(2011-08-25) Previously, the only way to display mathematical symbols on the Web.
The World Wide Web was originally developed at CERN to facilitate Internationalscientific communications. In the early days, only the 7-bit characters in theASCII setwere unambiguously understood. (EBCDIChas always been limited to IBM's mainframe computers). Only 95 codes in the ASCII set correspond to ordinary printable characters (the most common of which is the blank space). The remaining 33 other codes in the 128-character ASCII set are assigned to so-called control characters meant tocontrol either the flow of information or the output device (the most common being the end-of-line indicators; carriage-return and/or line-feed).
That humble starting point has led to a rather sorry state of affairs whenever theoriginal character set is clearly insufficient, as is the case for scientific communications. (As of June 2011, IT professionals in at least one big German organisation were testingcritical web pages on no fewer than 42 slightly different delivery platforms.)
Big Browser :
There is no excuse for not supporting the legacy Symbol font in modern browsers. Doing so does not interfere at all withproper support, for example. I argue that browsers that do not support legacy standards toinsure the readability of yesteryear's valuable information simply do notdeserve our trust in the long run. On that basis alone, I recommend Internet Explorer and Google Chrome and must, regretfully, advise against the latest versions ofOpera, Safari and Firefox (not a single Web author who has everused the Symbol font has ever meant it to be renderedlike those browsers do, by mistakingly using a "standard"character-encoding for it).
Thanks to Philippe Verdy for background information (private messages).
The Lambda-Nu Test (2016-02-07) :
To some extend, it's possible to check on-the-fly whether the Symbol font is well-handled by the browser being usedand take corrective action if it's not.
Due to privacy concerns, browsers aren't allowed to check directly whether or nota given font is installed (since that would involve reading fileson the user machine, with the ability to report some of the contents to everybody). However, we may fairly reliably test whether the Symbol font is correctly loaded with the proper encodingby checking that a lowercase lambda is wider thana lowercase nu (whereas the corresponding "l" is narrower than "n" in allnon-monospaced latin fonts).
If that's not the case, advanced capabilities of the more recent versionsof JavaScript can be used to replace individual characters by theirclosest counterparts. That's not a perfect solution butit does make mathematical expressions readable (albeit with poorly-rendered details).
This is used in the patch at the bottom of Numericana's main script to correct browserswhose designers have messed up with the (admittedly ugly) original Symbol encoding (for semi-religious reasons). However, some mobile devices don't yet support the aforementioned advancedcapabilities which allow the patch to work.
There's a chance that better versions of JavaScript will be supportedon mobile devices in the future, which will fix the issue (kinda). There's also a (slim) chance that all browser designers willeventually become aware of their God-given duty to cherish all types of evergreeninformation, which can't be economically re-encoded to follow the latest fashion!
One advantage of the lambda-nu test is that it puts no demands onfuture maintainers of a site, without penalizing at all the users of browsers withproper legacy support of the Symbol font. Be it right now (e.g., Google Chrome, Internet Explorer) or in the future... If FireFox is ever fixed, Numericana will look slightly betterto all Firefox users. Instantly.
Emily Guerin (2004-06-18; e-mail) Who was the first person to use the modern equal sign?
A very elongated form of the modern equality symbol (=) was introduced in printinThe Whetstone of Witte (1557) byRobert Recorde(1510-1558) the man who first introduced algebra into England. He justified the new symbolby stating that no two things can be more equal than a pair of parallel lines.
I've been told that a manuscript from the University of Bologna, dated between 1550 and 1568, features the same notation for equality, apparently independently of the work of Robert Recorde (and possibly slightly earlier).
William Oughtred(1574-1660) was instrumental in the subsequent popularizationof the equal sign, which appearsnext in 1618, in the appendix [attributed to him]of the English translation by Edward Wright of John Napier'sDescriptio(whereearly logarithms were first described in 1614). The same mathematical glyph is then seen again, and perhaps more importantly,in Oughtred's masterpieceClavis Mathematicae (1631) in whichother scientificsymbols are experimented with, which are still with us today (including for multiplication). It was independently established by the Traité d'algèbre (1690) of Michel Rolle (1652-1719).
Instead of the now familiar equal sign,many mathematicians used words or abbreviations (including "ae" for theLatinaequalis) well into the 18th century. Thomas Harriot (1560-1621) was using a different symbol (), while some others used a pair ofvertical lines ( || ) instead.
