# 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 (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 (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.

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. 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. 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.

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.

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 also the logo of Ballantine beer.

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

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. An example with three distinct elements is the logo of Sport Club Internacional.

## Molecular Borromean rings

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 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