Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, a link is a collection of knots which do not intersect but which may be linked (or knotted together In Knot theory, a branch of Mathematics, a Brunnian link is a nontrivial link that becomes trivial if any component is removed 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, the Brunnian property means they cannot (see "References"). It is, however, true that one can use ellipses of arbitrarily small eccentricity (center picture). In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section.

The Borromean rings give examples of several interesting phenomena in mathematics. One is that the cohomology of the complement supports a non-trivial Massey product. In Mathematics, particularly in Algebraic topology but also in Geometric topology and Differential topology, the Massey product is a Cohomology Another is that it is a hyperbolic link: the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume. In Mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative The canonical (Epstein-Penner) polyhedral decomposition of the complement consists of two ideal octahedra.

## History of origin and depictions

The Borromean rings as a symbol of the Christian Trinity, from a 13th-century manuscript. SSC RF "Troitsk Institute of Innovative and Termonuclear Research" or TRINITY for shprt Троицкий Институт инновационных и термоядерных

The name "Borromean rings" comes from their use in the coat of arms of the aristocratic Borromeo family in Italy. A coat of arms or armorial bearings (often just arms for short in European tradition is a design belonging to a particular person (or group of people Aristocracy is a form of Government, where rule is established through an internal struggle over who has the most status and influence over society and internal relations Italy (Italia officially the Italian Republic, (Repubblica Italiana is located on the Italian Peninsula in Southern Europe, and on the two largest 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. The Valknut ( Old Norse valr, "slain warriors" + knut, "knot" is a Symbol consisting of three interlocked Norsemen 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 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. SSC RF "Troitsk Institute of Innovative and Termonuclear Research" or TRINITY for shprt Троицкий Институт инновационных и термоядерных The psychoanalyst Jacques Lacan famously found inspiration in the Borromean rings as a model for his topology of the human mind, with each ring representing a fundamental Lacanian component of reality (the "real", the "imaginary", and the "symbolic"). Jacques-Marie-Émile Lacan (French ʒak lakɑ̃ ( April 13, 1901 &ndash September 9, 1981) was a French Psychoanalyst

The Borromean rings were also the logo of Ballantine beer.

A monkey's fist knot is essentially a 3-dimensional representation of the Borromean rings, albeit with 3 layers, in most cases. A monkey's fist or monkey paw is a type of Knot, so named because it looks somewhat like a small bunched fist/paw

It should also be noted that the Borromean rings appear in Ghandarva (Afghan) Buddhist art from around the second century C. E.

### Partial Borromean rings emblems

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. The 9th century Runestone at Snoldelev, Ramsø, Denmark, is decorated with a design of three Drinking horns interlocking as incomplete Diane de Poitiers ( September 3, 1499 - April 25, 1566) was a noblewoman and a fixture at the courts of Francis I and Henri II

## Molecular Borromean rings

Crystal structure of molecular Borromean rings reported by Stoddart et al. Molecular Borromean rings are an example of a Mechanically-interlocked molecular architecture in which three Macrocycles are interlocked in such a way that breaking Sir James Fraser Stoddart (born May 24, 1942) is a Scottish Chemist currently at the Department of Chemistry at Northwestern University, one Science 2004, 304, 1308–1312.

Molecular Borromean rings are the molecular pendants of Borromean rings, which are mechanically-interlocked molecular architectures. Molecular Borromean rings are an example of a Mechanically-interlocked molecular architecture in which three Macrocycles are interlocked in such a way that breaking Mechanically-interlocked molecular architectures are connections of molecules not through traditional bonds but instead as a consequence of their topology

In 1997, biologists Chengde Mao and coworkers of New York University succeeded in constructing molecular Borromean rings from DNA (Nature, vol 386, page 137, March 1997). Year 1997 ( MCMXCVII) was a Common year starting on Wednesday (link will display full 1997 Gregorian calendar Foundations of modern biology There are five unifying principles New York University ( NYU) is a private, Nonsectarian, Coeducational Research University in New York City. Molecular Borromean rings are an example of a Mechanically-interlocked molecular architecture in which three Macrocycles are interlocked in such a way that breaking Deoxyribonucleic acid ( DNA) is a Nucleic acid that contains the genetic instructions used in the development and functioning of all known Nature is a prominent Scientific journal, first published on 4 November 1869

In 2003, chemist Fraser Stoddart and coworkers at UCLA utilised coordination chemistry to construct molecular Borromean rings in one step from 18 components. Year 2003 ( MMIII) was a Common year starting on Wednesday of the Gregorian calendar. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Sir James Fraser Stoddart (born May 24, 1942) is a Scottish Chemist currently at the Department of Chemistry at Northwestern University, one The University of California Los Angeles (generally known as UCLA) is a public research university located in Westwood Los Angeles, California, United The term complex in Chemistry is usually used to describe molecules or ensembles formed by the combination of Ligands and metal Ions. Molecular Borromean rings are an example of a Mechanically-interlocked molecular architecture in which three Macrocycles are interlocked in such a way that breaking This work was published in Science 2004, 304, 1308-1312. Science is the Academic journal of the American Association for the Advancement of Science and is considered one of the world's most prestigious Scientific Abstract