In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the
Borromean rings consist of three topological
circleA circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre. The common distance of the points of a circle from its center is called its radius....
s which are
linkIn mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...
ed and form a
Brunnian linkIn knot theory, a branch of mathematics, a Brunnian link is a nontrivial link which becomes trivial if any component is removed. In other words, cutting any loop frees all the other loops ....
, 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. proves why a certain class of links including the Borromean links cannot be exactly circular.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the
Borromean rings consist of three topological
circleA circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre. The common distance of the points of a circle from its center is called its radius....
s which are
linkIn mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...
ed and form a
Brunnian linkIn knot theory, a branch of mathematics, a Brunnian link is a nontrivial link which becomes trivial if any component is removed. In other words, cutting any loop frees all the other loops ....
, i.e., removing any ring results in two unlinked rings.
Mathematical properties
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. 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 .
It is, however, true that one can use ellipses (center picture). These may be taken to be of arbitrarily small
eccentricityIn mathematics, the eccentricity, denoted e or , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,* The eccentricity of a circle is zero....
, 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
There are a number of ways of seeing that the Borromean rings cannot be unlinked.
Simplest is that the
fundamental groupIn mathematics, more specifically algebraic topology, the fundamental group or Poincaré group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of the complement of two unlinked circles is the
free groupIn mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses .A related but different notion is a free abelian group.- History...
on two generators,
a and
b, by the
Seifert–van Kampen theoremIn mathematics, the Seifert–van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X...
, and then the third loop has the class of the
commutatorIn mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
, [
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
cohomologyIn mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
of the complement supports a non-trivial
Massey productIn algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product.-Massey triple 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 linkIn mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry...
: 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 linkIn knot theory, a branch of mathematics, a Brunnian link is a nontrivial link which becomes trivial if any component is removed. In other words, cutting any loop frees all the other loops ....
and Brunnian braid, respectively.
In the standard link diagram, the Borromean rings are ordered non-transitively, in a
rock-paper-scissorsRock-Paper-Scissors, or Paper-Rock-Scissors, is a two-person hand game....
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
The name "Borromean rings" comes from their use in the
coat of armsA coat of arms, more properly called an armorial achievement, armorial bearings or often just arms for short, in European tradition, is a design belonging to a particular person and used by them in a wide variety of ways. Historically, they were used by knights to identify them apart from enemy...
of the
aristocraticAristocracy is a form of government, in which a few of the most prominent citizens rule. This may be a hereditary elite, or it may be by a system of cooption where a council of prominent citizens add leading soldiers, merchants, land owners, priests, and lawyers to their number...
BorromeoThe aristocratic Borromeo family, said to date from before the twelfth century, were counts of Arona from the mid-fifteenth century and played important roles in the politics of the Duchy of Milan and in the Catholic Reformation...
family in
ItalyItaly , officially the Italian Republic , is a country located on the Italian Peninsula in Southern Europe and on the two largest islands in the Mediterranean Sea, Sicily and Sardinia. Italy shares its northern, Alpine boundary with France, Switzerland, Austria and Slovenia...
. The link itself is much older and has appeared in Ghandarva (Afghan) Buddhist art from around the second century C.E., and in the form of the
valknutThe Valknut is a symbol consisting of three interlocked triangles, and appears on various Germanic objects. A number of theories have been proposed for its significance....
on
NorseNorsemen is used to refer to the group of people as a whole who speak one of the North Germanic languages as their native language. The meaning of Norseman was "people from the North"...
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
TrinityThe Christian doctrine of the Trinity teaches the unity of Father, Son, and Holy Spirit as three persons in one Godhead. The doctrine states that God is the Triune God, existing as three persons, or in the Greek hypostases, but one being. Each of the persons is understood as having the one...
. The psychoanalyst
Jacques LacanJacques-Marie-Émile Lacan was a French psychoanalyst and psychiatrist who made prominent contributions to psychoanalysis, philosophy, and literary theory. He gave yearly seminars, in Paris, from 1953 to 1981, mostly influencing France's intellectuals in the 1960s and the 1970s, especially the...