(2010-05-05) Equilibrium can be denoted by a right over left double-harpoon.
Somechemical reactions proceed until one of the reactanthas virtually disappeared. Ihis is denoted by a simple rightward-arrow symbol:
2 H2 + O2 2 H2O
However, as the rate of a chemical reaction depends on the concentrationof the reactants, a dynamic equiibrium is often reachedwhereby the concentrations of all the compounds involved are such that both directions of thechemical reaction proceeed at equal rates. Several symbols have been used to indicate this. The most symmetrical such symbolis the double-headed arrow sign ( )
However, the preferred scientific symbol for chemical equilibriumconsists of two superposed arrows (the rightward arrow is always above the leftward one) This has evolved graphically into the following stylishsign, affectionately known as the double-harpoon symbol:
This is the so-called right-left version of the symbol (). In chemistry, it's considered bad form to use its left-right mirror image.
An ancient symbol meant to evoke dynamic equilibrium is the caduceus (symbol of trade and alchemy,commonly used by pharmacists and often wrongly associatedwithmedicine).
(Monica of Glassboro, NJ.2001-02-08) What's the correct terminology for the linebetween the numerator and denominator of a fraction?
When the numerator is written directly above the denominator,the horizontal bar between them is best called a vinculum.
Theoverbar part of a square-root sign or aguzintais also called avinculum, so is the full weightsuperbar or overscore used to tie several symbols together (in particular, groups of letters with a numerical meaning in Greek orLatin, where such explicit groupings may also imply multiplication by 1000). The thinner diacritical mark placed over a single characteris called a macron. (e.g., macrons are used over long vowels in some modern Latin transcriptions).
When the numerator and denominator appear at the same level,separated by a slanted line (e.g., "1/2")such a line is best called asolidus. It's also calledslash or stroke [British] and, more formally,virgule oroblique [British]. In the German language, this symbol was the predecessor of the modern comma punctuation symbol (virgule is French for comma).
The noun solidus originates from theRoman gold coin of the same name(the ancestor of theshilling, of the Frenchsol orsou, etc.).The sign was originally a monetary symbol, which was still used for the Britishshilling in 1971 (when British money was decimalized). See discussionbelow.
The related symbol "" is called anobelus. It was introduced as a division symbol in 1659 by the Swiss mathematician Johann Rahn(1622-1676) who is also credited for the "therefore" symbol (). Today, the obelus symbol is rarely used to separate both parts of a ratio,but it remains very familiar as the icon identifying thedivision key oncalculators...
I was at Uni in 1971 andcan't remember ever using "/" instead of "s"for shillings. Beforeanother meaningcame along, the acronym Lsd (or £sd ) referred to the old British coinage system based on the ancient Roman currency names (libra,solidus,denarius) as opposed to the new decimal " £p " system.
Although one pound and two shillings could, indeed, have been denoted £1/2 I remember thinking of the solidussymbol only as a separator : Two-and-sixpence would have been 2/6d. One pound, two shillings and sixpence would have been £1/2/6d. In shops, a price of one pound was often marked this way: 1 / - / - The symbol was pronounced stroke (oblique wasposh).
Both meanings of the solidus sign (i.e., currency prefix and/or separator) are compatibleand havecoexistedpeacefully. Arguably, thedefinition presented by John Harmer became dominant with the passage of time.
(2003-08-08) The infinity symbol introduced by John Wallis in 1655.
This sign was first given its current mathematical meaning in"Arithmetica Infinitorum" (1655)by the British mathematician John Wallis (1616-1703).
(resp. ) is the mathematical symbol used to denote the "limit" of a real quantity thateventually remains above (resp. below)any preset bound.
Incanonical mapsbetween thecomplex planeand a sphere minus a point, theunsigned symbol () corresponds tothe "missing point" of the sphere, but is not aproper complex number... It's just a convenient way to denote the fictitious"infinite circle" at the horizon of the complex plane.
The symbol itself is properly called alemniscus,a latin noun which means "pendant ribbon"and was first used in 1694 by Jacob Bernoulli (1654-1705)to describe a planar curve now called Lemniscate of Bernoulli.
The design appeared in Western iconography before modern times. It's found on the cross ofSaint Boniface(bishop and martyr, English apostle of Germany, néWinfrid c.675-755).
