please keep in mind that the pictures of pillars in this blog are from a temple taken with permission of the temple authorities. Although I dislike any form of copyright on things I did not create, I ask you to use these if you must with care.

Borromean triangles and prime knots in an ancient temple

The ancient Marundheeswarar temple in Thiruvanmayur, South Chennai,

has a series of pillars with beautiful geometric designs that are quite surprisingly fairly sophisticated mathematical motifs of contemporary scientific interest. The motifs surround the sanctum of the goddess "Tripurasundari", the "belle of the three cities". The number three is crucial in the motifs under advertisement and the irreducible tripartite nature of the divinity is emphasized through links and knots, which have their usual meanings as well as precise mathematical ones.

has a series of pillars with beautiful geometric designs that are quite surprisingly fairly sophisticated mathematical motifs of contemporary scientific interest. The motifs surround the sanctum of the goddess "Tripurasundari", the "belle of the three cities". The number three is crucial in the motifs under advertisement and the irreducible tripartite nature of the divinity is emphasized through links and knots, which have their usual meanings as well as precise mathematical ones.

The first of the patterns is a set of three identical overlapping equilateral triangles at whose center is a four petalled flower. Unlike the two-dimensional yantras which typically have several overlapping triangles, this one is sculpted with the third dimension in mind.

we can make out when one triangle goes ``over'' another. The three triangles overlap in a very specific and remarkable way: no two of the three triangles are linked to each other, but the three are inextricably collectively linked; if any one of the triangles is removed the other two fall apart as well.

Borromean Triangles at Marundheeswarar temple, Thiruvanmayur, Chennai.

Modern mathematics classifies this object as a "Brunnian link". Formally a link is a collection of "knots" that do not intersect each other, but may otherwise be linked, such as simply two interlinked hoops. A "knot" conforms to our conventional notion of a string looping around itself, but mathematicians prefer that the ends of the string be joined together.

Thus the single hoop is a string that is not knotted at all, and is therefore called an "unknot". A Brunnian link (after H. Brunn, German mathematician who published his work on knot theory in the late nineteenth century) is a link such that if any one of the component knots is removed the remaining ones become ``trivial'' and fall apart into unlinked knots. It is clear that the three triangles of the Tripurasundari temple form precisely such a link.

However the best known and simplest example of a Brunnian link are the

{\em Borromean circles}, three circles interlinked in such a manner that no two of them

are linked but all three are simultaneously linked. The name derives from the medieval

aristocratic Borromeo family from northern Italy who used this symbol extensively,

including in their coat of arms. It is presumed that it signified the inseparable

union of three powerful families at that time. The symbol however has been found in several

other places and predates the medieval Italian family.

It appears that a version of the Borromean links with three triangles appears on 7th century

scandinavian rune stones where the god Odin is shown with these symbols called ``valknuts", meaning slain warriors' knots. The links found on the pillars of the Marundheeswarar temple appear to be a symmetrical version of this symbol. Borromean motifs have also been found in Japanese shrines and family emblems, and in medieval Christian iconography where it reconciles monotheism with the potential polytheism implied by the Trinity of the Father, the Son and the Holy Ghost.

While the Borromean circles and triangles are topologically similar, in the sense that a cirlce can be deformed into a triangle, they are geometrically quite different. It has been established mathematically, fairly recently, that Borromean circles of any relative sizes are impossible. Thus there cannot be an actual three-dimensional realization of the Borromean circles. This constraint of geometry does not forbid Borromean configurations for other shapes such as ellipses, triangles or golden rectangles. Golden rectangles are those whose sides are that of the golden ratio; three such rectangles can be inscribed in a regular icosahedron (polyhedra with 20 faces, each an equilateral triangle).

The Institute of Mathematical Sciences at Chennai, not far from the Marundheeswarar temple, have recently taken up this ``golden'' version of the Borromean links for their logo. The International Mathematical Union adopted a three component Borromean link which minimized the link length when tied with a thick rope, as their logo in August 2006.

The universality of the Borromean symbol is very remarkable as it is found in disparate cultures and times. Ancient Indian uses of the symbol which must indeed be quite prevalent however do not seem to be documented. The Australian artist John Robinson has made Borromean sculptures using various shapes including triangles. It is indeed a lot fun and challenge to take up some cardboard and create ones own three-dimensional version of the Borromean triangles whose projection is found in the temple. Once created it is an object that is often the nucleus of conversations on Borromean matters! Actually making and feeling one brings home more clearly the singular and beautiful nature of such links.

The Borromean symbolism is in use in quantum physics in at least two different contexts.

