simple closed curve
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
s in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of a
Venn diagram
A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships ...
, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing.
The Borromean rings are named after the Italian
House of Borromeo
), type= Noble family, country=, estates= Rocca d'Angera Palazzo Borromeo, Milan Castel of Peschiera BorromeoBorromean Islands Villa Borromeo, Arcore, titles=* Prince of Angera
* Marquess of Romagnano
* Count of Arona
* Count of Peschiera
* Lor ...
, who used the circular form of these rings as a
coat of arms
A coat of arms is a heraldic visual design on an escutcheon (i.e., shield), surcoat, or tabard (the latter two being outer garments). The coat of arms on an escutcheon forms the central element of the full heraldic achievement, which in its ...
, but designs based on the Borromean rings have been used in many cultures, including by the
Norsemen
The Norsemen (or Norse people) were a North Germanic ethnolinguistic group of the Early Middle Ages, during which they spoke the Old Norse language. The language belongs to the North Germanic branch of the Indo-European languages and is the pr ...
and in Japan. They have been used in Christian symbolism as a sign of the
Trinity
The Christian doctrine of the Trinity (, from 'threefold') is the central dogma concerning the nature of God in most Christian churches, which defines one God existing in three coequal, coeternal, consubstantial divine persons: God th ...
, and in modern commerce as the logo of Ballantine beer, giving them the alternative name Ballantine rings. Physical instances of the Borromean rings have been made from linked DNA or other molecules, and they have analogues in the
Efimov state The Efimov effect is an effect in the quantum mechanics of few-body systems predicted by the Russian theoretical physicist V. N. Efimov in 1970. Efimov’s effect is where three identical bosons interact, with the prediction of an infinite series ...
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 rectangle
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618.
Golden rectangles exhibit a special form of self-similarity ...
s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In knot theory, the Borromean rings can be proved to be linked by counting their Fox -colorings. As links, they are Brunnian, alternating, algebraic, and
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
. In
arithmetic topology 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.
Analogies
The following are some of the analogies used ...
, certain triples of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s have analogous linking properties to the Borromean rings.
Definition and notation
It is common in mathematics publications that define the Borromean rings to do so as a
link diagram
In the mathematical field of 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 so it cannot ...
, a drawing of curves in the plane with crossings marked to indicate which curve or part of a curve passes above or below at each crossing. Such a drawing can be transformed into a system of curves in three-dimensional space by embedding the plane into space and deforming the curves drawn on it above or below the embedded plane at each crossing, as indicated in the diagram. The commonly-used diagram for the Borromean rings consists of three equal
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s centered at the points of an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
, close enough together that their interiors have a common intersection (such as in a
Venn diagram
A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships ...
or the three circles used to define the Reuleaux triangle). Its crossings
alternate
Alternative or alternate may refer to:
Arts, entertainment and media
* Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki''
* ''The Alternative'' (film), a 1978 Australian television film
* ''The Alternative ...
between above and below when considered in consecutive order around each circle; another equivalent way to describe the over-under relation between the three circles is that each circle passes over a second circle at both of their crossings, and under the third circle at both of their crossings. Two links are said to be equivalent if there is a continuous deformation of space (an ambient isotopy) taking one to another, and the Borromean rings may refer to any link that is equivalent in this sense to the standard diagram for this link.
In ''
The Knot Atlas
''The Knot Atlas'' is a website, an encyclopedia rather than atlas, dedicated to knot theory. It and its predecessor were created by mathematician Dror Bar-Natan, who maintains the current site with Scott Morrison. According to Schiller, the si ...
'', the Borromean rings are denoted with the code "L6a4"; the notation means that this is a link with six crossings and an alternating diagram, the fourth of five alternating 6-crossing links identified by
Morwen Thistlethwaite
Morwen Bernard Thistlethwaite is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville. He has made important contributions to both knot theory and Rubik's Cube group theory.
Biography
Morwen Thistlethwait ...
in a list of all
prime link
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s with up to 13 crossings. In the tables of knots and links in Dale Rolfsen's 1976 book ''Knots and Links'', extending earlier listings in the 1920s by Alexander and Briggs, the Borromean rings were given the
Alexander–Briggs notation
In the mathematical field of 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 so it cannot ...
