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	Borromean Rings
		A motif of three interlaced crescent moons, similar to the Borromean Rings, can be seen at the Palace of Fontainebleau; designed by the architect Philibert de l'Orme, it is based on the moon emblem used by Diane de Poitiers (1499-1566), mistress of King Henry II of France.
		The Borromean Rings are commonly used as a symbol of the Christianity Trinity.
		Borromean Rings can be also be found on Japanese family emblems, at a Japanese Shinto shrine north of Sakurai in the province of Nara, and in the sculptures of the Australian artist John Robinson.
	www.daviddarling.info /encyclopedia/B/Borromean_Rings.html (532 words)

	Borromean rings - Wikipedia, the free encyclopedia
		The Borromean rings as a symbol of the Christian Trinity, from a 13th-century manuscript.
		The name "Borromean rings" comes from their use in the coat of arms of the aristocratic Borromeo family in Italy.
		The psychoanalyst Jacques Lacan famously found inspiration in the Borromean rings as a model for his topology of the human mind, with each ring representing a fundamental Lacanian component of reality (the "real", the "imaginary", and the "symbolic").
	en.wikipedia.org /wiki/Borromean_rings (398 words)

	Basic polyhedron 6* (Site not responding. Last check: 2007-10-20)
		Such a three-component link is named "Borromean rings" after the Borromeas, an Italian family from the Renaissance that used them as their family crest symbolizing the value of collaboration and unity.
		Borromean circles are impossible: B.Lindström and H.O.Zetterström [39] proved using geometrical arguments that three flat circles cannot be Borromean, but that Borromean triangles are possible.
		Borromean rings you could find in geometry as the regular octahedron {3,4}, in Venn diagrams, in DNA, and in other various areas.
	www.mi.sanu.ac.yu /vismath/sl/l30.htm (429 words)

	Borromean Rings \|\| The Liberal Arts \|\| Dean's Welcome \|\| College of Arts and Sciences \|\| George Mason University (Site not responding. Last check: 2007-10-20)
		The three interlocking rings at the center of the logo of the College of Arts and Sciences are known as the Borromean Rings.
		If the Borromean Rings are stretched or pulled, they cease to be circles, but their special linking property remains.
		The Borromean Rings at the center of the College logo remind us also that the division of areas of study according to disciplines is an artificial means of organizing learning.
	cas.gmu.edu /deans_welcome/what_are_the_liberal_arts/borromean_rings (513 words)

	Borromean Rings (Site not responding. Last check: 2007-10-20)
		Borromean rings are three interlinked rings such that no two rings are linked.
		In other words, the three rings cannot be separated and yet no two of them are linked.
		It can be shown that there is no solution for flat rings, it can only be constructed by perturbing the rings in a third dimension.
	astronomy.swin.edu.au /~pbourke/curves/borromean (208 words)

	Constructing Borromean Links (Site not responding. Last check: 2007-10-20)
		Borromean is named after an Italian family of the Renaissance period who had a pattern of three, inter-linked circles as their crest.
		However, the inherent beauty of these rings which has caused their abundance in history is not the primary concern of knot theorists and chemists.
		For example, in the Tait series, the Borromean Rings are the element of the series where m = 1.
	www.ma.utexas.edu /users/katerman/a/essays/math321/borromean.html (1458 words)

	Cook's Borromean Ring Links
		To the Norse people of Scandinavia, a drawing of the Borromean rings using triangles is known as "Odin's triangle" or the "Walknot" (or "valknut" -- the knot of the slain).
		If the disks bounded by two rings don't intersect, then the rings clearly won't touch as they shrink, so the only interesting case is when the disks do intersect.
		The disc bounded by any ring (A) must intersect another (B) (since otherwise it could be shrunk to a point without hitting the others), and therefore must intersect it twice.
	paradise.caltech.edu /~cook/Workshop/Math/Borromean/Borrring.html (870 words)

