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Final Answers
© 2000-2019   Gérard P. Michon, Ph.D.

Scientific Symbols and Icons

 (1646-1716)  Coat-of-arms of the 
 famous Bernoulli family 
 (Swiss mathematicians).
Symbols are more than cultural artefacts:  They
 address our intellect, emotions, and spirit.

  David Fontana (The Secret Language of Symbols, 1993)

Related articles on this site:

Related Links (Outside this Site)

symbols.com.  Merged with Symbols.net, from  George F. Sutton (b. 1956).
Mathematical Symbols  by Robin Whitty  (Theorem of the Day).
History of Mathematical Symbols by Douglas Weaver & Anthony D. Smith.
Flags with Mathematical Symbols  |  Table of Mathematical Symbols
Symbols in the  Mathematical Association'coat of arms.
Sixty Symbols:  Videos about the symbols of physics and astronomy.
Mathematical Unicode characters and associated HTML entities.
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)
Previously, the only way to display 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 the ASCII set were unambiguously understood.  (EBCDIC has 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 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 the original 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 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 readability 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 font  has ever meant it to be rendered like 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 used and take corrective action if it's not.

Due to privacy concerns, browsers aren't allowed to check directly whether or not a given font is installed  (since that would involve reading files on 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 encoding by checking that a lowercase  lambda  is wider than a lowercase  nu  (whereas the corresponding "l" is narrower than "n" in all non-monospaced latin fonts).

If that's not the case, advanced capabilities of the more recent versions of JavaScript can be used to replace individual characters by their closest  UNICODE  counterparts.  That's not a perfect solution but it 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 browsers whose 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 advanced capabilities which allow the patch to work.

There's a chance that better versions of JavaScript will be supported on mobile devices in the future, which will fix the issue  (kinda).  There's also a  (slim)  chance that all browser designers will eventually become aware of their God-given duty to cherish all types of evergreen information, which can't be economically re-encoded to follow the latest fashion!

Well, at least, users of the desktop versions of FireFox are thus accomodated right now,  in spite of the ruthlessness  (or rudeness)  of the people who deliberately modified the details of their browser.

One advantage of the  lambda-nu test  is that it puts no demands on future maintainers of a site, without penalizing at all the users of browsers with proper 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 better to all Firefox users.  Instantly.

W3C recommendations for mathematical symbols in HTML 4.0  (December 1997)
Symbol font (Wikipedia)   |   ASPpdef (Special characters tables)
Special <Font Face="Symbol"> 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 font and nonstandard symbolic typefaces  by  Ian C. G. Bell  (1998-2006)
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).

 Equality Symbol 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 in The Whetstone of Witte (1557) by Robert 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.

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 popularization of the equal sign, 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 same mathematical glyph is then seen again, and perhaps more importantly, in Oughtred's masterpiece Clavis Mathematicae (1631) in which other scientific symbols 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 the Latin aequalis) well into the 18th century.  Thomas Harriot (1560-1621) was using a different symbol  ( Harriot's Equality Symbol ),  while some others used a pair of vertical 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.

Some chemical reactions proceed 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 consists of 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:

 Double-Harpoon Equilibrium Symbol

This is the so-called  right-left  version of the symbol  (UNICODE: 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 by pharmacists and often  wrongly  associated with medicine).

 one over two 
 (one half)(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  bar  between them is best called 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 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 the Roman gold coin of 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 British shilling in 1971  (when British money was decimalized).  See discussion below.


The related symbol "¸" is called an obelus.  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 the division key on calculators...

On 2009-08-07, John Harmer wrote:       [edited summary]
I was at Uni in 1971 and can't remember  ever  using "/" instead of "s" for shillings.  Before another meaning came 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 solidus symbol 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  was posh).
Cheers,   John Harmer
Chippenham, UK

Both meanings of the  solidus  sign  (i.e., currency prefix  and/or separator)  are compatible and have coexisted peacefully.  Arguably, the definition presented by   John Harmer  became dominant with the passage of time.

 Infinity (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 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 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 way to denote the fictitious "infinite circle" at the horizon of the complex plane.

 of Bernoulli

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.  Cross of
 St. Boniface

The design appeared in Western iconography before modern times.  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 graphic 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   |   Black Arts Diary   |   Unsigned infinity and signed infinities on calculators

 Aleph 0  omega (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 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 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 infinity symbol  Ào  (pronounced "aleph zero", "aleph null", or "aleph nought") was defined 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 own diagonal argument proves that reals and integers have different cardinalities.  (Actually, because the powerset of 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 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 infinity symbol  w  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

 cap, intersection  wedge, chevron, logical and, gcd, hcf (2005-04-10)   Cap: Ç   Cup: È   Wedge: Ù   Vee: Ú
Intersection (greatest below) & Union (lowest above).
 wedge, chevron, logical and, gcd, hcf

The wedge is  also  the standard symbol for an  exterior product,  (which is the true nature of the product of  differential forms  found within multivariate  integrals, where it's customary to use terse  multiplicative  notations instead).  Relatedly, it's also used as a substitute for the cross symbol to denote a  vector product.  Neither usage is within the advertised category,  discussed below.