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 also the logo of Ballantine beer.
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 stoneThe 9th century runestone at Snoldelev, Ramsø, Denmark, is decorated with a design of three drinking horns interlocking as incomplete Borromean rings , and a swastika. The triple horn motif has been compared to a triskelion, or to the valknut symbol...
horns and the Diana of Poitiers crescents. An example with three distinct elements is the logo of
Sport Club InternacionalSport Club Internacional, known simply as Internacional or Inter, is a Brazilian football team from Porto Alegre, Rio Grande do Sul, founded on April 4, 1909. They play in red shirts, white shorts and socks, the first of which gives the team its nickname of O Colorado, or The Reds...
.
Similarly, a
monkey's fistA monkey's fist or monkey paw is a type of knot, so named because it looks somewhat like a small bunched fist/paw. It also looks somewhat like a volleyball or an older style football. It is tied at the end of a rope to serve as a weight, making it easier to throw, and also as an ornamental knot...
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.
Multiple Borromean rings
Some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol in
DiscordianismDiscordianism is a modern religion centered on the idea that chaos is all that there is, and that disorder and order, the former considered a concept distinct from chaos, are both illusions that are imposed on chaos...
, based on a depiction in the
Principia DiscordiaPrincipia Discordia is a Discordian religious text written by Greg Hill and Kerry Thornley . It was originally published under the title "Principia Discordia or How The West Was Lost" in a limited edition of 5 copies in 1965...
.
Molecular Borromean rings
Molecular Borromean ringsMolecular Borromean rings are an example of a mechanically-interlocked molecular architecture in which three macrocycles are interlocked in such a way that breaking any macrocycle allows the others to disassociate. They are the smallest examples of Borromean rings. The synthesis of molecular...
are the molecular counterparts of Borromean rings, which are
mechanically-interlocked molecular architecturesMechanically-interlocked molecular architectures are connections of molecules not through traditional bonds, but instead as a consequence of their topology. This connection of molecules is analogous to keys on a key chain loop. The keys are not directly connected to the key chain loop but they...
.
In 1997,
biologistBiology is the natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy...
s Chengde Mao and coworkers of
New York UniversityNew York University is a private, nonsectarian, research university in New York City. NYU's main campus is situated in the Greenwich Village section of Manhattan...
succeeded in constructing
molecular Borromean ringsMolecular Borromean rings are an example of a mechanically-interlocked molecular architecture in which three macrocycles are interlocked in such a way that breaking any macrocycle allows the others to disassociate. They are the smallest examples of Borromean rings. The synthesis of molecular...
from
DNADeoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms and some viruses. The main role of DNA molecules is the long-term storage of information...
(
NatureNature is a prominent British scientific journal, first published on 4 November 1869. Most scientific journals are now highly specialized, and Nature is among the few journals that still publish original research articles across a wide range of scientific...
, volume 386, page 137, March 1997).
In 2003,
chemistChemistry is the science concerned with the composition, behavior, structure, and properties of matter, as well as the changes it undergoes during chemical reactions...
Fraser StoddartSir James Fraser Stoddart is a Scottish chemist currently at the Department of Chemistry at Northwestern University. He works in the area of supramolecular chemistry and nanotechnology...
and coworkers at UCLA utilised coordination chemistry to construct
molecular Borromean ringsMolecular Borromean rings are an example of a mechanically-interlocked molecular architecture in which three macrocycles are interlocked in such a way that breaking any macrocycle allows the others to disassociate. They are the smallest examples of Borromean rings. The synthesis of molecular...
in one step from 18 components. This work was published in
ScienceScience is the academic journal of the American Association for the Advancement of Science and is considered one of the world's most prestigious scientific journals. The peer-reviewed journal, first published in 1880 is circulated weekly and has a print subscriber base of around 130,000...
2004,
304, 1308–1312.
Abstract
External links