SCIENCE SYMBULS

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Final Answers

2000-2012Grard P. Michon, Ph.D.

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)

Adobe's Symbol font: Endangered standard HTML mathematical symbols. The equality symbol( = ). The "equal sign" dates back to the 16th century. The double-harpoon symbol denotes chemical equilibrium. Line components: Vinculum, bar, solidus, virgule, slash, macron, etc.

The infinity symbol( ) introduced in 1655 by John Wallis (1616-1703). Transfinite numbers and the many faces of mathematical infinity. Chrevron symbols: Intersection (highest below) or union (lowest above).

Disjoint union. Square "U" or inverted symbol. Blackboard bold:Doublestruck characters denote sets of numbers.

The integration sign( ) introduced by Leibniz at the dawn of Calculus. The end-of-proof box(or tombstone) is called a halmos symbol (QED).

Two "del" symbols: for partial derivatives, and for Hamilton's nabla. The rod of Asclepius: Medicine and the 13th zodiacal constellation. The Caduceus: Scepter of Hermes, symbol of commerce (not medicine). Tetractys: Mystical Pythagorean symbol, "source of everflowing Nature". The Borromean Rings: Three interwoven rings which are pairwise

separate.

The Tai-Chi Mandala: The taiji (Yin-Yang) symbol was Bohr's coat-of-arms.

Dangerous-bend symbol: Introduced by Bourbaki, popularized by Knuth.
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Related articles on this site:

Style and Usage. Free monoid: Strings of characters.

Scientific notation. Scientific calculators. Chemical elements. The periodic table.

("Pi"): Circumference to diameter ratio. Names of operandsin common operations. The h-bar symbol ( ). Quantum of spin. Legendre symbol and its generalizations. The diamondon US tape measures. Roman numeralsand Roman numeration system(s). Etruscan numerals. Notation for sesterce: HS or IIS (2 asses, "unus et unus et semis"). Alchemical Glyphs: Gold, acidum salis, aqua fortis, aqua regia, etc.

Escutcheons of Science(Armorial): Coats-of-arms of famous scientists. Tree of Knowledge(or Conscience): An overloaded heraldic design.

Related Links (Outside this Site)

Scientific Symbol Resourcesatsymbols.net, byGeorge F. Sutton(b. 1956).

Graphic Symbol Indexatsymbols.com

Mathematical Symbols byRobin Whitty (Theorem of the Day).

History of Mathematical Symbolsby Douglas Weaver and Anthony D. Smith

Flags with Mathematical Symbols | Table of Mathematical Symbols

Symbolsin the Mathematical Association's coat of arms.

Sixty Symbols: Videos about the symbols of physics and astronomy.

Greek letters used in mathematics, science, and engineering (Wikipedia)

Mathematical Symbols and Scientific Icons

(2011-08-25) The Symbol Font of Yesteryear's Web (HTML 4.0)

Once the proper way to display most mathematical symbols on the Web.

The World Wide Web was originally developed at CERN to facilitate

International scientific communications. In the early days, only the 7-bit

characters in theASCIIset were 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 to

control either the flow of information or the output device (the most

common of those are the end-of-line indicators; carriage-return and/or line-
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feed).

As of June 2011, IT professionals in at least one big organisation are testing

critical 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 with proper UNICODE support,

for example. I argue that browsers that do not support legacy standards to

insure the readibility of yesteryear's valuable information simply do not

deserve 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 of Opera, Safari and Firefox (not a single

Web author who has ever used the Symbol has ever meant it to be

rendered the way those browsers do, by mistakingly using a "standard"

character encoding for it).

Thanks to Philippe Verdy for background information (private messages).

W3C recommendationsfor mathematical symbols in HTML 4.0 (December 1997)Symbol font(Wikipedia) | ASPpdef(Special characters tables)

Special Characters by Ted M. Montgomery (1998-2011)Using Symbol font to display Greek letters with Firefox or Netscape 6+ (The Modern Jesus Army)

Enabling Symbol font for Mozilla on Windows TeX to HTML Translator (2005)Getting Symbol Font to Display in Firefox by Dave (2006)

Symbol fontand nonstandard symbolic typefaces by Ian C. G. Bell (1998-2006)Symbol font by David W. Knight, G3YNH (Nov. 2009, Sept. 2011)

Symbol font - Unicode alternatives for Greek and special characters in HTML by Alan Wood (1997-2010)Google I/O 2011: HTML5 & What's Next (2011 Video).

