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# Scientific Symbols and Icons

Symbols are more than just cultural artefacts:
[They address] our intellect, emotions, and spirit.
David Fontana (The Secret Language of Symbols, 1993)
 Are you interested in math lessons? Math can be a difficult subject but the proper curriculum can make math just as easy as any other subject.

### Related Links (Outside this Site)

Scientific Symbol Resources at symbols.net
Graphic Symbol Index at symbols.com
History of Mathematical Symbols by Douglas Weaver and Anthony D. Smith
Flags with Mathematical Symbols  |  Table of Mathematical Symbols
Symbols in the  Mathematical Association's  coat of arms.

## Mathematical Symbols and Scientific Icons

Emily Guerin (2004-06-18; e-mail)   The Equality Symbol
Who was the first person to use the = (equal) sign?

A very elongated form of the modern equality symbol (=) was first introduced in print in The Whetstone of Witte (1557) by Robert Recorde (1510-1558) the man who first introduced algebra into England.  He justified the symbol by stating that no two things can be more equal than a pair of parallel lines...

We've been told that a manuscript from the University of Bologna, dated between 1550 and 1568, features the same equality symbol, apparently independently of the work of Robert Recorde (and possibly slightly earlier).

William Oughtred (1574-1660) was instrumental in the subsequent popularization of the symbol, which appears next in 1618, in the appendix [attributed to him] of the English translation by Edward Wright of John Napier's Descriptio (where early logarithms were first described in 1614).  The mathematical symbol is then seen again, and perhaps more importantly, in Oughtred's masterpiece Clavis Mathematicae (1631) in which other mathematical symbols are experimented with, which are still with us today  (including ´ for multiplication).

Instead of the now familiar equality symbol, many mathematicians of that era used words or abbreviations (including "ae" for the Latin aequalis) well into the 18th century.  Thomas Harriot (1560-1621) was using a slightly different symbol (), while some others used a pair of vertical lines ( || ) instead.

Earliest Uses of Symbols of Relation   (Jeff Miller)

(Monica of Glassboro, NJ. 2001-02-08)
What's the correct terminology for the line between
the numerator and denominator of a fraction?

When the numerator is written directly above the denominator, the horizontal line between them is called [either a bar or] a vinculum

The overbar part of a square-root sign or a guzinta is also called a vinculum,  so is the full weight superbar or overscore used to tie several symbols together  (in particular, groups of letters with a numerical meaning in Greek or Latin, where such explicit groupings  may  also imply multiplication by 1000).  The thinner diacritical mark placed over a single symbol  (e.g., over long vowels in some modern Latin transcriptions)  is called a  macron.

When the numerator and denominator appear at the same level, separated by a slanted line (e.g., "1/2") such a line is called a solidus (also virgule or slash).  The noun  solidus  originates from the Roman gold coin of the same name (the ancestor of the shilling, of the French sol or sou, etc.). The symbol was originally a monetary symbol, which was still being used for the British shilling in 1971  (when British money was decimalized).

The related symbol "¸" is called an obelus. Today, this symbol is virtually unused to separate both parts of a ratio, but it remains very familiar as the icon which identifies the division key on electronic calculators...

(2003-08-08)   ¥
The infinity symbol introduced by John Wallis in 1655.

This symbol was first given its current mathematical meaning in "Arithmetica Infinitorum" (1655) by the British mathematician John Wallis (1616-1703).

(resp. )  is simply the mathematical symbol used to denote the "limit" of a real quantity which eventually remains above (resp. below) any preset bound.

Incidentally, he above illustrates the proper mathematical usage of "resp." (which is best construed as a mathematical symbol, as discussed elsewhere on this site).  This remark was prompted by an entry (2006-11-18) in the blog of a professional translator  (Margaret Marks)  who used this very prose as an example of a usage she was discovering with the help of her readers...

In canonical maps between the complex plane and a sphere minus a point, the unsigned symbol (¥) corresponds to the "missing point" of the sphere,  but  ¥  is not a proper complex number...  It's just a convenient symbol for the fictitious "circle at infinity" beyond the horizon of the complex plane, so to speak.

The symbol itself is properly called a lemniscus, 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 the  Lemniscate of Bernoulli.

The design was a part of Western iconography well before modern times.  In particular, it's found on the cross of Saint Boniface (bishop and martyr, English apostle of Germany, né Winfrid c.675-755).