Theinfinity snake, theouroboros symbol(also,uroboros or uroborus)is a serpent or a dragon biting its own tail (ómeans "tail swallower"). The graphic appeared in Egypt as early as 1600 BC, and independentlyin Mesoamerica (see a Mayan version at left). It has been associated with the entire Zodiac and the eternity of time. It's the symbol of the perpetual cyclic renewal of life. It has been found in Tibetan rock carvings and elsewheredepicted in the shape of a lemniscate, although aplain circle is more common(the circle symbolizes infinity in Zen Buddhism).
(2003-11-10) and, theotherinfinity symbols.
As discussedabove, theinfinity symbol of Wallis() isnot a number...
However, there are two different definitions that makegood mathematical sense of actual infinite numbers. Both of those were first investigated byGeorg Cantor (1845-1918):
Two sets are said to have the samecardinal number of elementsif they can be put in one-to-one correspondence with each other. For finite sets, the natural integers (0,1,2,3,4 ...) areadequatecardinal numbers, buttransfinite cardinalsare needed forinfinite sets. The infinity symbol (pronounced "aleph zero", "aleph null", or "aleph nought") was defined by Cantor todenote the smallest of these (the cardinal of the set of the integers themselves).
Cantor knew that more than one transfinite cardinal was needed because his owndiagonal argumentproves that reals and integers have different cardinalities. (Actually, because thepowersetof a set is always strictly larger than itself,there are infinitely many different types of infinities,each associated with a different transfinite cardinal number.)
The second kind of infinite numbers introduced by Cantorare calledtransfinite ordinals. Observe that a natural integer may be represented bytheset of all nonnegative integers before it,starting with the empty set ( )for 0 (zero) because there areno nonnegative integers before it. So, 1 corresponds to the set {0}, 2 is {0,1}, 3 is {0,1,2}, etc.For the ordinal corresponding to the set ofall the nonnegative integers {0,1,2,3...}the infinity symbol was introduced.
Cantor did not stop there, since {0,1,2,3...}corresponds toanother transfinite ordinal, which is best "called" +1. {0,1,2,3...+1} is +2, etc. Thus, is much more like an ordinary number than. In fact, within the context ofsurreal numbersdescribed by John H. Conway around 1972, most of the usual rules of arithmetic apply toexpressions involving (whereas Cantor's scheme foradding transfinite ordinals is not even commutative). Note that 1/ is anothernonzerosurreal number, aninfinitesimal one. By contrast, adding one element to an infinity of elementsstill yields just elements, and 1/ismeaningless.
The chevron (wedge) and inverted chevron (vee) are the generic symbols usedto denote the basic binary operators induced by a partial ordering on a lattice. Those special characters have the following meanings:
The chevron symbol (wedge) denotes the highest element "less" than (or equal to)both operands. ab = inf(a,b) is called the greatest lower bound, the infimum or meet ofa andb. The operation is well-defined only in what's called a meet semilattice, a partially ordered set where two elements alwayshave at least one lower bound (i.e., an element which is less than or equal to both).
The inverted chevron symbol (vee) denotes the lowest element "greater" than(or equal to) both operands. ab = sup(a,b) is called the least upper bound, the supremum or join ofa andb. The operation is well-defined only in what's called a join semilattice, a partially ordered set where two elements alwayshave at least one upper bound (i.e., an element which is greater than or equal to both).
A set endowed with a partial ordering relation which makes it both a meet-semilattice and a join-semilattice is called a lattice (French: treillis).
In the special case of a total ordering (like the ordering of real numbers) two elements can always be compared (if they're not equal, one is larger and one is smaller) so either operation will always yield one of the two operands:
By contrast, consider the relation among positive integers (usually denoted by a vertical bar) which we may call "divides" or "is a divisor of". It's indeed an ordering relation (because it'sreflexive,antisymmetric andtransitive) but it's only apartial ordering relation (for example, 2 and 3 can't be "compared" to each other, as neither divides the other). In that context, pq is the greatest common divisor (GCD) of p and q, more rarely dubbed highest common factor (HCF). Conversely, pq is their lowest common multiple (LCM).
pq = gcd(p,q) [ = (p,q) ] (*)
pq = lcm(p,q)
(*)
In the context of Number Theory, the above use of the "wedge" and "vee"mathematical symbols needs little or no introduction, except to avoid confusion with the meaningthey have in predicate calculus (the chevron symbol stands for "logical and", whereasthe inverted chevron is "logical or", also called "and/or").
