Borromean rings

Borromean rings

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.

Mathematical properties

Although the typical picture of the Borromean rings (above right picture) may lead one to think the link can be formed from geometrically round circles, they cannot be. (Freedman & Skora 1987) proves that 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).

A realization of the Borromean rings as ellipses

It is, however, true that one can use ellipses (right 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.

Linking

For more details on this topic, see Link group.

In knot theory, the Borromean rings are a simple example of a Brunnian link: although each pair of rings are unlinked, the whole link cannot be unlinked. There are a number of ways of seeing this.

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).

In arithmetic topology, there is an analogy between knots and prime numbers in which one considers links between primes. The triple of primes (13, 61, 937) are linked modulo 2 (the Rédei symbol is −1) but are pairwise unlinked modulo 2 (the Legendre symbols are all 1). Therefore these primes have been called a "proper Borromean triple modulo 2" or "mod 2 Borromean primes".

Hyperbolic geometry

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 regular ideal octahedra. The volume is 16Л(π/4) = 7.32772… where Л is the Lobachevsky function.

Connection with braids

The standard 3-strand braid corresponds to the Borromean rings.

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 cyclic 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

Valknut on Stora Hammars I stone

The Borromean rings as a symbol of the Christian Trinity, from a 13th-century manuscript.

A monkey's fist knot.

The Discordian "mandala", containing five Borromean rings configurations.

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 Gandhara (Afghan) Buddhist art from around the 2nd century^{[citation needed]}, 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").

The Borromean rings were formerly used as the logo of the German Krupp industrial concern and are used as part of the logo for the successor ThyssenKrupp. The rings were used as the logo of Ballantine beer and are still used by the Ballantine brand beer, now produced by successor Falstaff.

In 2006, the International Mathematical Union decided at the 25th International Congress of Mathematicians in Madrid, Spain to use a new logo based on the Borromean rings.

A stone pillar at Marundeeswarar Temple in Thiruvanmiyur, Chennai, Tamil Nadu, India, has such a figure dating to before 6th century.^{[citation needed]}

Partial rings

In medieval and renaissance Europe, a number of visual signs are found that consist of three elements 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 of Sport Club Internacional. Less-related visual signs include the Gankyil and the Venn diagram on three sets.

Similarly, a monkey's fist knot is essentially a 3-dimensional representation of the Borromean rings, albeit with three layers, in most cases.

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.

Multiple rings

Some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol in Discordianism, based on a depiction in the Principia Discordia.

Realizations

Crystal structure of molecular Borromean rings reported by Stoddart et al. Science 2004, 304, 1308–1312.

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 a set of rings from DNA. In 2003, chemist Fraser Stoddart and coworkers at UCLA utilised coordination chemistry to construct a set of rings in one step from 18 components.

A quantum-mechanical analog of Borromean rings, called an Efimov state, was predicted by physicist Vitaly Efimov in 1970. A team of physicists led by Randall Hulet of Rice University in Houston achieved this with a set of three bound lithium atoms and published their findings in the online journal Science Express. In 2010, a team led by K. Tanaka created an Efimov state within a nucleus.

See also

• Triquetra

References

• P. R. Cromwell, E. Beltrami and M. Rampichini, "The Borromean Rings", Mathematical Intelligencer 20 no 1 (1998) 53–62.

• Freedman, Michael H.; Skora, Richard (1987), "Strange Actions of Groups on Spheres", Journal of Differential Geometry 25: 75–98 

• Lindström, Bernt; Zetterström, Hans-Olov (1991), "Borromean Circles are Impossible", American Mathematical Monthly 98 (4): 340–341, doi:10.2307/2323803, JSTOR 2323803 (subscription required). This article explains why Borromean links cannot be exactly circular. 

• Brown, R. and Robinson, J., "Borromean circles", Letter, American Math. Monthly, April, (1992) 376–377. This article shows how Borromean squares exist, and have been made by John Robinson (sculptor), who has also given other forms of this structure.

• Chernoff, W. W., "Interwoven polygonal frames". (English summary) 15th British Combinatorial Conference (Stirling, 1995). Discrete Math. 167/168 (1997), 197–204. This article gives more general interwoven polygons.

External links

• Site devoted to the Borromean Rings.
• The Borromean link and related entities in knot theory
• The Borromean Rings at the wiki Knot Atlas.
• History of the Borromean rings
• Borromean rings and John Robinson (sculptor).
• African Borromean ring carving
• Borromean rings spinning as a group
• Video animation: International Mathematical Union logo