The infinity snake, the ouroboros symbol (also, uroboros or  uroborus) is a serpent or a dragon biting its own tail  (ourobóroV means "tail swallower").  The symbol appeared in Egypt as early as 1600 BC, and independently in 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 elsewhere depicted in the shape of a lemniscate, although a plain circle is more common (the circle symbolizes infinity in Zen Buddhism).

The Lemniscate or Infinity Symbol

(2003-11-10)   Symbols of Infinite Numbers
w and Ào, the other infinity symbols.

As discussed above, the infinity symbol of Wallis (¥) is not a number...

However, there are two different approaches that make sense of actual infinite numbers.  Both of them were first investigated by Georg Cantor (1845-1918).

Two sets are said to have the same cardinal number of elements if they can be put in one-to-one correspondence with each other.  For finite sets, the natural integers (0,1,2,3,4 ...) are adequate cardinal numbers, but transfinite cardinals are needed for infinite sets.  The symbol Ào (pronounced "aleph zero", "aleph null", or "aleph nought") was introduced by Cantor to denote the smallest of these (the cardinal of the set of the integers themselves).  He knew that more than one transfinite cardinal was needed because his famous diagonal argument proves that reals and integers have different cardinality.

The second kind of infinite numbers introduced by Cantor are called transfinite ordinals.  Observe that a natural integer may be represented by the set of all nonnegative integers before it, starting with the empty set ( Æ ) for 0 (zero) because there are no 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 of all the nonnegative integers {0,1,2,3...} the w infinity symbol was introduced.

Cantor did not stop there, since {0,1,2,3...w} corresponds to another transfinite ordinal, which is best "called"  w+1.  {0,1,2,3...w,w+1} is w+2, etc.  Thus, w is much more like an ordinary number than Ào.  In fact, within the context of surreal numbers described by John H. Conway around 1972, most of the usual rules of arithmetic apply to expressions involving w (whereas Cantor's scheme for adding transfinite ordinals is not even commutative).  Note that 1/w is another nonzero surreal number, an infinitesimal one.  By contrast, adding one element to an infinity of Ào elements still yields just Ào elements, and 1/Ào is meaningless.

Infinite Ordinals and Transfinite Cardinals   |   The Surreal Numbers of John H. Conway

(2005-04-10)   Cap: Ç   Cup: È   Wedge: Ù   Vee: Ú
Intersection (greatest below) & Union (lowest above).

The chevron (wedge) and inverted chevron (vee) are the generic symbols used to denote the basic binary operators induced by a partial ordering on a  lattice.

• The chevron symbol (wedge) denotes the highest element "less" than (or equal to) both operands.   aÙb = inf(a,b)   is called the  greatest lower bound,  the  infimum  or  meet  of a and b.  The operation is well-defined only in what's called a  meet semilattice,  a partially ordered set where two elements always have 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.   aÚb = sup(a,b)   is called the  least upper bound,  the  supremum  or  join  of a and b.  The operation is well-defined only in what's called a  join semilattice,  a partially ordered set where two elements always have 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:

 pÙq   =   min(p,q) Î  {p,q} pÚq   =   max(p,q) Î  {p,q}

On the other hand, consider the relation mong  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's reflexive, antisymmetric and transitive)  but it's only a partial ordering relation  (for example, 2 and 3 can't be "compared" to each other,  as  neither divides the other).  In that context,  pÙq is the  greatest common divisor  (GCD)  of p and q, more rarely called their  highest common factor  (HCF).  Conversely, pÚq is their  lowest common multiple  (LCM).

 pÙq   =   gcd(p,q)     [ = (p,q) ]   (*) pÚq   =   lcm(p,q)

(*) We do not recommend the widespread but dubious notation  (p,q)  for the GCD of p and q.  It's unfortunately dominant in English texts.

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 meaning they have in predicate calculus (the chevron symbol stands for "logical and", whereas the inverted chevron is "logical or", also called "and/or").

In Set Theory, the fundamental ordering relation among sets may be called "is included in"  (Ì or, more precisely, Í).  In this case, and in this case only, the corresponding symbols for the related binary operators assume rounded shapes and cute names:  cap (Ç) and cup (È).  AÇB and AÈB are respectively called the intersection and the union of the sets A and B.

The intersection AÇB is the set of all elements that belong to both A and B.  The union AÈB is the set of all elements that belong to A and/or B ("and/or" means "either or both"; it is the explicitly inclusive version of the more ambiguous "or" conjunction, which normally does mean "and/or" in any mathematical context).

The chevron symbol is also used for the  exterior product  (the wedge product).

In an international context, the same mathematical symbol may be found to denote the vectorial cross product as well...