InSet Theory, the fundamental ordering relation among sets may becalled "is included in" ( or, more precisely,). In this case, and in this caseonly,the corresponding symbols for the related binary operators assumerounded shapes and cute names: cap () andcup (). AB and ABare respectively called theintersection and theunion of the sets A and B.
The intersection AB is the set of all elements thatbelong toboth A and B. The union AB is the set of all elements thatbelong to A and/or B ("and/or" means "either or both";it is the explicitlyinclusiveversion of the more ambiguous "or" conjunction, which normallydoes mean "and/or"in any mathematical context).
The chevron symbol is also used as a sign denoting the exterior product (thewedge product).
In an international context, the same mathematical symbol may be found to denotethe vectorialcross productas well...
(2007-11-12) Unionof distinct copies of sets in an indexed family.
The concept of disjoint union coincides with the ordinaryunion for sets that are pairwise disjoint. In modern usage, the term disjoint union is almost alwaysused to denote the ordinary union of sets that are pairwise disjoint.
In that particular case it coincides with the concept of what's best calleda discriminated union, as discussed below. However, that notion is all but obsolete;you can live a happy mathematical life without it.
Formally, the discriminated union of an indexed family ofsets Ai is:
Ai =
{ (x,i) | xAi }
I
I
However, such an indexed family is often treated as a mere collection of sets. The existence of an indexation is essential in the above formulation, but theusual abuse of notation is to omit the index itself, which is considered mute. This makes it possible to use simple notations like A+B or AB for the disjoint union of two sets A and B. The squared "U" symbol () is the preferred one (because the plus sign is so overloaded). In handwriting and in print, that "squared U" is best drawnas an "inverted pi", to avoid any possible confusion withthe "rounded U" symbol (cup) denoting an ordinary union of sets.
) and vice-versa.
Additive notations are [somewhat] popular for discriminated unions because thecardinal of a discriminated union is always the sum of the cardinals of its components. Denoting |E| the cardinal of the set E :
| A | = |A |
From acategorial perspective,the disjoint union is the dual of the categorial product. It's called either coproduct or sum.
(2005-09-26) Letters enhanced with double lines are symbols forsetsof numbers.
Such symbols are attributed toNicolas Bourbaki,although they don't appear in the printed work of Bourbaki... Some Bourbakists like Jean-Pierre Serre advise against them, except in handwriting (including traditional blackboard use).
One advantage of using the doublestruck symbols, even in print (against the advice of Jean-Pierre Serre) is that they do not sufferfrom anyoverloading. This makes them usable without the need forbuilding up a context (with the possible exception of , which some authors use for the projective plane).
The group formed by theinvertibleelements of a multiplicative monoid M is denoted M*. That's compatible with the common usage ofstarring the symbol of a set of numbers to denote the nonzero numbers in it (the two definitions areequivalent for *, *, * and *). In particular:
* is undefined (arguably, that symbol might denote the odd primes ).
(2003-08-03) The integration sign of Leibniz (29 October 1675).
Gottfried Wilhelm Leibniz (1646-1716)viewedintegration as a generalized summation,and he was partial to the name "calculus summatorius"for what we now call [integral]calculus. He eventually settled on the familiar elongated "s" for the sign of integration, after discussing the matter withJacob Bernoulli(1654-1705) who favored the name "calculus integralis"and the symbol I for integrals.
Eventually, what prevailed was the symbol of Leibniz, with the name advocated by Bernoulli...
(2021-07-15) Evaluation bar (appearing after an expression). Value at one point, or difference between the values at two points.
This glyph is normally taller than the regular pipe symbol ("|") found on modern keyboards.
The evaluation bar can be used in two distinct ways; specifyingone or two points of evaluation. In both cases, the expression to be evaluated is written before the bar. In the more common two-point version, the symbol denotes the difference betweenthe value of the expression at the point given as superscript and the value at thepoint given as subscript.