Emily Guerin (2004-06-18; e-mail) The Equal Sign

Who was the first person to use the modern equal sign?

A very elongated form of the modern equality symbol (=) was

introduced in print inThe Whetstone of Witte(1557) byRobert Recorde

(1510-1558) the man who first introduced algebra into England. He

justified the new symbol by stating that no two things can be more equal

than a pair of parallel lines.
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We'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 ofRobert Recorde (and possibly slightly earlier).

William Oughtred(1574-1660) was instrumental in the subsequentpopularization of the equal sign, which appears nextin 1618, in the

appendix [attributedto him] of the English translation by Edward Wright

of John Napier'sDescriptio(whereearly logarithmswere first described in

1614). The same mathematical glyph is then seen again, and perhaps more

importantly, in Oughtred's masterpiece Clavis Mathematicae (1631) in

which otherscientific symbols are experimented with, which are still with

us today (including for multiplication).

Instead of the now familiar equal sign, many mathematicians used words or

abbreviations (including "ae" for the Latin aequalis) well into the 18th

century. Thomas Harriot(1560-1621) was using a different symbol (

), while some others used a pair ofvertical lines ( || ) instead.

Earliest Uses of Symbols of Relation (Jeff Miller)

(2010-05-05) Chemical reaction & chemical equilibrium

Equilibrium can be denoted by a right over left double-harpoon.

Somechemical reactionsproceed until one of the reactant has 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 concentration

of the reactants, a dynamic equiibrium is often reached whereby the

concentrations of all the compounds involved are such that both directions

of the chemical reaction proceeed at equal rates. Several symbols have

been used to indicate this. The most symmetrical such symbol is the

double-headed arrow sign ( )

However, the preferred scientific symbol for chemical equilibrium consistsof two superposed arrows (the rightward arrow is always above the

leftward one) This has evolved graphically into the following stylish sign,

affectionately known as the double-harpoon symbol:

This is the so-called right-left version of the symbol (UNICODE:
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21CC). 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 bypharmacists and often wrongly associated withmedicine).

(Monica of Glassboro, NJ.2001-02-08)What's the correct terminology for the line betweenthe 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.

The overbarpart of a square-root sign or aguzintais 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 orLatin, where such explicit groupings may also imply

multiplication by 1000). The thinner diacritical mark placed over a single

character is 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 a

solidus. It's also called slash or stroke [British] and, more formally,virgule or oblique [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 coinof the same name

(the ancestor of the shilling, of the French sol or sou, etc.). The sign was

originally a monetary symbol, which was still used for the Britishshilling

in 1971 (when British money was decimalized). See discussion

below.

The related symbol "" is called an obelus. It was introduced as adivision symbol in 1659 by the Swiss mathematician Johann Rahn

(16221676) 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 the division key on

calculators...

On 2009-08-07,John Harmerwrote: [edited summary]

I was at Uni in 1971 and can't remember ever using "/" instead of "s" for

shillings. Beforeanother meaningcame along, the

acronym Lsd (or sd ) referred to the old British coinage
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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 solidus symbol only asa 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 1 / - / -

The symbol was pronounced stroke (oblique wasposh).

Cheers, John Harmer

Chippenham, UK

Both meanings of the solidus sign (i.e., currency prefix and/or

separator) are compatible and havecoexistedpeacefully. Arguably, the

definition presented by John Harmer became dominant with the passageof 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 areal quantity that 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 discussedelsewhere on this site). This remark was prompted by an entry (2006-11-18) in the blog of a professional translator (Margaret Marks) who usedthis very prose as an example of a usage she was discovering with thehelp of her readers...

Incanonical mapsbetween thecomplex planeand a sphere minus a point,

the unsigned symbol () corresponds to the "missing point" of the

sphere, but is not apropercomplex number... It's just a convenientway to denote the fictitious "infinite circle" at the horizon of the complex

plane.

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 Lemniscate of Bernoulli.

The design appeared in Western iconography before modern

times. It's found on the cross ofSaint Boniface(bishop and
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martyr, English apostle of Germany, n Winfridc.675-755).

The infinity snake, theouroborossymbol (also, uroboros

or uroborus) is a serpent or a dragon biting its own

tail ( means "tail swallower"). The graphicappeared 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

aplain circle is more common (the circle symbolizes

infinity in Zen Buddhism).