(2007-11-12)   Disjoint Union  =  Discriminated Union
Union of distinct copies of sets in an  indexed  family.

Formally, the  disjoint union  of an indexed family of sets  Ai  is:

 Ai   = È { (x,i)  |  x Î Ai } iÎI iÎI

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 the usual 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 symbol  (because the  plus symbol  is so overloaded).  In handwriting and in print, that "squared U" symbol is best drawn as an "inverted pi", to avoid any possible confusion with the "rounded U" symbol  (cup symbol)  denoting an  ordinary  union of sets.

What makes additive symbols [somewhat] popular to denote  discriminated unions  is the fact that the cardinal of a discriminated union is  always  the sum of the cardinals of its components.  Denoting  |E|  the cardinal of the set  E,  we have:

å A  |   =   å  | A |

(2005-09-26)   "Blackboard Bold" or Doublestruck Symbols
Letters enhanced with double lines are symbols for sets of numbers.

Such symbols are attributed to Nicolas 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).

Some Doublestruck Symbols and their Meanings
DoublestruckBold EtymologySymbol's Meaning
PPrime Numbers2, 3, 5, 7, 11, 13, 17, 19, 23 ...
NNatural NumbersMonoid of Natural Integers
ZZahl [German]Ring of Signed Integers
Q QuotientField of Rational Numbers
R RealField of Real Numbers
C ComplexField of Complex Numbers
H HamiltonSkew Field of Quaternions
O OctonionsAlternative  Division Algebra

One advantage of using those doublestruck letters even in print  (against the advice of Jean-Pierre Serre)  is that they do not suffer from any overloading, which makes it unnecessary to build up a  context  when using them.

 ln(x)   = óõ x  1 dt t
(2003-08-03)   ò
The integration symbol introduced by Leibniz.

Gottfried Wilhelm Leibniz thought of integration 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 with Johann Bernoulli, 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...

(2002-07-05)   Q.E.D.   [ QED = Quod Erat Demonstrandum ]
What's the name of the end-of-proof box, in a mathematical context?

Mathematicians call it a halmos symbol, after Paul R. Halmos (1916-2006) who is also credited with the "iff" abbreviation for "if and only if".  Typographers call it a  tombstone, which is the name of the symbol in any non-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 a tombstone is especially useful for an article which spans 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 logo in lieu of a plain tombstone symbol.

See Math Words...  Here's a halmos 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 actually pronounced "del".  Which is it?  I thought "del" was reserved for [Hamilton's nabla operator]   Ñ  =  < ¶/¶x, ¶/¶y, ¶/¶z > ...

"Del" is indeed a correct name for both    and  Ñ.  Some authors present these two as the lowercase and uppercase versions of the same mathematical symbol, although the terms "small del" and "big del" [sic!] are rarely used, if ever...

Physicists and others often pronounce y/x "del y by del x".  In an international scientific context, the possible confusion between    and Ñ  is probably best avoided by calling the  Ñ  mathematical symbol "nabla del", or simply  nabla.

William Robertson Smith (1846-1894) coined the name "nabla" for the Ñ mathematical symbol, whose shape is reminiscent of a Hebrew harp by the same name (also spelled "nebel").  The term was first adopted by Peter Guthrie Tait (1831-1901) by Hamilton and also by Heaviside.  Maxwell apparently never used the name in a scientific context.

The question is moot for many mathematicians, who routinely read a    symbol like 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".  It corresponds to the cursive "dey" of the Cyrillic alphabet and is sometimes also known as Jacobi's delta, because Carl Gustav Jacobi (1804-1851) is credited with the popularization of the symbol's modern mathematical meaning (starting in 1841, with the introduction of  Jacobians  in the epoch-making paper entitled "De determinantibus functionalibus").  Historically, this mathematical symbol was first used by Condorcet in 1770, and by Legendre around 1786.

Geetar  (2007-07-18)   The Staff of Aesculapius
What's the symbol for the 13th zodiacal constellation, Ophiuchus?

Ophiuchus  is the name  (abbreviated Oph)  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 arts by 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 that he could bring people back from the dead.  This made the underworld ruler (Hades) complain to Zeus, who struck Asclepius with a bolt of lightning but decided to honor him with a place in the sky, as Ophiuchus.  The Greeks identified Asclepius with the deified Egyptian doctor  Imhotep  (27th century BC).