This is most often used when a definite integral is computed as the differenceof the values of a primitive at the two bounds of integration. For example, the integration by parts which establishes the recurrence relation between Wallis integrals can be written:
sinn+2t dt =
sinn+1t cos t
+ (n+1)
sinnt cos2 t dt
I prefer putting the upper and lower points as superscript and subscript on a closingsquare bracket (matching an earlier opening square bracket). For an expression involving more than one variable, it's prudent to indicate which variableis fixed, in the superscript or the subscript (or both):
x sinn+2t dt =
x sinn+1t cos t
t = 0
+ (n+1)
x sinnt cos2 t dt
According to Florian Cajori (1859-1930) a similar notation was first introduced by Frédéric Sarrus (1798-1861) in 1823 and subsequently popularized by Augustin Cauchy (1789-1857) and his student Abbé Moigno (1804-1884). Early on, the bar described above may have come before the expression, or the expression was between two barsthe second of which being as described above. Both are obsolete.
In its single-point version, the evaluation bar is simply read evaluated at. The point where the evaluation takes place is just given to the right of the evaluation bar (possibly as a subscript or a superscript, it doesn't matter).
In some computer languages or calculators, like the Voyage 200, the glyph thus denotes the handy evaluation operator, which has lower precedence than any other operator except the assignment operators. For example, the following line assigns the value 4 to the variable x:
x + x | x = 2 x
(2002-07-05) [QED =Quod Erat Demonstrandum ] What's the name of the end-of-proof box, in a mathematical context?
Mathematicians call it ahalmos symbol, afterPaulR. Halmos(1916-2006). Typographers call it a tombstone, which is the name of the symbol in anynon-scientific context.
Before Halmos had the idea to use the symbol in a mathematical context,it was widely used to mark the end of an article in popular magazines (it still is). Such atombstone is especially useful for an article whichspans a number of columns on several pages,because the end of the article may not otherwise be so obvious... Some publications use a small stylized logoin lieu of a plain tombstone symbol.
SeeMath Words... Here's ahalmos symbol, at the end of this last line!
Jacob Krauze(2003-04-20; e-mail) As a math major, I had been taught that the symbol (used for partial derivatives) was pronounced "dee",but a chemistry professor told me it was pronounced "del". Which is it? I thought "del" was reserved for [Hamilton's nabla operator] = x,y,z
"Del" is a correct name forboth and . Some authors present these two signs as the lowercase and uppercase versions of the samemathematical symbol (the terms"small del" and "big del" [sic!] are rarely used, if ever).
Physicists and others often pronouncey/x"del y by del x". A better way to read this aloud in front of a classroomis either "partial of y with respect to x" or "partial of y along x" (especially when x is a space orspacetime coordinate).
In an international scientific context, the confusion between and is best avoided by calling "nabla del",or simply nabla. Some practitioners also read it "grad" (since nabla can be construed as denoting ageneralized gradient ).
mathematical symbol, whose shape is reminiscentof a Hebrew harp by the same name (also spelled "nebel"). The term was first adopted by Peter Guthrie Tait (1831-1901)byHamiltonand also byHeaviside. Maxwell apparently never used the name in a scientific context.
The question is moot for many mathematicians, who routinely reada symbollike a "d" (mentally or aloud). I'm guilty of this myself, but don't tell anybody!
When it's necessary to lift all ambiguities without sounding overly pedantic,"" is also routinely called"curly d", "rounded d" or "curved d". The signcorresponds to the cursive "dey" of theCyrillic alphabet and is sometimesalso known asJacobi's delta, becauseCarl Gustav Jacobi(1804-1851)is credited with the popularization of the modern mathematical meaning ofthis special character (starting in 1841,with the introduction of Jacobians in theepoch-making paper entitled"De determinantibus functionalibus"). Historically, this lowercase mathematical symbol was first used by Condorcet in 1770,and byLegendre around 1786.
(2021-08-11) Traditional dedicated symbol (since 1890, at least).
This glyph is actually a very stylized lowercase "p".
Geetar (2007-07-18) What's the symbol for the 13th zodiacal constellation, Ophiuchus?
Ophiuchus is the name (abbreviatedOph) of a constellation also known as Serpentarius (French:Serpentaire). The serpent bearer.
This "snake handler" is actually the demigod Asclepios/Aesculapius,the Greek/Roman god of medicine, a son of Apollo who was taught the healing artsby the centaur Chiron. Asclepius served aboard Argo as ship's doctor of Jason (in the quest for the Golden Fleece) and became so good at healing thathe could bring people back from the dead. This madethe underworld ruler (Hades) complain to Zeus, whostruck Asclepius with a bolt of lightning but decided to honor him with a placein the sky, as Ophiuchus. The Greeks identified Asclepius with the deified Egyptian official Imhotep whe probably never practiced medicine himself (27th century BC).
