Tuesday, October 26, 2021

What are the Borromean rings?

12:37 AM | ,

Borromean rings - a fascinating mathematical concept where three interlinked circles create an impossible object in three-dimensional space.
Borromean rings are a set of three interlinked, simple closed curves in three-dimensional space. If any one of the three is cut, the others break apart into two unknotted and unlinked loops. The rings are usually depicted as three circles that cross over and under each other in a Venn diagram pattern. Other triples of curves are also considered Borromean rings as long as they have the same topological properties. However, three-dimensional circular Borromean rings are considered an impossible object because it is not possible to construct them from circles in three-dimensional space.

Michael H. Freedman and Richard Skora also showed using four-dimensional hyperbolic geometry that no Brunnian link can be exactly circular.

For three rings in their conventional Borromean arrangement, this can be seen from considering the link diagram. If one assumes that two of the circles touch at their two crossing points, then they lie in either 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. Another argument for the impossibility of circular realizations, by Helge Tverberg, uses inversive geometry to transform any three circles so that one of them becomes a line, making it easier to argue that the other two circles do not link with it to form the Borromean rings.

However, the Borromean rings can be realized using ellipses. These may be taken to be of arbitrarily small eccentricity: 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 are named after the Italian House of Borromeo, who used the circular form of these rings as a coat of arms, but designs based on the Borromean rings have been used in many cultures, including by the Norsemen and in Japan.

Matthew Cook has conjectured that any three unknotted simple closed curves in space, not all circles, can be combined without scaling to form the Borromean rings.

Physical instances of the Borromean rings have been made from linked DNA or other molecules, and they have analogues in the Efimov state and Borromean nuclei, both of which have three components bound to each other although no two of them are bound.

Geometrically, the Borromean rings may be realized by linked ellipses, or (using the vertices of a regular icosahedron) by linked golden rectangles.

In knot theory, the Borromean rings can be proved to be linked by counting their Fox n-colorings. As links, they are Brunnian, alternating, algebraic, and hyperbolic. In arithmetic topology, certain triples of prime numbers have analogous linking properties to the Borromean rings.

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