Borromean rings are three simple closed curves in three-dimensional space .These rings are topologically linked and they cannot be separated from each other. But when any one of the three rings break or is removed ,we get two unlinked and unknotted loops.

These rings can be drawn as three circles on a plane. It looks like the pattern of a Venn diagram,or we can say the rings are crossing over and under each other

History of Borromean rings

The name Borromean rings evolved from the Italian House of Borromeo, who used the circular form of these rings as a coat of arms.

But other cultures like norsemen ,Japanese also used these kind of designs .In christainity it symbolises as a sign of the Trinity, and in modern commerce as the logo of Ballantine beer.

The Borromean rings are made from linked DNA or other molecules. They have analogues in the Efimov state( is an effect in the quantum mechanics of few-body systems ) and Borromean nuclei .Both of which have three components bound to each other although no two of them are bound.

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

Properties of Borromean rings

Linkedness

According to knot theory, the Borromean rings are a small example of a Brunnian link, a link that cannot be separated but that breaks apart into separate unknotted loops when any one of its components is removed.

Ring shape

The Borromean rings are typically drawn as rings projecting to circles in the plane. But it is impossible to draw three-dimensional circular Borromean rings.

the Borromean rings can also be realized using ellipses which is not a perfect circle, but still they can form Borromean links if suitably positioned. We can realise the Borromean rings by three mutually perpendicular golden rectangles ,that can be found within a regular icosahedron by connecting three opposite pairs of its edges.

Ropelength

According to knot theory, the length of a rope of a knot (link )is the shortest length of flexible rope (of radius one). Mathematically, the realization is described as a smooth curve whose radius-one tubular neighborhood avoids self-intersections.The minimum rope length of the Borromean rings has not been proven yet. But the smallest value attained is realized by three copies of a 2-lobed planar curve.

Hyperbolic geometry

The Borromean rings are a hyperbolic in geometry. The surrounding space of the Borromean rings admits a complete hyperbolic metric of finite volume.

Borromean decomposition consists of two ideal regular octahedra.The space is a quotient space of a uniform honeycomb of ideal octahedra. The order-4 octahedral honeycomb, making the Borromean rings one of at most 21 links that correspond to uniform honeycombs in this way.

Number theory

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 , but are pairwise unlinked modulo 2 . Therefore, these primes have been called a “proper Borromean triple modulo 2” or mode 2 Borromean primes.

A monkey’s fist knot is a 3-dimensional representation of the Borromean rings, albeit with three layers.

Sculptor John Robinson has made artworks with three equilateral triangles made using sheet metal, linked to form Borromean rings and resembling a three-dimensional version of the valknut.

Another example of the Borromean rings in quantum information theory involves the entanglement of three qubits in the Greenberger–Horne–Zeilinger state.

Borromean rings are three simple closed curves in three-dimensional space .These rings are topologically linked and they cannot be separated from each other

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