Borromean RingsArunn Narasimhan
My colleague Prof. Arul Lakshminarayanan recently wrote in Resonance about the Borromean triangles embossed on the stone pillars of Marundeeswarar temple, a place of worship few miles from my institute. [article pdf]. Following is my extended note on Borromean rings.
Observe the accompanying picture of a stone inscription at the Marundheeswarar Temple [front picture]. At first glance, it looks like Sri-Chakra, the familiar Hindu symbol. It isn’t. The overlapping triangles do something more interesting. Observe how if each triangle is traced, it goes “under” or “above” the other two triangles, although it forms a coupling on the whole. Pull out one triangle from the inscription, the other two fall off. They can be taken out separately, as they lie one over the other.
The basic idea of a Borromean ring is that the three rings are inseparable when taken together, but if one of the ring is taken out, the other two fall apart. It is an idea where the sum is more than the parts. An idea where things that individually don’t stand up to make sense, when combined not in pairs but in three, makes something meaningful.
A possible example is the model of the atom, with its usual proton, electron and the neutron. A combination of all these three forms a stable atom, while individually these components are unstable.
Arul gives another example in his article
The adjective Borromean is in use in few-body quantum systems it describes the situation where a 2-body configuration is not stable, while a 3-body configuration may be. An example is provided by halo nuclei with some neutrons loosely bound to a core, such as in the case of 6He, which is stable against dissociation while 5He is not. Thus while the 3-body configuration involving (&alpha, ,n,n) is stable (there exists a bound state), the 2-body ones (&alpha,n) and (n,n) involving a Helium nucleus (&alpha) and a neutron (n) or just two bare neutrons are unstable (there are no bound states).
The etymology of Borromean rings from wikipedia reads
The name “Borromean rings” comes from their use in the coat of arms of the aristocratic Borromeo family in Italy. The link itself is much older and has appeared in the form of the valknut on Norse image stones dating back to the 7th century.
A detailed web page dedicated to Borromean rings now include the pictures taken by Arul.
The late Australian sculptor John Robinson (4/5/1935 - 6/4/2007) used extensively such mathematical curiosities in his sculptures. Accompanying picture is one of his Borromean rings sculpture, using three rhombuses made of stainless steel offering nice light reflections. The sculpture called Genesis is located in the green lawns of the University of Wales. You can visit the website that discusses his works extensively.
Intrigued, I attempted to create the Borromean ring at home. Overlapping circles will not do, as proved rigorously in the “Borromean circles are impossible,” Amer. Math. Monthly, 98 (1991) 340-341, by B.Lindstrom and H.-O. Zetterstrem. So I started with ellipses. From household rubber bangles suitably squashed to form ellipses, here is my attempt to create Borromean rings in pictures.
Obviously, one of the three bangles has to be compromised of its topology. I hope the Mystery of the Three Missing Rubber Bangles will not be solved by my daughter too quickly.
The home-made borromean rings is also shown below in almost (forced) 2 dimension. Compare it with the 2-D rings and the 3-D orthogonal rings (formed with bangles) shown above. The second picture below shows how a “cut” is required to form the borromean ring at home.
Borromean rings are used as symbolisms for the Christian Trinity. Even in the Marundheeswarar Temple, as Arul observes, the inscription of the Borromean triangles is found near the sanctum of Tri-pura-sundari, the female deity, perhaps symbolizing Shakthi, the female omniscience as the unification of the Trinity of the Hindu Gods, Brahma, Vishnu and Shiva.
As Arul writes in his article, Marundheeswarar temple
[...] has been in existence from about the 6th century AD (it has been sung by Saivite saints of the 8th century) and it has at least 11th century inscriptions, I cannot comment on the era inwhich 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 (yantras) is prevalent in both Hindusim 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 may be part of a larger spectrum.
Recent ripples of this idea in scientific community include the molecular borromean reported in 2004 by the research group of J. Fraser Stoddart in the Science Vol 304, Issue 5675, 1308-1312 , 28 May 2004 [Abstract]. A recent 2007 paper in the J. of Chemical Education reports [Abstract] how to make these molecular borromean rings in the undergraduate laboratory.
I am now on the look out for more such intrigues in our temples. A nice reason to visit them I guess.