"6", meaning that this is the second of three 6-crossing 3-component links to be listed. The Conway notation for the Borromean rings, ".1", is an abbreviated description of the standard link diagram for this link.
History and symbolism
The name "Borromean rings" comes from the use of these rings, in the form of three linked circles, in the
coat of arms
A coat of arms is a heraldic visual design on an escutcheon (i.e., shield), surcoat, or tabard (the latter two being outer garments). The coat of arms on an escutcheon forms the central element of the full heraldic achievement, which in its ...
of the
aristocratic
Aristocracy (, ) is a form of government that places strength in the hands of a small, privileged ruling class, the aristocrats. The term derives from the el, αριστοκρατία (), meaning 'rule of the best'.
At the time of the word' ...
Borromeo Borromeo is a surname. Notable people with the surname include:
* House of Borromeo, an aristocratic family in Milan
Members of the House of Borromeo
* Andrea Borromeo (c. 1615 – 1683), Theatine priest
* Charles Borromeo (1538 – 1584), cardina ...
family in Northern Italy. The link itself is much older and has appeared in the form of the , three linked
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
image stone
A picture stone, image stone or figure stone is an ornate slab of stone, usually limestone, which was raised in Germanic Iron Age or Viking Age Scandinavia, and in the greatest number on Gotland.The article ''Bildstenar'' in ''Nationalencykloped ...
s dating back to the 7th century. The
Ōmiwa Shrine
, also known as , is a Shinto shrine located in Sakurai, Nara Prefecture, Japan. The shrine is noted because it contains no sacred images or objects because it is believed to serve Mount Miwa, the mountain on which it stands. For the same re ...
in Japan is also decorated with a motif of the Borromean rings, in their conventional circular form. A stone pillar in the 6th-century
Marundeeswarar Temple
Marundeeswarar Temple is a temple dedicated to Hindu deity Shiva, located in Thiruvanmiyur, Chennai adjacent to the beach of Bay of Bengal. It is one of the 275 ''Paadal Petra Sthalams'' where two of the most revered Nayanars (Saivite Saints), ...
in India shows three equilateral triangles rotated from each other to form a regular enneagram; like the Borromean rings these three triangles are linked and not pairwise linked, but this crossing pattern describes a different link than the Borromean rings.
The Borromean rings have been used in different contexts to indicate strength in unity. In particular, some have used the design to symbolize the
Trinity
The Christian doctrine of the Trinity (, from 'threefold') is the central dogma concerning the nature of God in most Christian churches, which defines one God existing in three coequal, coeternal, consubstantial divine persons: God th ...
. A 13th-century French manuscript depicting the Borromean rings labeled as unity in trinity was lost in a fire in the 1940s, but reproduced in an 1843 book by
Adolphe Napoléon Didron
Adolphe Napoléon Didron (1806–1867) was a French art historian and archaeologist.
Biography
Adolphe Napoléon Didron was born in Hautvillers on 13 March 1806. He began his education as a student of law. He then completed his early studies at t ...
. Didron and others have speculated that the description of the Trinity as three equal circles in canto 33 of
Dante
Dante Alighieri (; – 14 September 1321), probably baptized Durante di Alighiero degli Alighieri and often referred to as Dante (, ), was an Italian people, Italian Italian poetry, poet, writer and philosopher. His ''Divine Comedy'', origin ...
's ''Paradiso'' was inspired by similar images, although Dante does not detail the geometric arrangement of these circles. 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").
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
The Pabst Brewing Company () is an American company that dates its origins to a brewing company founded in 1844 by Jacob Best and was, by 1889, named after Frederick Pabst. It is currently a holding company which contracts the brewing of over ...
. For this reason they have sometimes been called the "Ballantine rings".
The first work of knot theory to include the Borromean rings was a catalog of knots and links compiled in 1876 by
Peter Tait Peter Tait may refer to:
* Peter Tait (physicist) (1831–1901), Scottish mathematical physicist
* Peter Tait (footballer) (1936–1990), English professional footballer
* Peter Tait (mayor) (1915–1996), New Zealand politician
* Peter Tait (radio ...
. In
recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
, the Borromean rings were popularized by
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...