The Lemniscate or Infinity Symbol | Black Arts Diary

(2003-11-10) Symbols of InfiniteNumbers

and o, the otherinfinity symbols.

As discussedabove, the infinity symbol of Wallis () isnota number...

However, there are two different definitions that make good mathematical

sense of actual infinite numbers. Both of those were first investigated by

Georg Cantor(1845-1918):

Two sets are said to have the same cardinal numberof 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 infinity

symbol o (pronounced "aleph zero", "aleph null", or "aleph nought") wasdefined by Cantor to denote 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 differentcardinalities. (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 Cantor are called

transfinite ordinals. Observe that a natural integer may be represented by

the setof all nonnegative integers before it, starting with the empty set ( )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 ofall the nonnegative integers {0,1,2,3...} the
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infinity symbol was introduced.

Cantor did not stop there, since {0,1,2,3...} corresponds to anothertransfinite ordinal, which is best "called" +1. {0,1,2,3...+1} is +2,etc. Thus, is much more like an ordinary number than o. In fact, withinthe context ofsurreal numbersdescribed by John H. Conway around 1972,

most of the usual rules of arithmetic apply to expressions involving (whereas Cantor's scheme for adding transfinite ordinals is not even

commutative). Note that 1/ is another nonzero surreal number, aninfinitesimal one. By contrast, adding one element to an infinity ofoelements still yields just o elements, and 1/o is meaningless.

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

(2005-04-10) Cap: Cup: Wedge: Vee:

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. 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 calledthe greatest lower bound, the infimum or meet ofa and b. Theoperation 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. ab = sup(a,b) iscalled the least upper bound, the supremum or join ofa 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:

pq = min(p,q) {p,q}
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9/21

pq = max(p,q) {p,q}

For example, astochastic process Xt stopped at time T is equal to Xt;T

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's reflexive, antisymmetric and

transitive) 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, morerarely called their highest common factor (HCF). Conversely, pq istheir lowest common multiple (LCM).

pq = gcd(p,q) [ = (p,q) ] (*)

pq = lcm(p,q)

(*) We do notrecommend the widespread but dubiousnotation (p,q) for the GCD of p and q. It's unfortunately dominant in

English texts.

In the context ofNumber 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").

InSet Theory, the fundamental ordering relation among sets may be called

"is included in" ( or, more precisely, ). In this case, and in this caseonly, the corresponding symbols for the related binary operators assume

rounded shapes and cute names: cap () and cup (). AB and AB arerespectively called the intersection and the union of the sets A and B.

The intersection AB is the set of all elements that belong to both A andB. The union AB is the set of all elements that belong to A and/or B("and/or" means "either or both"; it is the explicitlyinclusiveversion of the

more ambiguous "or" conjunction, which normally does mean "and/or" inany 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

denote the vectorialcross productas well...
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(2007-11-12) Disjoint Union = Discriminated Union

Union of distinct copies of sets in an indexed family.

The concept of disjoint union coincides with the

ordinary union for sets that are pairwise disjoint. In modern usage, the

term disjoint union is almost always used to denote the ordinary union of

sets that are pairwise disjoint.

In that particular case it coincides with the concept of what's best called

a 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) | x Ai }iI iI

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 A B for the disjoint union of two sets A and B. Thesquared "U" symbol ( ) is the preferred (because the plus sign is so

overloaded). In handwriting and in print, that "squared U" is best drawn as

an "inverted pi", to avoid any possible confusion with the "rounded U"

symbol (cup) denoting an ordinary union of sets.

A symbol is said to be overloaded if its meaning depends on thecontext. Mathematical symbols are very often overloaded. Theoverloading of a symbol usually implies the overloading of relatedsymbols. For example, the overloading of the addition sign (+) implies

an overloading of the summation sign () and vice-versa.

Additive notations are [somewhat] popular for discriminated

unions because thecardinalof a discriminated union is always the sum of

the cardinals of its components. Denoting |E| the cardinal of the set E :

| A | = | A |

(2005-09-26) "Blackboard Bold" or Doublestruck Symbols

Letters enhanced with double lines are symbols forsets of 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
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traditional blackboard use).

Those symbols are also called "doublestruck" because mechanicaltypewriters could be coaxed into producing them by striking a capital letter

twice (pushing the carriage out of alignment the second time).