TheRod of Asclepius, symbol of medicine,is asingle snake entwined around a stick. Originally, the symbol may have depicted the treatment of dracunculiasis (very common in the Ancient World) in which the long parasitic worm wastraditionally extracted through the patient's skinby wrapping it around a stick over a period of days or weeks (because a faster procedure might break the worm).
Any symbol involving a snake would seem natural for medicine: The snake is a symbol of renewed life out of old shedded skin, not to mention the perpetual renewal of life evoked by theouroborossymbol (a snake feeding on its own tail). A snake around a walking stick is also an ancient symbol of supernatural powerswhich can triumph over death, like medicine can (biblically, the symbol of Moses' divine mission was his ability to change hiswalking stick into a snake).
The large Ophiuchus constellationis one of the 88 modern constellations. It was also oneof the 48 traditional constellations listed by Ptolemy. In both systems, it's one of only 13zodiacal constellations. By definition, a zodiacal constellation is a constellation whichis crossed by theecliptic (the path traced by the Sun on the celestial sphere, which is so named because that'swhere solareclipses occur).
Ophiuchus is the only zodiacal constellation which has not givenits name to one of the 12 signs of the zodiac associated withthe 12 traditionalequal subdivisions of the solar year, which form thecalendar used by astrologers. However, somemodernastrologers are advocating a reformed system withuneven zodiacal signs, where Ophiuchus has found its place...
Astrological belief systems are not proper subjects for scientific investigation. Nevertheless, we must point out that it's a plain error to associate Ophiuchuswith thecaduceus symbol (two snakes around awinged staff) since that symbol of Hermes (messenger of the gods) is associated with commerce, not medicine.
The proper symbol for Ophiuchus is indeed the Rod of Asclepius or Staff of Asclepius (one snake around aplain stick) the correct symbol of the medical profession, which is mythologically tied to the Ophiuchus constellation. Period.
(2007-11-25) Image of dynamic equilibrium. Symbol of commerce.
Several explanations exist for this ancientoverloaded symbol.
In Greek mythology, the kerykeion symbol (latin:Caduceus) which was ultimately inherited by Hermes (called Mercury by the Romans) is often said to have originatedwith the blind seerTiresias,the prophet who had experienced both sexes.
The caduceus symbol evokes a dynamic equilibrium emerging from a confrontation of opposing principles (male and female). As an alleged symbol of peace,the kerykeion represents a balance of powersrather than a lack of tensions.
The two facing serpents are also said to symbolize water and fire,two opposing elements entwined around the axis of the Earth. The wings evoke the spiritual or spatial dimension of the fourth element : sky, wind or air.
Also, the copulating serpents have been construed as a fertility symbol involving twocomplementary forces revolving around a common center. This makes the caduceusa western counterpart of the oriental taiji.
Hermes was the god of alchemists, who were fascinated by theelementmercury and held as fundamental the unification of opposites. By extension, the caduceus became associated with chemistry and pharmacy.
The caduceus is also associated with communication, eloquence,trade and commerce, the traditional attributions of Hermes, messenger for the gods and protector ofall merchants, thieves, journalists, tricksters and... inventors.
(2008-05-03) Symbol of quantized Pythagorean harmony.
Him who brought us the tetractys, the Source of everflowing Nature.
The Pythagorean musical system was based on the harmonyof the simple ratios 4:3, 3:2 and 2:1. Many detailed explanations have been devised about the many meanings of the tetractys symbol. Most such details are dubious. The tetractys is essentially a symbol for the counting numbers themselves (1, 2, 3, 4...). This sign evokes the Pythagorean belief systemwhich puts small whole numbers at the core of every fundamental explanation.
(2003-06-10) What are Borromean rings?
These are 3 interwoven rings which are pairwise separated (see picture). 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 converseseems to be true: Three simple unknotted closed curves may always be placedin a Borromean configuration unless they're all circles(no other counterexamples are known at this time).
The design was once the symbol of the alliance between the Visconti, Sforza and Borromeo families. It's been named after the Borromeo family who has perused the three-ring symbol, withseveral other interlacing patterns! The three rings are found among the many symbols featured on theBorromeo coat of arms(they're not nearly as prominent as one would expect).
The Borromean interlacing is also featured in symbols which don't involve rings. One example, pictured at left, is [one of the two versions of] Odin's triangle.