, who featured
Seifert surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link.
Such surfaces can be used to study the properties of the associated knot or link. For example ...
s for the Borromean rings in his September 1961 "
Mathematical Games
A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games ne ...
" column in ''
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...
''. In 2006, the
International Mathematical Union
The International Mathematical Union (IMU) is an international non-governmental organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports ...
In medieval and renaissance Europe, a number of visual signs consist of three elements interlaced together in the same way that the Borromean rings are shown interlaced (in their conventional two-dimensional depiction), but with individual elements that are not closed loops. Examples of such symbols are the
Snoldelev stone
The Snoldelev Stone, listed as DR 248 in the Rundata catalog, is a 9th-century runestone that was originally located at Snoldelev, Ramsø, Denmark.
Description
The Snoldelev Stone, which is 1.25 meters in height, is decorated with painted scrat ...
horns and the
Diana of Poitiers
Diane de Poitiers (9 January 1500 – 25 April 1566) was a French noblewoman and prominent courtier. She wielded much power and influence as King Henry II's royal mistress and adviser until his death. Her position increased her wealth and family' ...
crescents.
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
The ''Principia Discordia'' is the first published Discordian religious text. It was written by Greg Hill ( Malaclypse the Younger) with Kerry Wendell Thornley (Lord Omar Khayyam Ravenhurst) and others. The first edition was printed allegedly usi ...
''.
Mathematical properties
Linkedness
In knot theory, the Borromean rings are a simple example of a
Brunnian link
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 (so that no two loops can be ...
, a link that cannot be separated but that falls apart into separate unknotted loops as soon as any one of its components is removed. There are infinitely many Brunnian links, and infinitely many three-curve Brunnian links, of which the Borromean rings are the simplest.
There are a number of ways of seeing that the Borromean rings are linked. One is to use Fox -colorings, colorings of the arcs of a link diagram with the integers modulo so that at each crossing, the two colors at the undercrossing have the same average (modulo ) as the color of the overcrossing arc, and so that at least two colors are used. The number of colorings meeting these conditions is a
knot invariant
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...
, independent of the diagram chosen for the link. A trivial link with three components has colorings, obtained from its standard diagram by choosing a color independently for each component and discarding the colorings that only use one color. For standard diagram of the Borromean rings, on the other hand, the same pairs of arcs meet at two undercrossings, forcing the arcs that cross over them to have the same color as each other, from which it follows that the only colorings that meet the crossing conditions violate the condition of using more than one color. Because the trivial link has many valid colorings and the Borromean rings have none, they cannot be equivalent.
The Borromean rings are an alternating link, as their conventional
link diagram
In the mathematical field of 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 so it cannot ...
has crossings that alternate between passing over and under each curve, in order along the curve. They are also an
algebraic link
In the mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
, a link that can be decomposed by Conway spheres into 2-tangles. They are the simplest alternating algebraic link which does not have a diagram that is simultaneously alternating and algebraic. It follows from the
Tait conjectures
The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conje ...
that the crossing number of the Borromean rings (the fewest crossings in any of their link diagrams) is 6, the number of crossings in their alternating diagram.
Ring shape
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
An impossible object (also known as an impossible figure or an undecidable figure) is a type of optical illusion that consists of a two- dimensional figure which is instantly and naturally understood as representing a projection of a three-d ...
: it is not possible to form the Borromean rings from circles in three-dimensional space. More generally proved using four-dimensional
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
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
In the mathematical field of 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 so it cannot ...
. 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
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
: 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. A realization of the Borromean rings by three mutually perpendicular
golden rectangle
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618.
Golden rectangles exhibit a special form of self-similarity ...
s can be found within a regular icosahedron by connecting three opposite pairs of its edges. Every three unknotted
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
s 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. In particular, because the Borromean rings can be realized by three triangles, the minimum number of sides possible for each of its loops, the stick number of the Borromean rings is nine.
More generally,
Matthew Cook
Matthew Cook (born February 7, 1970) is a mathematician and computer scientist who is best known for having proved Stephen Wolfram's conjecture that the Rule 110 cellular automaton is Turing-complete.
Biography
Cook was born in Morgantown, West ...
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.