One advantage of using the doublestruck symbols, even in

print (against the advice of Jean-Pierre Serre) is that they do not suffer

from anyoverloading. This makes them usable without the need for

building up a context.

Some Doublestruck Symbols and their Meanings

Symbol Bold Etymology Symbol's Meaning

P Prime Numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

q Fq Finite Field Galois field of q = pn

elements (p prime)

N Natural NumbersNonnegative Integers [additivemonoid]

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

Z Zahl [German]Ringof Signed Integers (countable)

... -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...

/n Z/nZ Quotient Ring Ringof Integersmodulon (finite)

p Zp Zahl, radix p Ringofp-adic Integers(uncountable)

Q Quotient

Fieldof Rational Numbers (countable)p Qp Quotient, radix p Fieldofp-adic Numbers(uncountable)

R Real Field ofReal Numbers(uncountable)

C Complex Field ofComplex Numbers

H Hamilton Skew FieldofQuaternions

O Octonions Alternative Division Algebra

The group formed by theinvertible elementsof

a multiplicative monoid M is denoted M*. That's compatible with thecommon usage of starring the symbol of a set of numbers to denote

the nonzero numbers in it (the two definitions are equivalent for *,

*, * and *). In particular:

= { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ... } (Natural numbers,A001477)

* = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ... } (Counting numbers,A000027)

Unfortunately, this international usage is sometimes butchered in the US,where the locution "natural numbers" may mean positive integers.

* is undefined (arguably, that symbol might denote the odd primes ).
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(2003-08-03) The integration sign of Leibniz (29 October 1675).

Gottfried Wilhelm Leibniz(1646-1716) viewedintegrationas 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...

BBC, 1986 : The Birth Of Calculus [Jan Huddewas quoted by both Newton and Leibniz ]

(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, afterPaul R. Halmos(1916-

2006). Typographers call it a tombstone, which is the name of the symbol

in any non-scientific context.

Paul Halmos also invented the "iff" abbreviation for "if and only if".

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 ofa plain tombstone symbol.

SeeMath Words... Here's a halmos symbol, at the end of this last line!

To Euclid and the ancient Greeks, the end of a demonstration was indicated by the

acronym (transliterated "Oper Edei Deixai"). SeeRobin Whitty'sTheorem #149.

Jacob Krauze(2003-04-20; e-mail) Del & Nabla

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] = x, y, z

"Del" is a correct name for both and . Some authors present these two

Log x = x

1

dt

t
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signs as the lowercase and uppercase versions of the same mathematical

symbol (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". A better wayto read this aloud in front of a classroom is either "partial of y with respectto x" or "partial of y along x" (especially when x is a space orspacetime

coordinate).

In an international scientific context, the confusion between and isbest avoided by calling "nabla del", or simply nabla. Somepractitioners also read it "grad" (since nabla can be construed as denoting

a generalized gradient).

William Robertson Smith (1846-1894) coined the name "nabla" for the mathematical symbol, whose shape is reminiscent of a Hebrew harp by

the same name (also spelled "nebel"). The term was first adopted byPeter Guthrie Tait (1831-1901) by Hamilton and also byHeaviside. Maxwellapparently never used the name in a scientificcontext.

The question is moot for many mathematicians, who routinely read

a symbol like a "d" (mentally or aloud). I'm guilty of this myself, butdon'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 sign

corresponds to the cursive "dey" of the Cyrillic alphabet and is sometimesalso known asJacobi's delta, becauseCarl Gustav Jacobi(1804-1851) is

credited with the popularization of the modern

mathematical meaning of this special character (starting in

1841, with the introduction ofJacobians in the epoch-

making paper entitled "De determinantibus

functionalibus"). Historically, this lowercase

mathematical symbol was first used by Condorcet in 1770, and by

Legendrearound 1786.

Geetar (2007-07-18) Rod of Asclepius (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.
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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. Asclepiusserved 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 deifiedEgyptian

doctor Imhotep (27th century BC).

TheRod 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 theouroborossymbol (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 13zodiacal constellations. Bydefinition, 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 aremarkably stable line (since it's tied to the orbital motion of the Earth, notits wobbling spin). It does notvary with the relatively rapid precession ofequinoxes (whose period is roughly 25772 years). What does vary is thelocation on the ecliptic of the so-called "gamma point" (the position of theSun at the vernal equinox).