	New Scientist Breaking News - Lord of the molecular rings created (Site not responding. Last check: 2007-10-20)
		The Borromean ring, an icon of Nordic and Christian traditions, has been self-assembled at the molecular scale level for the first time.
		Borromean rings have symbolised the holy trinity in Christian iconography, the heart of a giant in Nordic mythology, and the crest of the Borromeo family in 15th-century Tuscany.
		The result was a Borromean ring with a diameter of 2.5 nanometres and an inner chamber lined with 12 oxygen atoms.
	www.newscientist.com /article.ns?id=dn5050 (524 words)

	C&EN: LATEST NEWS - THREE RINGS IN AN INSEPARABLE UNION
		A macrocyclic molecule having the topology of Borromean rings has been prepared from 18 components using a “mix the pieces together and shake them all about” approach by chemists at the University of California, Los Angeles, and the University of Missouri, Columbia [Science, 304, 1308 (2004)].
		Borromean rings, whose use on the crest of the Italian Borromeo family can be traced back to the 15th century, comprise three mechanically interlinked rings that are inseparably united, although no two rings are linked.
		The authors suggest that their Borromean ring compounds could be exploited as highly organized nanoclusters in spin-electronic materials or for medical imaging.
	pubs.acs.org /cen/news/8222/8222notw1.html (574 words)

	Molecular Borromean rings unlocked
		Chemists in the US have provided verification of the structure of their acclaimed nanoscale Borromean rings, and suggest that they can now find practical applications for the rings.
		The attractive Borromean ring motif, dating from the 15th century, is characterised by three inseparable, interlocked rings.
		Stoddart describes the Borromean ring complex as a ‘precisely defined nanoparticle with the opportunity to introduce functionality’, suggesting that the Borromeand represents not only an elaborate molecule, but also has real potential for applications.
	www.rsc.org /chemistryworld/Issues/2005/August/Borromean.asp (274 words)

	Borromean rings (Site not responding. Last check: 2007-10-20)
		The design of interlocked rings on the lefthand side of the picture above is called the Borromean rings.
		A web search for the Borromean rings reveals that they are sometimes used to symbolize the value of collaboration.
		Not surprisingly to mathematicians, the rings are of interest to knot theorists.
	www.math.ntnu.no /borromean.html (155 words)

	Illumine Me: Borromean Rings
		Notice those rings in the small blue portion of the crest just behind the horse toward the bottom.
		Throughout the estate these rings occur in the design of the flower gardens and in the inlaid wooden floors.
		But because there are three rings, the identity of each one is inseparable from the other two.
	www.illumineme.com /archives/2005/12/borromean_rings.html (733 words)

	Survey of Venn Diagrams -- Borromean Rings (Site not responding. Last check: 2007-10-20)
		The Borromean rings consists of three interlocking rings, with the property that if any one of them is removed, then all three separate.
		The shadow, a minimal projection of the rings, is the familiar 3-Venn diagram.
		For further information, particularly of a mathematical nature, about Borromean rings and their generalizations, see the web page [SJ] of Slavick Jablan.
	www.math.helsinki.fi /EMIS/journals/EJC/Surveys/ds5/borromean.html (295 words)

	Chemists make molecular interlocked rings
		Named for a noble Italian family, the Borromean rings first appeared on the family's coat of arms in the 15th century.
		Examples of the rings can be seen in buildings on three islands in northern Italy's Lake Maggiore, which are still owned by the Borromeo family.
		Each molecule of the Borromean ring compound is 2.5 nanometers across and contains an inner chamber that is a quarter of a cubic nanometer in volume and is lined by 12 oxygen atoms.
	www.eurekalert.org /pub_releases/2004-05/uoc--cmm052604.php (990 words)

	EUROPA - Research - RTD info -Special Issue - March 2004 - The enigma of knots
		Two ways of presenting the Borromean Rings, the heraldic symbol of the Borromea family at the time of the Renaissance, at the Castello Sforzesco in Milan (IT).
		No one ring links with any other within the Borromean Rings, yet the group that they form cannot be unknotted.
		These rings can also be seen on stone sculptures in Gotland, an island off the coast of Sweden, which are believed to date from around the ninth century.
	europa.eu.int /comm/research/rtdinfo/special_as/article_812_en.html (408 words)