Ropelength
In knot theory, the
ropelength
In physical knot theory, each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are ...
of a knot or link is the shortest length of flexible rope (of radius one) that can realize it. Mathematically, such a realization can be described by a smooth curve whose radius-one
tubular neighborhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the ...
avoids self-intersections. The minimum ropelength of the Borromean rings has not been proven, but the smallest value that has been attained is realized by three copies of a 2-lobed planar curve. Although it resembles an earlier candidate for minimum ropelength, constructed from four
circular arc
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circular ...
s of radius two, it is slightly modified from that shape, and is composed from 42 smooth pieces defined by
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s, making it shorter by a fraction of a percent than the piecewise-circular realization. It is this realization, conjectured to minimize ropelength, that was used for the
International Mathematical Union
The International Mathematical Union (IMU) is an international non-governmental organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports ...
logo. Its length is , while the best proven lower bound on the length is .
For a discrete analogue of ropelength, the shortest representation using only edges of the
integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ...
, the minimum length for the Borromean rings is exactly . This is the length of a representation using three integer rectangles, inscribed in Jessen's icosahedron in the same way that the representation by golden rectangles is inscribed in the regular icosahedron.
Hyperbolic geometry
The Borromean rings are a
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.
As a ...
: the space surrounding the Borromean rings (their
link complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...
) admits a complete
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
metric of finite volume. Although hyperbolic links are now considered plentiful, the Borromean rings were one of the earliest examples to be proved hyperbolic, in the 1970s, and this link complement was a central example in the video '' Not Knot'', produced in 1991 by the Geometry Center.
Hyperbolic manifolds can be decomposed in a canonical way into gluings of hyperbolic polyhedra (the Epstein–Penner decomposition) and for the Borromean complement this decomposition consists of two
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
uniform honeycomb
In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of face ...
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. The
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
Catalan's constant
In mathematics, Catalan's constant , is defined by
: G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots,
where is the Dirichlet beta function. Its numerical value is approximately
:
It is not known whether is irra ...
. The complement of the Borromean rings is universal, in the sense that every closed 3- manifold is a branched cover over this space.
Number theory
In
arithmetic topology 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.
Analogies
The following are some of the analogies used ...
, there is an analogy between
knots
A knot is a fastening in rope or interwoven lines.
Knot may also refer to:
Places
* Knot, Nancowry, a village in India
Archaeology
* Knot of Isis (tyet), symbol of welfare/life.
* Minoan snake goddess figurines#Sacral knot
Arts, entertainme ...
and
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s in which one considers links between primes. The triple of primes 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" or "mod 2 Borromean primes".
Physical realizations
A
monkey's fist
A monkey's fist or monkey paw is a type of knot, so named because it looks somewhat like a small bunched fist or paw. It is tied at the end of a rope to serve as a weight, making it easier to throw, and also as an ornamental knot. This type of ...
knot is essentially a 3-dimensional representation of the Borromean rings, albeit with three layers, in most cases. Sculptor John Robinson has made artworks with three
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
s made out of
sheet metal
Sheet metal is metal formed into thin, flat pieces, usually by an industrial process. Sheet metal is one of the fundamental forms used in metalworking, and it can be cut and bent into a variety of shapes.
Thicknesses can vary significantly; ex ...
, linked to form Borromean rings and resembling a three-dimensional version of the valknut. A common design for a folding wooden tripod consists of three pieces carved from a single piece of wood, with each piece consisting of two lengths of wood, the legs and upper sides of the tripod, connected by two segments of wood that surround an elongated central hole in the piece. Another of the three pieces passes through each of these holes, linking the three pieces together in the Borromean rings pattern. Tripods of this form have been described as coming from Indian or African hand crafts.
In chemistry,
molecular Borromean rings
In chemistry, molecular Borromean rings are an example of a mechanically-interlocked molecular architecture in which three macrocycles are interlocked in such a way that breaking any macrocycle allows the others to dissociate. They are the smal ...
are the molecular counterparts of Borromean rings, which are
mechanically-interlocked molecular architectures
In chemistry, mechanically-interlocked molecular architectures (MIMAs) are molecules that are connected as a consequence of their topology. This connection of molecules is analogous to keys on a keychain loop. The keys are not directly connected ...