In one it describes the situation where two objects by themselves cannot be bound together, while three of them can. An example is provided by ``halo nuclei'' with some neutrons loosely bound to a core, such as in the case of some isotopes of Helium. To put it colloquially two neutrons cannot live with each other, neither can an Helium nucleus live with a neutron, but two neutrons and the Helium nucleus together form a companionable triple. Another use of the Borromean links as a descriptive metaphor in quantum physics is in the study of quantum entanglement which is a peculiar quantum correlation that we do not observe in our everyday ``classical" world, but which could be a crucial resource for quantum computers, computers that are presumably much more powerful than the ones we use today. Thus there could be three objects such that no two of these are entangled, but all three are simultaneously entangled.

There are many other contexts in which Borromean rings have appeared in science, for instance DNA and other molecules have been knotted into Borromean configurations recently. The French psychoanalyst Jacques Lacan used the Borromean links in his lectures, with the elements being the ``real'', the ``symbolic'' and the ``imaginary'', apparently he considered that when any one of these aspects was absent it resulted in psychosis.

Returning to the Thiruvanmayur temple we could ask why such links were carved in their pillars. Indeed a moderate acquaintance with the ``esoteric'' aspects of Hinduism gives us several plausible answers. As mentioned earlier, the pillars of interest are near the sanctum of the goddess Tri-pura-sundari, The recurrent theme of the triad is therefore to be expected. The archetypal mantra AUM contains three parts. The yogi's three principal nadis the {\em ida}, the {\em pingala } and the central {\em sushumna} form a core tantric triad, which incidentally is also the symbolism of the staff of Caduceus with two snakes intertwinning around a staff. There are three {\em sakthis} or powers that derive from the goddess, the

{\em iccha} (desire), the {\em gnana} (knowledge) and the {\em kriya} (action), and the symbol maybe emphasizing that without any one of these the other two are useless. There is of course the triad of Brahma, Vishnu and Shiva, or in a more esoteric sense there are three knots ({\em granthis}) in the human body, the brahma granthi in the lower body,the vishnu granthi in the region of the heart and the rudra granthi at the center of the eyebrows. Tripurasundari is the single goddess or power that devolves into these three knots that essentially creates, sustains and dissolves.

This theme of a single power differentiating into three while remaining essentially one (the {\em Brahman} of the Upanishads) seems to be brought out in another symbol in a pillar of the same sanctum.

The Stevedores knot at Marundheeswarar temple, Thiruvanmayur, Chennai

This represents what looks like a snake knotting itself up into three parts, thus carrying within it again the irreducible triad. If the snake swallows its head or the ends are joined, we get the ``stevedore's" knot, which is known from the mathematical theory of knots to be a ``prime knot" with six crossings. A prime knot cannot be composed of smaller simpler knots. Just as there are prime integers out of which all the whole numbers can be constructed, so also there are prime knots. Knots have been the object of mathematical and scientific interest since Lord Kelvin studied these in the latter half of the nineteeth century as a model of atoms (``vortex atoms''). This idea was soon abandoned, but the theory of knots stayed on and was developed into a beautiful subject. After a lull, there was a resurgence of interest in this subject from about twenty years ago, when seminal results were found and concrete connections to modern physics emerged. Again, it is very instructive to actually take a piece of string and construct the knot as found in the pillar.

Apart from the motifs of the Borromean triangles and the stevedore's knot, the Tripurasundari sanctum pillars are decorated with other simpler geometric designs, notably the well-known Yin-Yang symbol, having a circle divided in two halves by smaller semicircles representing the Yin (female, {\em sakthi})

and the Yang (male, {\em siva}) energies.

An Ying-Yang symbol at the Marundheeswarar temple, Thiruvanmayur, Chennai.

While the temple has been in existence from about the 6-th century A.D. (it has been sung

by Saivite saints of the 8-th century) and it has at least 11-th century inscriptions, I cannot comment on the era in which the pillars with the motifs discussed here were carved. Such motifs are also certainly not unique to this temple and the use of geometric patterns ({\em yantras}) is prevalent in both Hinduism and Buddhism. Further explorations may throw up more intriguing uses of mathematics to build bridges with the inner worlds that these temples seek to connect. The Borromean triangles or the Stevedore's knot as logos of Tripurasundari maybe part of a larger spectrum.

It is with pleasure that I record the cooperation of the Marundheeswarar Temple authorities who allowed me to take photographs within temple precincts, and thank several of my colleagues and friends who clarified ideas.

One definitive sort of website on Borromean matters from which I learnt much is:

http://www.liv.ac.uk/~spmr02/rings/

This article was written by:

Arul Lakshminarayan

Department of Physics

Indian Institute of Technology Madras

Chennai 600036.