Ophiuchus is the only zodiacal constellation which has not given its name
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to one of the 12 signs of the zodiac associated with the 12 traditional equal

subdivisions of the solar year, which form thecalendar used by

astrologers. However, somemodern astrologersare 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 thecaduceussymbol (two snakes around a

wingedstaff) 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.

In 1910, the House of Delegates of the American Medical Associationissued a resolution stating that "the true ancestral symbol of healing art isthe 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 ancientoverloadedsymbol.

In Greek mythology, the kerykeion symbol (latin:

Caduceus) which was ultimately inherited by Hermes (called Mercury by

the Romans) is often said to have originated with the blind seer Tiresias,

the prophet who had experienced both sexes.

Tiresias was a son of Zeus and the nymph Calypso (daughter of the titanAtlas). 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. Thisexperience of both sexes uniquely qualified him to settle a disputebetween Zeus (Jupiter) and his wife Hera (Juno). He sided with Zeus bystating that women experience ten times more sexual pleasure thanmen. This displeased Hera who made him blind (in another version, it's

Athena who blinded him, because he had surprised her bathing in thenude). Zeus tried to make up for this by giving Tiresias foresight andallowing him 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, the kerykeion represents a balance of powers rather than

a lack of tensions.
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The oldest depiction of two snakes entwined around an axial rod is in theLouvre museum. It appears on asteatitevase carved for Gudea ofLagash (who ruled from around 2144 to 2124 BC) and dedicated to theMesopotamian underworld deityNin-giz-zidawho is so represented. Thename means "Lord of the Good Tree" in Sumerian, which is reminiscent of

Zoroastrian righteousness (Good and Evil) and of the biblicalTree ofKnowledge of Good and Evil, also featuring a serpent...

Curiously, the gender of Nin-giz-zida seems as ambiguous as the sexualidentity ofTiresias. Coincidentally or not, Nin-giz-zida is associated withthe large vonstellation Hydra whose name happens toevoke Hydrargyrum, the latin name of the metal mercury (symbolHg). The Hydra constellation is either associated to the Hydra ofLerna (the multi-headed reptilian monster defeated by Heracles) or,interestingly, to the serpent cast into the heavens by Apollo (who endedup giving the caduceus emblem to his brother Hermes/Mercury).

The two facing serpents are also said to symbolize water and fire, twoopposing 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 two complementary forces revolving around a common

center. This makes the caduceus a western counterpart of the oriental taiji.

Hermes was the god of alchemists, who were fascinated by the

element mercury and held as fundamental the unification of opposites. By

extension, the caduceus became associated with chemistry and pharmacy.

It's a common mistake, dating back to the 16th century, to associate theCaduceus with medicine. The misguided heraldic use of the symbol bymilitary medicine started in the 19th century and culminated with theadoption of the symbol by the Medical Department of the US Army, in1902. It's still the official emblem of theUS Navy Hospital Corps. Yet, thecorrect symbol for medicine is definitely theStaff of Asclepius (no wingsand a singleserpent) 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.

(2008-05-03) The Pythagorean TetractysSymbol of quantized Pythagorean harmony.

In theiroath, Pythagoreans calledPythagoras:

Him who brought us the tetractys,

the Source of everflowing Nature.
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The Pythagorean musical system was based on the harmony of 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.

Tetractys Symbol(Wikipedia) | The Tetraktys Symbol by Robert ApatowPythagorean Harmony in "Week 266" by John Baez.

(2003-06-10) Borromean Symbol. Borromean Links.

What are Borromean rings?

These are 3 interwoven rings which are pairwiseseparated (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, withseveral other interlacing

patterns! The three rings are found among the many symbols featured ontheBorromeo coat of arms(they are not nearly as prominent

as one would expect, you may need acloser 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-calledOdin'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 ofRudolf Grimm at

the University of Innsbruck (Austria) in collaboration with Cheng

Chin of Chicago (Nature, March 16, 2006).