	Borromean rings - TheBestLinks.com - Circle, Mathematics, TheBestLinks.com:Find or fix a stub, Aristocracy, ...
		Borromean rings - TheBestLinks.com - Circle, Mathematics, TheBestLinks.com:Find or fix a stub, Aristocracy,...
		Borromean rings, Circle, Mathematics, TheBestLinks.com:Find or fix a stub...
		In mathematics, the Borromean rings consist of three topological circles that are linked despite the fact that no two of them are linked.
	www.thebestlinks.com /Borromean_rings.html (139 words)

	Molecular Borromean rings unlocked
		Fraser Stoddart and his group at the University of California, Los Angeles, were the first to design a completely synthetic molecular representation of the Borromean ring.
		The researchers have managed to remove zinc ion templates from their Borromean rings to produce the free structure, dubbed the Borromeand by Stoddart, who said, ‘People were constantly asking, "Can you demetallate the complex?" Now we can say, "Yes, we can".’
		David Leigh, an expert in functional molecular architectures at the University of Edinburgh, UK, explains: ‘The Borromean ring assembly is probably the most impressive designed one- pot synthesis of any molecular structure.’ He adds that the chemistry that Stoddart has now performed on this ‘extraordinary molecule’ is ‘chemical proof of the molecular topology’.
	www.rsc.org /Publishing/ChemScience/Volume/2005/8/Borromean.asp (287 words)

	think again! - comments (Site not responding. Last check: 2007-10-20)
		If one of the ring breaks, let's say the fl one, the blue is still attached to the yellow, and the green to the red.
		I would think not, because at least one of the rings is goint to have to be connected to two other rings, and if oneo f those other two broke, it would still be connected to the remaining ring.
		You could connect all of the colors to the fl ring only, and if the fl ring broke, then the rest would be disconnected, but that wouldn't hold true iof oneo fthe colored rings broke.
	simpler-solutions.net /pmachinefree/thinkagain/comments.php?id=946_0_3_0_C (674 words)

	Sculpture Maths - Borromean Rings
		The theorem stating Borromean rings to be impossible with flat circles is proved rigorously in the article "Borromean circles are impossible," Amer.
		Triangle form of the Borromean Rings which makes INTUITION is also shown by Heidi Burgiel in the Geometry Center Picture Archive.
		This material may be used freely for educational, artistic and scientific purposes, but may not be used for commercial purposes, for profit or in texts without the permission of the publishers.
	www.popmath.org.uk /sculpmath/pagesm/borings.html (546 words)

	[Sci.nanotech] Re: Interlocked Borromean Rings Molecules - Nanoscience, But Also Much More (Site not responding. Last check: 2007-10-20)
		Adjacent rings are joined either by a single C-C bond, an intermediate -N-, or an -O-.
		One of the diagrams shows the three rings as mutually orthogonal, in the XYZ planes and forming a globe, rather than lying in a single XY plane with Z-axis highway overpasses...
		>"Each molecule of the Borromean ring compound is >2.5 nanometers across and contains an inner chamber that is a quarter >of a cubic nanometer in volume and is lined by 12 oxygen atoms.".....hence, this passage seems to mean 12 oxygens total in the three-loop compound, with the 0.25-nm^3 chamber at the center of the globe.
	venusia.golgothe.net /pipermail/sci.nanotech/2004-June/001169.html (344 words)

	Amy Hayden
		Borromean-styled rings have been of interest for thousands of years in Christian, Norse, and Asian art and iconography.[1] They acquired their name from the wealthy Italian Borromeo family of the Renaissance period.
		Their family crest bears the symbol of these peculiar interlocking rings.[1] One sees on closer inspection that if one ring is severed, the other rings are no longer held together, serving as a symbol of the importance of strength in unity.
		Seeman4 was the first to have success in the synthesis of such rings, constructing them with the Borromean topology from three different strands of DNA.
	stoddart.chem.ucla.edu /personnel/grads/hayden/hayden.html (515 words)

