Borromean rings

Last updated
Borromean rings
Borromean Rings Illusion.png
L6a4
Braid length 6
Braid no. 3
Crossing no. 6
Hyperbolic volume 7.327724753
Stick no. 9
Unknotting no. 2
Conway notation [.1]
A-B notation 63
2
Thistlethwaite L6a4
Last /Next L6a3 /  L6a5
Other
alternating, hyperbolic

In mathematics, the Borromean rings consist of three topological circles which are linked but where removing any one ring leaves the other two unconnected. In other words, no two of the three rings are linked with each other as a Hopf link, but nonetheless all three are linked. The Borromean rings are one of a class of such links called Brunnian links.

Contents

Mathematical properties

Ring shape

Question, Web Fundamentals.svgUnsolved problem in mathematics:
Are there three unknotted curves, not all circles, that cannot form the Borromean rings?
(more unsolved problems in mathematics)

The Borromean rings are typically drawn with their rings projecting to circles in the plane of the drawing, but three-dimensional circular Borromean rings are an impossible object: it is not possible to form the Borromean rings from circles in three-dimensional space. [1] Michael H.Freedman andRichard Skora ( 1987 ) proved that a certain class of links, including the Borromean links, cannot be exactly circular. [2] 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. [3]

Realization of Borromean rings using ellipses 3d borromean rings by ronbennett2001.jpg
Realization of Borromean rings using ellipses

However, the Borromean rings can be realized using ellipses. [4] 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.

Three linked golden rectangles in a regular icosahedron Icosahedron-golden-rectangles.svg
Three linked golden rectangles in a regular icosahedron

A realization of the Borromean rings by three mutually perpendicular golden rectangles can be found within a regular icosahedron by connecting three opposite pairs of its edges. [4]

Every three unknotted polygons in Euclidean space may be combined, after a suitable scaling transformation, to form the Borromean rings. If all three polygons are planar, then scaling is not needed. More generally, 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. After Jason Cantarella suggested a possible counterexample, Hugh Nelson Howards weakened the conjecture to apply to any three planar curves that are not all circles. On the other hand, although there are infinitely many Brunnian links with three links, the Borromean rings are the only one that can be formed from three convex curves. [5]

Linkedness

In knot theory, the Borromean rings are a simple example of a Brunnian link: although each pair of rings is unlinked, the whole link cannot be unlinked. There are a number of ways of seeing that the Borromean rings are linked; one is to count their Fox n-colorings. A trivial link would have 125 Fox 5-colorings (one for each choice of color for each of the three links), but the Borromean rings have only five. [1]

Number theory

In arithmetic topology, 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 (the Rédei symbol is −1) but are pairwise unlinked modulo 2 (the Legendre symbols are all 1). Therefore, these primes have been called a "proper Borromean triple modulo 2" [6] or "mod 2 Borromean primes". [7]

Hyperbolic geometry

The Borromean rings are 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 regular octahedra. The volume is where is the Lobachevsky function. [8] This was a central example in the video Not Knot about knot complements, produced in 1991 by the Geometry Center. [9]

Name and history

Valknut
on Stora Hammars I stone Sacrificial scene on Hammars - Valknut.png
Valknut on Stora Hammars I stone

The name "Borromean rings" comes from their use in the coat of arms of the aristocratic Borromeo family in Northern Italy. [10] The link itself is much older and has appeared in the form of the valknut , three linked equilateral triangles with parallel sides, on Norse image stones dating back to the 7th century. [11] A stone pillar in the 6th-century Marundeeswarar Temple in India shows the Borromean rings in another form, three linked equilateral triangles rotated from each other to form a regular enneagram. [12] The Ōmiwa Shrine in Japan is also decorated with a motif of the Borromean rings, in their conventional circular form. [4]

The Borromean rings as a symbol of the Christian Trinity, from a 13th-century manuscript. BorromeanRings-Trinity.svg
The Borromean rings as a symbol of the Christian Trinity, from a 13th-century manuscript.