In North America, the pattern is sometimes called a ballantine because of

the3-ring logo(Purity, Body, Flavor) ofBallantine's Ale which was
http://en.wikipedia.org/wiki/Tetractyshttp://en.wikipedia.org/wiki/Tetractyshttp://users.ucom.net/~vegan/tetraktys.htmhttp://users.ucom.net/~vegan/tetraktys.htmhttp://math.ucr.edu/home/baez/week266.htmlhttp://math.ucr.edu/home/baez/week266.htmlhttp://www.numericana.com/fame/index.htm#baezhttp://www.numericana.com/fame/index.htm#baezhttp://www.numericana.com/fame/index.htm#baezhttp://paradise.caltech.edu/~cook/Workshop/Math/Borromean/Borrring.htmlhttp://paradise.caltech.edu/~cook/Workshop/Math/Borromean/Borrring.htmlhttp://www.liv.ac.uk/~spmr02/rings/islands.htmlhttp://www.liv.ac.uk/~spmr02/rings/islands.htmlhttp://www.liv.ac.uk/~spmr02/rings/islands.htmlhttp://www.liv.ac.uk/~spmr02/rings/islands.htmlhttp://www.borromeoturismo.it/scripts/loc.php?lang=en&loc=bofamhttp://www.borromeoturismo.it/scripts/loc.php?lang=en&loc=bofamhttp://www.borromeoturismo.it/scripts/loc.php?lang=en&loc=bofamhttp://www.liv.ac.uk/~spmr02/rings/crest.htmlhttp://www.liv.ac.uk/~spmr02/rings/crest.htmlhttp://www.liv.ac.uk/~spmr02/rings/crest.htmlhttp://www.liv.ac.uk/~spmr02/rings/vikings.htmlhttp://www.liv.ac.uk/~spmr02/rings/vikings.htmlhttp://www.liv.ac.uk/~spmr02/rings/vikings.htmlhttp://www.physorg.com/preview135.htmlhttp://www.physorg.com/preview135.htmlhttp://www.physorg.com/preview135.htmlhttp://www.world-science.net/othernews/060316_newstatefrm.htmhttp://www.world-science.net/othernews/060316_newstatefrm.htmhttp://www.rustycans.com/COM/month0104.htmlhttp://www.rustycans.com/COM/month0104.htmlhttp://www.rustycans.com/COM/month0104.htmlhttp://en.wikipedia.org/wiki/Ballantine_Brewery#Logohttp://en.wikipedia.org/wiki/Ballantine_Brewery#Logohttp://en.wikipedia.org/wiki/Ballantine_Brewery#Logohttp://en.wikipedia.org/wiki/Ballantine_Brewery#Logohttp://www.rustycans.com/COM/month0104.htmlhttp://www.world-science.net/othernews/060316_newstatefrm.htmhttp://www.physorg.com/preview135.htmlhttp://www.liv.ac.uk/~spmr02/rings/vikings.htmlhttp://www.liv.ac.uk/~spmr02/rings/crest.htmlhttp://www.borromeoturismo.it/scripts/loc.php?lang=en&loc=bofamhttp://www.liv.ac.uk/~spmr02/rings/islands.htmlhttp://www.liv.ac.uk/~spmr02/rings/islands.htmlhttp://paradise.caltech.edu/~cook/Workshop/Math/Borromean/Borrring.htmlhttp://www.numericana.com/fame/index.htm#baezhttp://math.ucr.edu/home/baez/week266.htmlhttp://users.ucom.net/~vegan/tetraktys.htmhttp://en.wikipedia.org/wiki/Tetractys


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popular in the WWII era. The term Ballantine rings is used byLouis H.

Kauffmanin his book Formal Knot Theory (Princeton University Press,

1983).

Borromean rings are but the simplest example of Brunnian links.

Borromean and Brunnian and Brunnian Rings (Belgrade, 2010). Animation by Dusan Zivaljevic ("duleziv").

(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 theYin-Yang symbol,this sign appears in the ancientI 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 (theTao). 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...

Both Yin and Yang are divided into greater and

lesser phases (or elements). A fifth central phase(earth) represents 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 highlighted in the following table:

Yin Yang

Etymology Dark Side (French: ubac) Bright Side (French: adret)