The Rod of Asclepius, symbol of medicine, is a single 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 was traditionally extracted through the patient's skin by 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 the ouroboros symbol  (a snake feeding on its own tail).  A snake around a walking stick is also an ancient symbol of supernatural powers which can triumph over death, like medicine can  (biblically, the symbol of Moses' divine mission was his ability to change his walking stick into a snake).

The large  Ophiuchus  constellation is one of the 88 modern constellations.  It was also one of the 48 traditional constellations listed by Ptolemy.  In both systems, it's one of only 13 zodiacal constellations.  By definition, a zodiacal constellation is a constellation which is crossed by the ecliptic  (the path traced by the Sun on the celestial sphere, which is so named because that's where solar eclipses occur).

As a path charted against the background of fixed stars, the ecliptic is a remarkably stable line (since it's tied to the orbital motion of the Earth, not its wobbling spin).  It does not vary with the relatively rapid precession of equinoxes  (whose period is roughly 25772 years).  What does vary is the location on the ecliptic of the so-called "gamma point"  (the position of the Sun at the vernal equinox).

Ophiuchus is the only zodiacal constellation which has  not  given its name to one of the 12  signs of the zodiac associated with the 12 traditional equal subdivisions of the solar year, which form the calendar used by astrologers.  However, some modern astrologers are advocating a reformed system with uneven 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 Ophiuchus with the caduceus symbol  (two snakes around a winged staff)  since that symbol of Hermes  (messenger of the gods)  is associated with  commerce,  not  medicine.

Ophiuchus is indeed properly associated with the  Staff of Asclepius  symbol  (one snake around a plain stick)  the correct symbol of the  medical profession,  which is mythologically tied to the Ophiuchus constellation.

In 1910, the House of Delegates of the American Medical Association issued a resolution stating that  "the true ancestral symbol of healing art is the knotty pine and the [single] serpent of Aesculapius".

Rod_of_Asclepius   |   Modernized Zodiac   |   Astrological Attributes of Ophiuchus  by  Betty Rhodes

(2007-11-25)   The Caduceus   (Scepter of Hermes)
Image of dynamic equilibrium.  Symbol of  commerce.

Several explanations exist for this ancient overloaded symbol.

In Greek mythology, the  Caduceus  symbol inherited by Hermes  (called  Mercury  by the Romans)  is often said to have originated with the  blind seer  Tiresias, the prophet who experienced both sexes.

Tiresias was a son of Zeus and the nymph Calypso  (daughter of the titan Atlas).  After he had separated two copulating serpents with a stick, Tiresias was changed into a woman for 7 years by Hera, experiencing marriage and childbirth before returning to his original male form.  This experience of both sexes uniquely qualified him to settle a dispute between Zeus (Jupiter) and his wife Hera (Juno).  He sided with Zeus by stating that women experience ten times more sexual pleasure than men.  This displeased Hera who made him blind  (in another version, it's Athena who blinded him, because he had surprised her bathing in the nude).  Zeus tried to make up for this by giving him foresight and allowing Tiresias to live 7 lives.

The caduceus symbol evokes a  dynamic equilibrium  emerging from a confrontation of opposing principles  (male and female).  As an alleged symbol of peace, it represents a balance of powers rather than a lack of tensions.

The oldest depiction of two snakes entwined around an axial rod is in the Louvre museum.  It appears on a steatite vase carved for  Gudea of Lagash  (who ruled from around 2144 to 2124 BC)  and dedicated to the Mesopotamian underworld deity Nin-giz-zida who is so represented.  The name means "Lord of the Good Tree" in Sumerian, which is reminiscent of Zoroastrian righteousness  (Good and Evil)  and of the biblical Tree of Knowledge of Good and Evil, also featuring a serpent...

Curiously, the gender of Nin-giz-zida seems as ambiguous as the sexual identity of Tiresias.  Coincidentally or not, Nin-giz-zida is associated with the large vonstellation  Hydra  whose name happens to evoke  Hydrargyrum, the latin name of the metal  mercury  (symbol Hg).  The Hydra constellation is either associated to the Hydra of Lerna  (the multi-headed reptilian monster defeated by Heracles)  or, interestingly, to the serpent cast into the heavens by Apollo  (who ended up giving the caduceus emblem to his brother Hermes/Mercury).

The two facing serpents have also been said to be symbols for water and fire, two opposing elements entwined around the axis of the Earth.  The wings evoke the spiritual or spatial dimension linked to the  fourth element :  sky, wind or air.

Also, the copulating serpents have been construed as a fertility symbol involving two complementary forces revolving around a common center.  This makes the caduceus a western counterpart of the oriental  Yin-Yang symbol.