The Borromean rings have been used in different contexts to indicate strength in unity. [13] In particular, some have used the design to symbolize the Trinity. [14] The psychoanalyst Jacques Lacan 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"). [15]

The rings were used as the logo of Ballantine beer, and are still used by the Ballantine brand beer, now distributed by the current brand owner, the Pabst Brewing Company. [16] [17] For this reason they have sometimes been called the "Ballantine rings". [14] [16]

Seifert surfaces for the Borromean rings were featured by Martin Gardner in his September 1961 "Mathematical Games column" in Scientific American . [17] In 2006, the International Mathematical Union decided at the 25th International Congress of Mathematicians in Madrid, Spain to use a new logo based on the Borromean rings. [4]

Partial and multiple rings

A monkey's fist knot Knot Monkey Fist.jpg
A monkey's fist knot

In medieval and renaissance Europe, a number of visual signs are found that consist of three elements 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 [18] and the Diana of Poitiers crescents. [14]

Similarly, a monkey's fist knot is essentially a 3-dimensional representation of the Borromean rings, albeit with three layers, in most cases. [19] Using the pattern in the incomplete Borromean rings, one can balance three knives on three supports, such as three bottles or glasses, providing a support in the middle for a fourth bottle or glass. [20]

The Discordian "mandala", containing five Borromean rings configurations Principia Discordia page00043 mandala (Brunnian link).svg
The Discordian "mandala", containing five Borromean rings configurations

Some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol in Discordianism, based on a depiction in the Principia Discordia . [21]

Physical realizations

Crystal structure of molecular Borromean rings reported by Stoddart et al. (Science 2004) Molecular Borromean Rings Atwood Stoddart commons.png
Crystal structure of molecular Borromean rings reported by Stoddart et al. (Science 2004)

Molecular Borromean rings are the molecular counterparts of Borromean rings, which are mechanically-interlocked molecular architectures. In 1997, biologist Chengde Mao and coworkers of New York University succeeded in constructing a set of rings from DNA. [23] In 2003, chemist Fraser Stoddart and coworkers at UCLA utilised coordination chemistry to construct a set of rings in one step from 18 components. [22] Borromean ring structures have been shown to be an effective way to represent the structure of certain atomically precise noble metal clusters which are shielded by a surface layer of thiolate ligands (-SR), such as Au25(SR)18 and Ag25(SR)18. [24] A library of Borromean networks has been synthesized by design by Giuseppe Resnati and coworkers via halogen bond driven self-assembly. [25] In order to access the molecular Borromean ring consisting of three unequal cycles a step-by-step synthesis was proposed by Jay S. Siegel and coworkers. [26]

A quantum-mechanical analog of Borromean rings is called a halo state or an Efimov state (the existence of such states was predicted by physicist Vitaly Efimov, in 1970). For the first time the research group of Rudolf Grimm and Hanns-Christoph Nägerl from the Institute for Experimental Physics (University of Innsbruck, Austria) experimentally confirmed such a state in an ultracold gas of caesium atoms in 2006, and published their findings in the scientific journal Nature. [27] A team of physicists led by Randall Hulet of Rice University in Houston achieved this with a set of three bound lithium atoms and published their findings in the online journal Science Express. [28] In 2010, a team led by K. Tanaka created an Efimov state within a nucleus. [29]

See also

Related Research Articles

John Milnor American mathematician

John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the six mathematicians to have won the Fields Medal, the Wolf Prize, and the Abel Prize.

Knot theory Study of mathematical knots

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knot (mathematics)

In mathematics, a knot is an embedding of a topological circle S1 in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies).

In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs. Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding.

3-manifold A space that locally looks like Euclidean 3-dimensional space

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

Link (knot theory) A collection of knots which do not intersect, but may be linked

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link.

Hyperbolic link

In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component.

Hopf link

In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.