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.   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}

For example, a stochastic process  Xt  stopped at time  T  is equal to  XtÙT

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 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 dubbed  highest common factor  (HCF).  Conversely,  pÚq  is their  lowest common multiple  (LCM).

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

(*)  Because  (p,q)  is the  only  possible way to denote an ordered pair  (namely,  an element of a  cartesian product)  we don't recommend the dubious notation  (p,q)  for the  GCD of  p  and  q.  It's unfortunately widespread 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 as a sign denoting 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...

 squared U  inverted pi (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 of sets  Ai  is:

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

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 disjoint union B  for the disjoint union of two sets  A  and  B.  The squared "U" symbol  ( disjoint union )  is the preferred one  (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 the context.  Mathematical symbols are very often overloaded.  The overloading of a symbol usually implies the overloading of related symbols.  For example, the overloading of the addition sign  (+)  implies an overloading of the summation sign  (S)  and vice-versa.

Additive notations are [somewhat] popular for  discriminated unions  because the cardinal of a discriminated union is  always  the sum of the cardinals of its components.  Denoting  |E|  the cardinal of the set  E :

å A  |   =   å  | A |

From a categorial perspective, the disjoint union is the dual of the  categorial product.  It's called either  coproduct  or  sum.

(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).

Those symbols are also called "doublestruck" because mechanical typewriters 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 any overloading.  This makes them usable without the need for building up a  context.

Some Doublestruck Symbols and their Meanings
SymbolBold EtymologySymbol's Meaning
PPPrime Numbers2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
PqFqFinite FieldGalois field of q = pn elements (p prime)
NNNatural Numbers Nonnegative Integers  [additive monoid]
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
ZZZahl [German]Ring of Signed Integers (countable)
... -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...
Z / nZZ/nZ Quotient Ring Ring of Integers modulo n (finite)
ZpZpZahl, radix pRing of p-adic Integers (uncountable)
QQ QuotientField of Rational Numbers (countable)
qpQpQuotient, radix pField of p-adic Numbers (uncountable)
RR RealField of Real Numbers (uncountable)
CC ComplexField of Complex Numbers
HH HamiltonSkew Field of Quaternions
HO OctonionsAlternative  Division Algebra

The group formed by the invertible elements of a  multiplicative  monoid  M  is denoted  M*.  That's  compatible  with the common usage of starring the symbol of a set of numbers to denote the  nonzero  numbers in it  (the two definitions are equivalent for  Q*,  R*,  C*  and  H*).  In particular:

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

P*  is  undefined  (arguably, that symbol might denote the  odd primes ).

  Log x   =  
ò  x  
(2003-08-03)   ò
The integration sign of Leibniz  (29 October 1675).

 Integration symbol   Gottfried Wilhelm Leibniz (1646-1716) viewed 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 Jacob 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 Hudde was quoted by both Newton and Leibniz ]

 Halmos tombstone = QED (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).  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 of a plain tombstone symbol.

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

To Euclid and the ancient Greeks, the end of a demonstration was indicated by the acronym  oed  (transliterated "Oper Edei Deixai"). See Robin Whitty's Theorem #149.

 Nabla  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 pronounced "del".  Which is it?  I thought "del" was reserved for [Hamilton's nabla operator]   Ñ  =  < ¶/¶x, ¶/¶y, ¶/¶z >

"Del" is a correct name for both    and  Ñ.  Some authors present these two 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 way to read this aloud in front of a classroom is either  "partial of y with respect to x"  or  "partial of y along x"  (especially when  x  is a space or spacetime coordinate).

In an international scientific context, the confusion between    and Ñ  is best avoided by calling  Ñ  "nabla del", or simply  nabla.  Some practitioners 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 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 HeavisideMaxwell 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".  The sign 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 modern mathematical meaning of  Condorcet 
 (1743-1794)  Legendre  
(1752-1833)  this special character  (starting in 1841, with the introduction of  Jacobians  in the epoch-making paper entitled "De determinantibus functionalibus").  Historically, this lowercase mathematical symbol was first used by Condorcet in 1770, and by Legendre around 1786.

 Rod of 
Asclepius 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.


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 official  Imhotep  whe probably never practiced medicine himself  (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.

The proper symbol for Ophiuchus is indeed the  Rod of Asclepius  or  Staff of Asclepius  (one snake around a plain 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 Association issued a resolution stating that  "the true ancestral symbol of healing art is the knotty pine and the [single] serpent of Aesculapius".