In North America, a Borromean pattern is sometimes called a ballantine because of the3-ring logo(Purity, Body, Flavor) ofBallantine's Ale which was popular in the WWII era. The term Ballantine rings is used by Louis H. Kauffman in his book Formal Knot Theory (Princeton University Press, 1983).
Borromean rings are but the simplest example of Brunnian links.
At the 25th International Congress of Mathematicians in Madrid, Spain (August 2006) the International MathematicalUnion (IMU) adopted for itself the logoat left, which shows three congruent Borromean rings.
(2003-06-23) Niels Bohr's coat-of-arms (Argent, a taiji Gules and Sable) illustrateshis motto: Contraria sunt complementa.
The Chinese Taiji symbol (Tai-Chi, ortaijitu)predates the Song dynasty (960-1279). Known in the West as theYin-Yang symbol,this sign appears in the ancientI Ching (orYiJing,the "Book of Changes"). It is meant to depict the two traditional types of complementaryprinciplesfrom which all things are supposed to come from, Yin and Yang,whirling within an eternally turning circle representing the primordial void(theTao). TheConfucianTai-Chi symbol represents actual plenitude,whereas the Taoist Wu-Chi symbol (an empty circle) symbolizes undifferentiated emptiness,but also the infinite potential of the primordial Tao, as the journey begins...
Act on it before it begins. Handle it before it becomes chaotic. [...] A journey of a thousand miles begins with a single step.
Both Yin and Yang are divided into greater and lesser phases (or elements). A fifthcentral phase (earth)represents perfect transformation equilibrium.
To a Western scientific mind, this traditional Chinese classification may seem entirelyarbitrary, especially the more recent "scientific" extensions to physics and chemistryin the following table:
Extensive quantities Volume, Entropy Charge Magnetic induction
Intensive quantities Pressure, Temperature Voltage Electric field
Yin
Yang
The traditional Chinese taiji symbol became a scientificiconwhen Niels Bohr made it his coat-of-arms in 1947(with the motto:contraria sunt complementa)but the symbol was never meant to convey any precise scientific meaning...
The oldest known Tai-Chi symbolwas carved in the stone of a Korean Buddhist temple in AD 682. A stylized version of the Yin-Yang symbol(Eum-Yang to Koreans) appears on the modern [South] Korean Flag(T'aeGuk-Ki)which was first used in 1882, by the diplomat Young-Hyo Park on a mission to Japan. The flag was banned during the Japanese occupation of Korea, from 1910 to 1945.
The decorative use of similar graphics is found much earlier,on the shields of several Roman military units recorded in the Notitia Dignitatum (c. AD 420). This includes, most strikingly, the pattern shown at right, which wassported by an infantry unit called armigeri defensores seniores (the shield-bearing veteran defenders).
(2012-08-11) (Bourbaki, Knuth) Announces a delicate point, possibly difficult or counterintuitive.
Certains passages sont destinés à prémunir le lecteur contre des erreurs graves, où il risquerait de tomber; ces passages sont signalés en marge par le signe ("tournant dangereux "). Nicolas Bourbaki (1935 - )
A single warning sign may also indicate ahazardous discussion of minute details, to skip on first reading.
A triple sign warns against possible crackpottery.
The design of the caution sign introducedby Bourbaki was inspired by the French roadsigns (at right) which were installed before 1949.
Now, those roadsigns have been replaced bythe international roadsigns below,which communicate much better to the driver which way the upcoming "dangerous bend" turns! Indeed.
International roadsigns of triangular shape signal a danger. Donald Knuth decided that a diamond shape would be more appropriate forthe mere mathematical caution sign he would use in his own books.
Unlike the unframed rendition of the caution sign (U+2621) which looks like a capital Z tothe uninitiated,D.E. Knuth's glyph (at right) really suggests a roadsign!
). Quantum of spin.
Emily Guerin (2004-06-18; e-mail) 
), while some others used a pair ofvertical lines ( || ) instead.
(Monica of Glassboro, NJ.
(2003-08-08) 
(2003-11-10) 
(2005-04-10) Cup: Wedge: Vee:
(2007-11-12) 
B for the disjoint union of two sets A and B. The squared "U" symbol (
, which some authors use for the 
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(2002-07-05) [
(2003-06-10) 
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The
![Modern [South] Korean Flag](/image.pl?url=http%3a%2f%2fwww.numericana.com%2fanswer%2ftaeguk.gif&f=jpg&w=240)