Molecular knot

In chemistry, a molecular knot is a mechanically interlocked molecular architecture that is analogous to a macroscopic knot. Naturally forming molecular knots are found in organic molecules like DNA, RNA, and proteins. It is not certain that naturally occurring knots are evolutionarily advantageous to nucleic acids or proteins, though knotting is thought to play a role in the structure, stability, and function of knotted biological molecules. The mechanism by which knots naturally form in molecules, and the mechanism by which a molecule is stabilized or improved by knotting, is ambiguous. The study of molecular knots involves the formation and applications of both naturally occurring and chemically synthesized molecular knots. Applying chemical topology and knot theory to molecular knots allows biologists to better understand the structures and synthesis of knotted organic molecules.

Brunnian link Interlinked multi-loop construction where cutting one loop frees all the others

In knot theory, a branch of topology, a Brunnian link is a nontrivial link that becomes a set of trivial unlinked circles if any one component is removed. In other words, cutting any loop frees all the other loops.

Massey product

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Unlink Link that consists of finitely many unlinked unknots

In the mathematical field of knot theory, an unlink is a link that is equivalent to finitely many disjoint circles in the plane.

Molecular self-assembly Molecules adopt a defined arrangement without guidance or management from an outside source

Molecular self-assembly is the process by which molecules adopt a defined arrangement without guidance or management from an outside source. There are two types of self-assembly. These are intramolecular self-assembly and intermolecular self-assembly. Commonly, the term molecular self-assembly refers to intermolecular self-assembly, while the intramolecular analog is more commonly called folding.

In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis,.

Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.

L10a140 link minimal Brunnian link which is not equivalent to the Borromean rings

In the mathematical theory of knots, L10a140 is the name in the Thistlethwaite link table of a link of three loops, which has ten crossings between the loops when presented in its simplest visual form. It is of interest because it is presumably the simplest link which possesses the Brunnian property — a link of connected components that, when one component is removed, becomes entirely unconnected — other than the six-crossing Borromean rings.

Vladimir Georgievich Turaev is a Russian mathematician, specializing in topology.

References

  1. 1 2 Aigner, Martin; Ziegler, Günter M. (2018), "Chapter 15: The Borromean Rings Don't Exist", Proofs from THE BOOK (6th ed.), Springer, pp. 99–106, doi:10.1007/978-3-662-57265-8_15, ISBN   978-3-662-57265-8
  2. Freedman, Michael H.; Skora, Richard (1987), "Strange Actions of Groups on Spheres", Journal of Differential Geometry , 25: 75–98, doi:10.4310/jdg/1214440725
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  4. 1 2 3 4 Gunn, Charles; Sullivan, John M. (2008), "The Borromean Rings: A video about the New IMU logo", in Sarhangi, Reza; Séquin, Carlo H. (eds.), Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture, London: Tarquin Publications, pp. 63–70, ISBN   9780966520194
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  13. Aravind, P. K. (1997), "Borromean Entanglement of the GHZ State" (PDF), in Cohen, R. S.; Horne, M.; Stachel, J. (eds.), Potentiality, Entanglement and Passion-at-a-Distance, Boston Studies in the Philosophy of Science, Springer, pp. 53–59, doi:10.1007/978-94-017-2732-7_4, they represent the motto 'united we stand, divided we fall', since if one of the rings is cut the other two fall apart
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  18. Baird, Joseph L. (1970), "Unferth the þyle", Medium Ævum, 39 (1): 1–12, doi:10.2307/43631234, JSTOR   43631234, the stone bears also representations of three horns interlaced
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  20. Comments on Knives And Beer Bar Trick: Amazing Balance
  21. "Mandala", Principia Discordia (4th ed.), March 1970, p. 43
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  29. K. Tanaka (2010), "Observation of a Large Reaction Cross Section in the Drip-Line Nucleus 22C", Physical Review Letters , 104 (6): 062701, Bibcode:2010PhRvL.104f2701T, doi:10.1103/PhysRevLett.104.062701, PMID   20366816