GeographyNorth of a mountain

South of a river

South of a mountain

North of a river

Gender Female, Feminine Male, Masculine

Celestial Moon, Planet, Night Sun, Star, Day

Ancient Symbol White Tiger Green Dragon

Colors Violet, Indigo, Blue Red, Orange, Yellow

Greater Phase

Equinox

West, Metal and Autumn

Potential Structure

East, Wood and Spring

Potential Action
http://www.liv.ac.uk/~spmr02/rings/beer.htmlhttp://www.liv.ac.uk/~spmr02/rings/beer.htmlhttp://en.wikipedia.org/wiki/Louis_Kauffmanhttp://en.wikipedia.org/wiki/Louis_Kauffmanhttp://en.wikipedia.org/wiki/Louis_Kauffmanhttp://en.wikipedia.org/wiki/Louis_Kauffmanhttp://en.wikipedia.org/wiki/Brunnian_linkhttp://en.wikipedia.org/wiki/Brunnian_linkhttp://www.youtube.com/watch?v=rSdMvkY3NAchttp://www.youtube.com/watch?v=rSdMvkY3NAchttp://www.chinesefortunecalendar.com/yinyang.htmhttp://www.chinesefortunecalendar.com/yinyang.htmhttp://www.chinesefortunecalendar.com/yinyang.htmhttp://gbog.free.fr/Chine/wg/wengu.php?l=Yijinghttp://gbog.free.fr/Chine/wg/wengu.php?l=Yijinghttp://gbog.free.fr/Chine/wg/wengu.php?l=Yijinghttp://gbog.free.fr/Chine/wg/wengu.php?l=Yijinghttp://www.artic.edu/taoism/tradition/introb.phphttp://www.artic.edu/taoism/tradition/introb.phphttp://www.artic.edu/taoism/tradition/introb.phphttp://www.jungtao.edu/resources/ccm/ccm.htmlhttp://www.numericana.com/arms/bohr.htmhttp://www.jungtao.edu/resources/ccm/ccm.htmlhttp://www.numericana.com/arms/bohr.htmhttp://www.jungtao.edu/resources/ccm/ccm.htmlhttp://www.numericana.com/arms/bohr.htmhttp://www.jungtao.edu/resources/ccm/ccm.htmlhttp://www.numericana.com/arms/bohr.htmhttp://www.artic.edu/taoism/tradition/introb.phphttp://gbog.free.fr/Chine/wg/wengu.php?l=Yijinghttp://gbog.free.fr/Chine/wg/wengu.php?l=Yijinghttp://www.chinesefortunecalendar.com/yinyang.htmhttp://www.youtube.com/watch?v=rSdMvkY3NAchttp://en.wikipedia.org/wiki/Brunnian_linkhttp://en.wikipedia.org/wiki/Louis_Kauffmanhttp://en.wikipedia.org/wiki/Louis_Kauffmanhttp://www.liv.ac.uk/~spmr02/rings/beer.html


19/21

Transition, Young Weak Nuclear Force Gravity

Lesser PhaseSolstice

Stability, Old

North, Water and Winter

Actual Structure

South, Fire and Summer

Actual Action

Strong Nuclear Force Electromagnetism

GeneralFeatures

Dark, Cold, Wet

Solid, Heavy, Slow

Curling, Deep

Soft voice, Sad

Yielding, Soft, Relaxed

Stillness, Passivity

Coming, Inward, Pull

Receive, Grasp, Listen

Descending, Low, Bottom

Contracting, PreservingSmall, Interior, Bone

Mental, Subtle

Buy

Bright, Hot, Dry

Gas, Light, Fast

Stretching, Shallow

Loud voice, Happy

Resistant, Hard, Tense

Motion, Activity

Going, Outward, Push

Transmit, Release, Talk

Ascending, High, Top

Expanding, ConsumingLarge, Exterior, Skin

Physical, Obvious

Sell

Food

Sweet, Bitter, Mild

Vegetable, Root

Red meat

Salty, Sour, Hot

Fruit, Leaf

Seafood

GeometryTopology

Space, Open angle

Finite, Discontinuous

Time, Closed circle

Infinite, Continuous

Logic Cause Effect

OrientationDexter, Negative, Loss

Front, Counterclockwise

Sinister, Positive, Gain

Back, Clockwise

Binary Arithmetic 0, Zero, Even, No 1, One, Odd, Yes

Chemistry Acidic, Cation, Oxidant Alkaline, Anion, Reductant

Genetic Code

Pyrimidines :

Cytosine (young)

Thymine or Uracil (old)

Purines :

Guanine (young)

Adenine (old)