	Introduction (Site not responding. Last check: 2007-10-20)
		B.Lindström and H.O.Zetterström [1] proved that "Borromean circles are impossible": three flat circles cannot construct them, but by triangles they can.
		P.Cromwel recognized it in a picture-stone from Gotland [2], and it are exactly these and other symmetrical combinations of three and four hollow triangles that were considered by H.S.M.Coxeter [3].
		In geometry, Borromean rings appear as the regular octahedron {3,4} [4], in Venn diagrams [5], in DNA [6], and in other various areas [7].
	members.tripod.com /vismath5/bor/bor1.htm (149 words)

	Science News: Chemical knot: scientists assemble legendary symbol by interlocking molecules
		In a feat of chemistry imitating art, researchers have created a molecular version of a Borromean knot, an attractive pattern of three interlocking rings that commonly adorned Viking art and Renaissance architecture.
		Making molecular versions of Borromean rings poses formidable challenges for chemists because no pair of rings is linked unless the third ring is present.
		When X-ray crystallographic analysis confirmed that the 2.5-nanometer-wide molecular structures were indeed Borromean rings, Cantrill and the rest of the UCLA team were elated.
	www.findarticles.com /p/articles/mi_m1200/is_22_165/ai_n6110377 (398 words)

	SummeResearch
		After starting the 3D printer on five sets of Borromean Rings this morning we went to have our tutorial on using dreamweaver so that we will be able to post a web journal about our research.
		A set of Borromean Rings is a set of three interlocking noncircular rings (it has been proved that circular Borromean Rings cannot exist) interlocked so that any two are not interlocked while the shape as a whole is locked (i.e.
		The goal was to construct a tighter model for the peanut shaped Borromean rings, but I realized this morning that I had been constructing the curve in a way completely different from what Cantarella had done.
	maven.smith.edu /~orourke/DTS/Sum05/WebPages/Gillian/SummerBlog.htm (5913 words)

	Sculpture Maths - African Borromean Rings (Site not responding. Last check: 2007-10-20)
		Here is a very fascinating and beautiful African object, a particularly intriguing interpretation of the Borromean Rings, yet probably rather common in African countries.
		Each component is topologically a ring (neglecting a small handle), and may be very roughly described as the union of 3 rectangles, one in the median position which is hollowed out in ring shape, and two full ones, connected by one small edge to opposite vertices of the median one.
		The median annular rectangle is set obliquely across the median third of the log, while the full rectangles are carved out from the external surface of the extreme thirds of the log.
	www.popmath.org.uk /sculpmath/pagesm/africa.html (225 words)

	Scientific Symbols, Icons, Mathematical Symbols - Numericana
		The Borromean Rings: Three interwoven rings which are pairwise separate.
		Interestingly, it can be shown that such rings cannot all be perfect circles (you'd have to bend or stretch at least one of them) and the converse seems to be true: three simple unknotted closed curves may always be placed in a Borromean configuration unless they are all circles [no other counterexamples are known].
		The three rings are found among the many symbols featured on the Borromeo coat of arms (they are not nearly as prominent as one would expect, you may need a closer look).
	home.att.net /~numericana/answer/symbol.htm (2260 words)

	Radioactive Nuclear Beams (Site not responding. Last check: 2007-10-20)
		The Borromean rings, heraldic symbol of the princes of Borromeo, are carved in the stone of their castle on an island in Lago Maggiore in Northern Italy.
		The three rings are interlocked in such a way that if any of them were removed, the other two would also fall apart.
		The Borromean property gives for bound states a unique and simple asymptotic behavior which is exponential (neutrons) in the hyperradius and governed by the two-nucleon separation energy.
	www.fi.uib.no /~rogde/borromean.html (1307 words)

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