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The Borromean rings, also called the Borromean links (Livingston 1993, p.10) are three mutually interlocked rings (left figure), named after the Italian Renaissance family who used them on their coat of arms. The configuration of rings is also known as a "Ballantine," and a brand of beer (right figure; Falstaff Brewing Corporation) has been brewed under this name. In the Borromean rings, no two rings are linked, so if any one of the rings is cut, all three rings fall apart. Any number of rings can be linked in an analogous manner (Steinhaus 1999, Wells 1991).
The Borromean rings are a prime link. They have link symbol 06-0302, braid word 
, and are also the simplest Brunnian link.
It turns out that rigid Borromean rings composed of real (finite thickness) tubes cannot be physically constructed using three circular rings of either equal or differing radii. However, they can be made from three congruent elliptical rings.
See also
Brunnian Link, Circle-Circle Intersection, Hopf Link, Johnson's Theorem, Link, Prime Link, Tube, Unlink, Venn Diagram, Whitehead Link
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References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp.58-59, 1989.Falstaff Brewing Corporation. "Ballantine Ale." http://www.falstaffbrewing.com/ballantine_ale.htm.Gardner, M. The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: University of Chicago Press, 1991.Jablan, S. "Borromean Triangles." http://members.tripod.com/~modularity/links.htm.Kauffman, L. Knots and Physics. Teaneck, NJ: World Scientific, p.12, 1991.Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., p.10, 1993.Pappas, T. "Trinity of Rings--A Topological Model." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p.31, 1989.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp.66 and 138, 1976.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp.266-267, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p.18, 1991.
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Cite this as:
Weisstein, Eric W. "Borromean Rings."From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BorromeanRings.html