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## All Comments

10Mark/ / 8 months agoBorromean Triangles at Marundheeswarar temple, Thiruvanmayur, Chennai.

This reminds me of when the conches views the sub conches from the un conches.

If you understand .. you need no explanation. Wow! A powerful simple-symbol.

4 + 3 = 7

Three Broken Reconciled OpenTriangle Points // Three separate triangles with Nine Points.

The simplest way I experience this living simple symbol is as an injunction of two active forces; the stabilised viewing force of: "type point" and of: "Instinctual Point"

This symbol is as much about the Law of four , as it is, the Law of three. Some numb nuts will insist it is the Law of severn, those who do will not be able to understand its operation it as: "Qe"

The number patterns identify I will withhold. But if you have six number patterns constituting a one movement, then we see one of ,the same pattern movements. For one of 18 of course and for for one of 108 of-course. Eighteen Sub types Six Instincts.

"two on - two off - one centre / one Soul"

Cheers.

seo-ul-hack/ / 5 yrs agoWhat would be interesting are some notes on the history of the temple, the type of sculptors and the kinds of experience and exposure they would have had. It could well be a simple exposure to certain of these motifs and the attendant fascination that could have resulted in these constructs with no more special meanings than a mere copy cat...just musing.

Nevertheless, for the thousands that are exposed to this temple, it probably evokes a sense of intrigue and a kickstart to a journey to explore the meanings and ponder about the abstract.

Having said all that, kudos to you for bringing it to the attention of sulekha readers as worthy of note.

Bokonon/ / 6 yrs agodear raghuram, sekhar, riverine & sampath

i am very happy to see the comments and appreciation.

thank you!

cheers

arul

kolipakkam/ / 6 yrs agoi am coming to this after nearly a year and a half after posting and thanks to dmr sekhar. the piece is absolutely wonderful and enlightening. it has to be read by children, if not to understand what you say but merely to recognize that one can glean intersting bits of knowledge from remote corners.

the way you have narrated, it deserves commendation. you have maintained a balance between science and metaphysics. extremely laudable. the photographs are more than explanatory. they engage your attention. i did not take your word that removing one ring would release the other two. i checked it. this is how engaging the photographs are. thank you so much.

raghuram ekambaram

dmrsekhar/ / 6 yrs agofirst i thank sreenivasarao s for bringing this blog to my notice. and then arul has done a remarkable thing by posting this blog. the idea of connecting three circles in itself is great!! thanks.

dmr sekhar.

riverine/ / 6 yrs agobokonon (shri arul lakshminarayan)

i feel small that the temple close to me (in either sense) has so much to reveal which i had no eyes to see! very very interesting indeed! of course, the yin-yang is familiar (what with bp pumps also using the same symbol) and i have certainly seen the "stevedores knot" with some interest, but the underlying details were not within one's superficial vision.

this woke one up from the dream state:

"it is indeed a lot fun and challenge to take up some cardboard and create ones own three-dimensional version of the borromean triangles whose projection is found in the temple. once created it is an object that is often the nucleus of conversations on borromean matters! actually making and feeling one brings home more clearly the singular and beautiful nature of such links."

there is a great possibility that one would translate this knowledge into action (the desire is there, of course).

thanks a lot!

DSampath/ / 6 yrs agodear arul

this is a very well written blog..

and the way three compltete wholes are linked in a

formation of interdependence..

where each one becomes the connecting link of other two..

this formations is the only stable and most reliable formation of linking up three complete natural conditions....

the reaserch done by arul is very imporatnt in understaning the role of symbols in the transactional life..

very well researched..

jayakumar2020/ / 7 yrs agodear sir,

today i realise the wonderfull world of maths and how it plays an important role in everyday life. i have also came to an understanding of the mathematical activity during the vedic period where geometry and arithmetic virtually essential for revenue administrators.

thank you so much for you to share and hope we can learn more about this ancient mathematics which are limitless and brings us to different dimensions

take care..

jayakumar2020/ / 7 yrs agodear sir,

today i realise the wonderfull world of maths and how it plays an important role in everyday life. i have also came to an understanding of the mathematical activity during the vedic period where geometry and arithmetic virtually essential for revenue administrators.

thank you so much for you to share and hope we can learn more about this ancient mathematics which are limitless and brings us to different dimensions

take care..

mahaabodhi/ / 7 yrs agowow!!!

that was extremely beautiful.it was news to me.there is more to temples than religion.

when people are so engrossed in praying to the almighty goddess or god to fulfill all their wishes & dreams ,where do people have the time or inclination to look at the beauty around them.they lose the very essence for which these places were built.

thanks to people like you,we the very uninitiated have hope,we can look around & learn more.

please keep us posted on more.