Particle Physics Matter, Particle, Fermion Energy, Force, Boson

Thermodynamics

Extensive quantities

Volume, Entropy

Charge

Magnetic induction

Intensive quantities

Pressure, Temperature

Voltage

Electric field

Yin Yang

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
http://www.circle-of-light.com/fengshui/bagua.htmlhttp://www.circle-of-light.com/fengshui/bagua.htmlhttp://www.cgvdb.org.tw/about_logo.phphttp://www.cgvdb.org.tw/about_logo.phphttp://www.numericana.com/answer/heat.htmhttp://www.numericana.com/answer/heat.htmhttp://www.numericana.com/answer/heat.htmhttp://www.cgvdb.org.tw/about_logo.phphttp://www.circle-of-light.com/fengshui/bagua.html


20/21

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-Yangsymbol (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, moststrikingly, the pattern shown at right, which

was sported by an infantry unit

called armigeri defensores seniores (the

shield-bearing veteran defenders).

Taijitu

(2012-08-11) Dangerous Bend Symbol (Bourbaki & Knuth)

Announces a delicate point, possibly difficult or counterintuitive.

Certains passages sont destins prmunir le lecteur

contre des erreurs graves, o il risquerait de tomber;

ces passages sont signals en marge par le signe

("tournant dangereux ").

Nicolas Bourbaki (1935 - )

That warning sign may also indicate a hazardous discussion of

minute details, to skip on first reading.

A double sign flags far out ideas. (Knuth)

Three signs would warn against delusional

stuff that must be exposed as crackpottery.
http://www.barrel.net/korean-flag.htmlhttp://www.barrel.net/korean-flag.htmlhttp://www.barrel.net/korean-flag.htmlhttp://en.wikipedia.org/wiki/Notitia_Dignitatumhttp://en.wikipedia.org/wiki/Notitia_Dignitatumhttp://en.wikipedia.org/wiki/Notitia_Dignitatumhttp://en.wikipedia.org/wiki/Taijituhttp://en.wikipedia.org/wiki/Taijituhttp://www.numericana.com/fame/bourbaki.htmhttp://www.numericana.com/fame/bourbaki.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://en.wikipedia.org/wiki/Taijituhttp://www.leestraditionaltaekwondo.com/south_korean_flag.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://en.wikipedia.org/wiki/Taijituhttp://www.leestraditionaltaekwondo.com/south_korean_flag.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://en.wikipedia.org/wiki/Taijituhttp://www.leestraditionaltaekwondo.com/south_korean_flag.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://en.wikipedia.org/wiki/Taijituhttp://www.leestraditionaltaekwondo.com/south_korean_flag.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://en.wikipedia.org/wiki/Taijituhttp://www.leestraditionaltaekwondo.com/south_korean_flag.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://en.wikipedia.org/wiki/Taijituhttp://www.leestraditionaltaekwondo.com/south_korean_flag.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://en.wikipedia.org/wiki/Taijituhttp://www.leestraditionaltaekwondo.com/south_korean_flag.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://en.wikipedia.org/wiki/Taijituhttp://www.leestraditionaltaekwondo.com/south_korean_flag.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/answer/info/fairuse.htmhttp://www.numericana.com/fame/bourbaki.htmhttp://en.wikipedia.org/wiki/Taijituhttp://en.wikipedia.org/wiki/Notitia_Dignitatumhttp://en.wikipedia.org/wiki/Notitia_Dignitatumhttp://www.barrel.net/korean-flag.html


21/21

The design of the caution sign introduced by

Bourbaki was inspired by the French

roadsigns (at right) which were installed before1949.

Now, those roadsigns have been replaced by

the 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

for the mere mathematical caution sign he would use in his own

books.

Unlike the unframed rendition of the UNICODE caution

sign (U+2621) which looks like a capital Z to the uninitiated,

D.E. Knuth's glyph (at right) really suggests a roadsign!

Knuth has collected many photos of diamond-shaped roadsigns for fun!

Bourbaki "dangerous bend" symbolThe "Dangerous Bend" Sign of Donald Knuth by Richard J. Kinch (January 2005).

www.numericana.com/answer/symbol.htm updated 2012-10-09 19:02
http://www-cs-faculty.stanford.edu/~knuth/http://www-cs-faculty.stanford.edu/~knuth/http://www-cs-faculty.stanford.edu/~knuth/http://www.charbase.com/2621-unicode-caution-signhttp://www.charbase.com/2621-unicode-caution-signhttp://www.charbase.com/2621-uni

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