Ophiuchus   |   Rod of Asclepius
Modernized Zodiac  by  Dr. Shepherd Simpson  (Astrological Historian).
Astrological Attributes of Ophiuchus  by  Betty Rhodes.

 Caduceus (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  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 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 Tiresias foresight and allowing 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.

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 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 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 the Caduceus with medicine.  The misguided heraldic use of the symbol by military medicine started in the 19th century and culminated with the 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.

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

In their  oath Pythagoreans called  Pythagoras

Him who brought us the tetractys,
the Source of everflowing Nature

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 system which puts small whole numbers at the core of every fundamental explanation.

Tetractys Symbol (Wikipedia)   |   The Tetraktys Symbol  by  Robert Apatow (1999).
Pythagorean Harmony  in  "Week 266"  by  John Baez.

Borromean rings(2003-06-10)   Borromean Symbol.  Borromean Links.
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'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,  with several other interlacing patterns!  The three rings are found among the many symbols featured on the Borromeo coat of arms (they're not nearly as prominent as one would expect).  One version of
 Odin's Triangle

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 a recent issue of the journal  Science  (May 28, 2004)  the synthesis of a molecule with Borromean topology  was reported by a group of chemists at UCLA,  led by  J. Fraser Stoddart (b. 1942, Nobel 2016).

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).

In North America, the pattern is sometimes called a  ballantine  because of the 3-ring logo (Purity, Body, Flavor) of Ballantine'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).

 IMU Logo  Borromean rings  are but the simplest example of  Brunnian links.

At the  25th  International Congress of Mathematicians  in Madrid, Spain  (August 2006)  the  International Mathematical Union  (IMU)  adopted for itself the logo at left,  which shows three  congruent Borromean rings.

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

 Niels Bohr's coat of arms 
 Motto: Contraria sunt complementa 
 (opposites are complementary)(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.

Taiji Mandala The Chinese Taiji symbol (Tai-Chi, or taijitu) predates the Song dynasty (960-1279).  Known in the West as the Yin-Yang symbol, this sign 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).  Wu-Chi 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, 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.
 "Tao Te Ching" (Book of the Way, #64)  by  Lao Tzu  (600-531 BC)  founder of Taoism.

 Yin and Yang 

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:

EtymologyDark Side  (French: ubac)Bright Side  (French: adret)
GeographyNorth of a mountain
South of a river
South of a mountain
North of a river
GenderFemale, FeminineMale, Masculine
CelestialMoon, Planet, NightSun, Star, Day
Ancient SymbolWhite TigerGreen Dragon
ColorsViolet, Indigo, BlueRed, Orange, Yellow
Greater Phase
Transition, Young
West, Metal and Autumn
Potential Structure
East, Wood and Spring
Potential Action
Weak Nuclear ForceGravity
Lesser Phase
Stability, Old
North, Water and Winter
Actual Structure
South, Fire and Summer
Actual Action
Strong Nuclear ForceElectromagnetism
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, Preserving
Small, Interior, Bone
Mental, Subtle
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, Consuming
Large, Exterior, Skin
Physical, Obvious
& Topology
Space, Open angle
Finite, Discontinuous
Time, Closed circle
Infinite, Continuous
OrientationDexter, Negative, Loss
Front, Counterclockwise
Sinister, Positive, Gain
Back, Clockwise
Annular gear, Rack, Wheel
Driven gear
ReasoningEffect, MaturationCause, Thought
Binary Arithmetic0, Zero, Even, No    No 1, One, Odd, Yes    Yes
FoodSweet, Bitter, Mild
Vegetable, Root
Red meat
Salty, Sour, Hot
Fruit, Leaf
ChemistryAcidic, Cation, OxidantAlkaline, 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
Magnetic induction
Intensive quantities
Pressure, Temperature
Electric field

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...

flagpole Modern [South] Korean Flag    

The oldest known Tai-Chi symbol was 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. 

Armigeri Defensores Seniores  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 was sported by an infantry unit called  armigeri defensores seniores  (the shield-bearing veteran defenders).


Caution Sign (2012-08-11)   Dangerous Bend  Symbol   (BourbakiKnuth)
Announces a delicate point, possibly difficult or  counterintuitive.

Certains passages sont destinés 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 - ¥)

Sign A single warning sign may also indicate a hazardous discussion of minute details, to skip on first reading.
Caution Sign A double sign flags  far out  ideas  (Knuth).
Caution Sign A triple sign warns against possible  crackpottery.
 Old French Roadsign for 
 Dangerous Bends

The design of the  caution sign  introduced by Bourbaki was inspired by the French roadsigns  (at right)  which were installed before 1949.

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 for dangerous bends.

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.  Donald Knuth's dangerous-bend symbol

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" symbol
The "Dangerous Bend" Sign of Donald Knuth  by  Richard J. Kinch  (January 2005).

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