. In 1997, biologist Chengde Mao and coworkers of
New York University
New York University (NYU) is a private research university in New York City. Chartered in 1831 by the New York State Legislature, NYU was founded by a group of New Yorkers led by then- Secretary of the Treasury Albert Gallatin.
In 1832, th ...
succeeded in constructing a set of rings from DNA. In 2003,
chemist
A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin ''alchemist'') is a scientist trained in the study of chemistry. Chemists study the composition of matter and its properties. Chemists carefully describe t ...
Fraser Stoddart
Sir James Fraser Stoddart (born 24 May 1942) is a British-American chemist who is Board of Trustees Professor of Chemistry and head of the Stoddart Mechanostereochemistry Group in the Department of Chemistry at Northwestern University in ...
and coworkers at
UCLA
The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California ...
utilised coordination chemistry to construct a set of rings in one step from 18 components. Borromean ring structures have been used to describe noble metal clusters shielded by a surface layer of thiolate ligands. A library of Borromean networks has been synthesized by design by Giuseppe Resnati and coworkers via
halogen bond A halogen bond occurs when there is evidence of a net attractive interaction between an electrophilic region associated with a halogen atom in a molecular entity and a nucleophilic region in another, or the same, molecular entity. Like a hydrogen ...
driven
self-assembly
Self-assembly is a process in which a disordered system of pre-existing components forms an organized structure or pattern as a consequence of specific, local interactions among the components themselves, without external direction. When the ...
. 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.
In physics, a quantum-mechanical analog of Borromean rings is called a halo state or an
Efimov state The Efimov effect is an effect in the quantum mechanics of few-body systems predicted by the Russian theoretical physicist V. N. Efimov in 1970. Efimov’s effect is where three identical bosons interact, with the prediction of an infinite series ...
, and consists of three bound particles that are not pairwise bound. The existence of such states was predicted by physicist
Vitaly Efimov
Vitaly N. Efimov (Russian: Вита́лий Никола́евич Ефи́мов) is a Russian theoretical physicist. He proposed the existence of a novel and exotic state of matter now dubbed the Efimov State as a researcher in A. F. Ioffe Phy ...
, in 1970, and confirmed by multiple experiments beginning in 2006. This phenomenon is closely related to a
Borromean nucleus
A Borromean nucleus is an atomic nucleus comprising three bound components in which any subsystem of two components is unbound. This has the consequence that if one component is removed, the remaining two comprise an unbound resonance, so that the ...
, a stable atomic nucleus consisting of three groups of particles that would be unstable in pairs. Another analog of the Borromean rings in
quantum information theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...
involves the entanglement of three
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
s in the
Greenberger–Horne–Zeilinger state
In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state (GHZ state) is a certain type of entangled quantum state that involves at least three subsystems (particle states, qubits, or qudits). It was first s ...
.
File:Knot Monkey Fist.jpg, A
monkey's fist
A monkey's fist or monkey paw is a type of knot, so named because it looks somewhat like a small bunched fist or paw. It is tied at the end of a rope to serve as a weight, making it easier to throw, and also as an ornamental knot. This type of ...
Molecular Borromean rings
In chemistry, molecular Borromean rings are an example of a mechanically-interlocked molecular architecture in which three macrocycles are interlocked in such a way that breaking any macrocycle allows the others to dissociate. They are the smal ...
Brady Haran
Brady John Haran (born 18 June 1976) is an Australian-British independent filmmaker and video journalist who produces educational videos and documentary films for his YouTube channels, the most notable being ''Periodic Videos'' and '' Numbe ...
Tadashi Tokieda
Tadashi Tokieda (Japanese: 時枝正; born 1968) is a Japanese mathematician, working in mathematical physics. He is a professor of mathematics at Stanford University; previously he was a fellow and Director of Studies of MathematicsNeon Knots and Borromean Beer Rings (
Clifford Stoll
Clifford Paul "Cliff" Stoll (born June 4, 1950) is an American astronomer, author and teacher.
He is best known for his investigation in 1986, while working as a systems administrator at the Lawrence Berkeley National Laboratory, that led to t ...
, 2018),
Numberphile
''Numberphile'' is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its ...