Hermes was the god of  alchemists, for which this unification of opposites was a fundamental credo  (in which the element  mercury  held center stage).  By extension, the  caduceus  has been associated with chemistry and pharmacy.

It's a common  mistake, dating back to the 16th century,  to associate the Caduceus with medicine.  The misguided heraldic use of the symbol by military medicine started in the 19th century and culminated with the unfortunate adoption of the symbol by the Medical Department of the US Army, in 1902.  It's still the official emblem of the US Navy Hospital Corps. Yet, the correct symbol for medicine is definitely the Staff of Asclepius  (no wings and a single serpent)  so recognized as a "true ancestral symbol" by the American Medical Association (AMA) in 1910.

The caduceus is also associated with communication, eloquence, trade and  commerce,  the traditional attributions of Hermes, messenger for the gods and protector of all merchants, thieves, journalists, tricksters and... inventors.

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 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 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, with several other interlacing patterns!  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).

The Borromean interlacing is also featured in other symbols which do not involve rings.  One example, pictured at left, is [one of the two versions of] the so-called Odin's triangle.

In a recent issue of the journal  Science  (May 28, 2004)  a group of chemists at UCLA reported the synthesis of a molecule with the Borromean topology.

At a more fundamental level, the logic of the Borromean symbol applies to a type of quantum entanglement first conjectured by  Vitaly Efimov  in 1970, where ternary stability may exist in spite of pairwise repulsion.  Such an  Efimov state  was first observed  (for three cesium atoms confined below 0.000000001 K)  by the group of  Rudolf Grimm  at the University of Innsbruck  (Austria)  in collaboration with  Cheng Chin  of Chicago  (NatureMarch 16, 2006).

(2003-06-23) The tai-chi mandala:  Taiji or Yin-Yang symbol.
Niels Bohr's coat-of-arms  (Argent, a taiji Gules and Sable)  illustrates his motto:  Contraria sunt complementa.

The Chinese Taiji symbol (Tai-Chi, or taijitu) predates the Song dynasty (960-1279).  Known in the West as the Yin-Yang symbol, it appears in the ancient I Ching (or YiJing, the "Book of Changes").  It is meant to depict the two traditional types of complementary principles from which all things are supposed to come from, Yin and Yang, whirling within an eternally turning circle representing the primordial void (the Tao).  The Confucian Tai-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...

Yin and Yang are each divided into greater and lesser "phases" (or "elements").  A fifth central phase (earth) represents a perfect transformation equilibrium.

To a Western scientific mind, this traditional Chinese classification may seem entirely arbitrary, especially the more recent "scientific" extensions to physics and chemistry, which are highlighted in the following table:

Yin Yang Etymology Dark Side  (French: ubac) Bright Side  (French: adret) Female, Feminine Male, Masculine Moon, Planet, Night Sun, Star, Day White Tiger Green Dragon Violet, Indigo, Blue Red, Orange, Yellow Greater PhaseEquinoxTransition, Young West, Metal and AutumnPotential Structure East, Wood and SpringPotential Action Weak Nuclear Force Gravity Lesser PhaseSolsticeStability, Old North, Water and WinterActual Structure South, Fire and SummerActual Action Strong Nuclear Force Electromagnetism Dark, Cold, WetSolid, Heavy, SlowCurling, DeepSoft voice, SadYielding, Soft, RelaxedStillness, PassivityComing, Inward, PullReceive, Grasp, ListenDescending, Low, BottomContracting, PreservingSmall, Interior, BoneMental, SubtleBuy Bright, Hot, DryGas, Light, FastStretching, ShallowLoud voice, HappyResistant, Hard, TenseMotion, ActivityGoing, Outward, PushTransmit, Release, TalkAscending, High, TopExpanding, ConsumingLarge, Exterior, SkinPhysical, ObviousSell Sweet, Bitter, MildVegetable, RootRed meat Salty, Sour, HotFruit, LeafSeafood Space, Open angleFinite, Discontinuous Time, Closed circleInfinite, Continuous Cause Effect Dexter, Negative, LossFront, Counterclockwise Sinister, Positive, GainBack, Clockwise 0, Zero, Even, No 1, One, Odd, Yes Acidic, Cation, Oxidant Alkaline, Anion, Reductant Pyrimidines: Cytosine, Thymine Purines: Guanine, Adenine Matter, Particle, Fermion Energy, Force, Boson

The traditional Chinese taiji symbol became a scientific icon when 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 symbol was carved in the stone of a Korean Buddhist temple in AD 682.  A stylized version of the Ying-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.