In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings.
Although the typical picture of the Borromean rings (left picture) may lead one to think the link can be formed from geometrically round circles, they cannot be. (Freedman & Skora 1987) proves why a certain class of links including the Borromean links cannot be exactly circular. Alternatively, this can be seen from considering the link diagram: if one assumes that circles 1 and 2 touch at their two crossing points, then they either lie in a plane or a sphere. In either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible; see (Lindström & Zetterström 1991).Mathematical properties
It is, however, true that one can use ellipses (center picture). These may be taken to be of arbitrarily small eccentricity, i.e. no matter how close to being circular their shape may be, as long as they are not perfectly circular, they can form Borromean links if suitably positioned: for example, Borromean rings made from thin circles of elastic metal wire will bend.
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Linking
There are a number of ways of seeing that the Borromean rings cannot be unlinked.
Simplest is that the fundamental group of the complement of two unlinked circles is the free group on two generators, a and b, by the Seifert–van Kampen theorem, and then the third loop has the class of the commutator, [a, b] = aba−1b−1, as one can see from the link diagram: over one, over the next, back under the first, back under the second. This is non-trivial in the fundamental group, and thus the Borromean rings are linked.
Another way is that the cohomology of the complement supports a non-trivial Massey product, which is not the case for the unlink. This is a simple example of the Massey product and further, the algebra corresponds to the geometry: a 3-fold Massey product is a 3-fold product which is only defined if all the 2-fold products vanish, which corresponds to the Borromean rings being pairwise unlinked (2-fold products vanish), but linked overall (3-fold product does not vanish).
Hyperbolic
The Borromean rings are a hyperbolic link: the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume. The canonical (Epstein-Penner) polyhedral decomposition of the complement consists of two ideal octahedra.
Connection to braid
If one cuts the Borromean rings, one obtains one iteration of the standard braid; conversely, if one ties together the ends of (one iteration of) a standard braid, one obtains the Borromean rings. Just as removing one Borromean ring unlinks the remaining two, removing one strand of the standard braid unbraids the other two: they are the basic Brunnian link and Brunnian braid, respectively.
In the standard link diagram, the Borromean rings are ordered non-transitively, in a rock-paper-scissors order. Using the colors above, these are red over yellow, yellow over blue, blue over red – and thus after removing any one ring, for the remaining two, one is above the other and they can be unlinked. Similarly, in the standard braid, each strand is above one of the others and below the other.
History of origin and depictions
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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 Gandharva (Afghan) Buddhist art from around the second century C.E., and in the form of the valknut on Norse image stones dating back to the 7th century.
The Borromean rings have been used in different contexts to indicate strength in unity, e.g. in religion or art. In particular, some have used the design to symbolize the Trinity. The psychoanalyst Jacques Lacan famously found inspiration in the Borromean rings as a model for his topology of human subjectivity, with each ring representing a fundamental Lacanian component of reality (the "real", the "imaginary", and the "symbolic").
Partial Borromean rings
In medieval and renaissance Europe, a number of visual signs are found which consist of three elements which are interlaced together in the same way that the Borromean rings are shown interlaced (in their conventional two-dimensional depiction), but the individual elements are not closed loops. Examples of such symbols are the Snoldelev stone horns and the Diana of Poitiers crescents. An example with three distinct elements is the logo ofSport Club Internacional.
Similarly, a monkey's fist knot is essentially a 3-dimensional representation of the Borromean rings, albeit with three layers, in most cases.
Balancing knives
Using the pattern in the incomplete Borromean rings, one can balance three knives on three supports, such as three bottles or glasses, providing a support in the middle for a fourth bottle or glass.[2]
Multiple Borromean rings
Some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol inDiscordianism, based on a depiction in the Principia Discordia.
Molecular Borromean rings
Molecular Borromean rings are the molecular counterparts of Borromean rings, which are mechanically-interlocked molecular architectures.
In 1997, biologists Chengde Mao and coworkers of New York University succeeded in constructing molecular Borromean rings from DNA(Nature, volume 386, page 137, March 1997).
In 2003, chemist Fraser Stoddart and coworkers at UCLA utilised coordination chemistry to construct molecular Borromean rings in one step from 18 components. This work was published in Science 2004, 304, 1308–1312. Abstract
Quantum-mechanical rings
A team of physicists led by Randy Hulet of Rice University in Houston achieved a quantum-mechanical version of Borromean rings predicted by physicist Vitaly Efimov and published their findings in the online journal Science Express.
Now let us check the news :
Houston, Dec 16 (THAINDIAN NEWS) Physicist Vitaly Efimov’s theory has finally been proved after almost 40 years. Efimov had predicted a quantum-mechanical version of Borromean rings, a symbol that first showed up in Afghan Buddhist art from around the second century. The symbol depicts three rings linked together and if any ring were removed, then they would all be undone.
Efimov had theorized an analog to the rings using particles: Three particles (such as atoms or protons or even quarks) could be bound together in a stable state, even though any two of them could not bind without the third. The physicist first proposed the idea, based on a mathematical proof, in 1970. Since then, people have tried their best to prove this theory but they didn’t succeed for almost 4 decades.
But now physicists at the Rice University have used atoms at temperatures colder than deep space. And they have delivered concrete proof for a once-ridiculed-at theory that’s become a hotbed for research some 40 years after it first appeared in 1970. In a paper available online in Science Express, Rice’s team offers experimental evidence for a universal quantum mechanism that allows trios of particles to appear and reappear at higher energy levels in an infinite progression. The triplets, often called trimers, form in special cases where pairs cannot.
“It’s such a remarkable phenomena,” said team leader Randy Hulet. “There are examples, like the Borromean rings, where having a third component is crucial. Any two of the rings will unbind if the third is removed, and these trimers are similar. The particles want to bind, but no two can do it. They need the third one to make it happen.”
source : http://www.thaindian.com/newsportal/sci-tech/quantum-mechanical-version-of-borromean-rings-finally-proved_100290497.html
Efimov had theorized an analog to the rings using particles: Three particles (such as atoms or protons or even quarks) could be bound together in a stable state, even though any two of them could not bind without the third. The physicist first proposed the idea, based on a mathematical proof, in 1970. Since then, people have tried their best to prove this theory but they didn’t succeed for almost 4 decades.
But now physicists at the Rice University have used atoms at temperatures colder than deep space. And they have delivered concrete proof for a once-ridiculed-at theory that’s become a hotbed for research some 40 years after it first appeared in 1970. In a paper available online in Science Express, Rice’s team offers experimental evidence for a universal quantum mechanism that allows trios of particles to appear and reappear at higher energy levels in an infinite progression. The triplets, often called trimers, form in special cases where pairs cannot.
“It’s such a remarkable phenomena,” said team leader Randy Hulet. “There are examples, like the Borromean rings, where having a third component is crucial. Any two of the rings will unbind if the third is removed, and these trimers are similar. The particles want to bind, but no two can do it. They need the third one to make it happen.”
source : http://www.thaindian.com/newsportal/sci-tech/quantum-mechanical-version-of-borromean-rings-finally-proved_100290497.html
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