Symbols are more than cultural artefacts: They _{ }address our intellect, emotions, and spirit. ^{ }
David Fontana (The Secret Language of Symbols, 1993)

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

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.

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 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).

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 of vertical lines ( || ) instead.

(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 H_{2} + O_{2}
® 2 H_{2}O

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:

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).

(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
1 / - / -
The symbol was pronounced stroke
(oblique was
posh).

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.

(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.

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 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).

(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.

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

The chevron (wedge) and inverted chevron (vee) are the generic symbols used
to denote the basic binary operators induced by a partial ordering on a
lattice. 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 X_{t}
stopped at time T is equal to
X_{t;ÙT}

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

pÙq = gcd(p,q)
[ = (p,q) ] (*)

pÚq = lcm(p,q)

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

In the context of
Number Theory, the above use of the "wedge" and "vee" mathematical
symbols needs little or no introduction, except to avoid confusion with the meaning
they have in predicate calculus (the chevron symbol stands for "logical and", whereas
the inverted chevron is "logical or", also called "and/or").

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

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

The chevron symbol is also used 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...

(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
A_{i} is:

A_{i} =

È

{ (x,i) |
x Î
A_{i} }

iÎI

iÎI

However, such an indexed family is often treated as a mere collection of sets.
The existence of an indexation is essential in the above formulation, but the
usual abuse of notation is to omit the index itself, which is considered
mute. This makes it possible to use simple notations like
A+B or
AB
for the disjoint union of two sets A and B.
The squared "U" symbol
() is the preferred
(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 |

(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.

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
*,
*,
* and
*).
In particular:

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 ).

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

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...

(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!

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.

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]
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 Heaviside.
Maxwell apparently never used the name in a scientific context.

The question is moot for many mathematicians, who routinely read
a ¶ symbol
like a "d" (mentally or aloud).
I'm guilty of this myself, but don't tell anybody!

When it's necessary to lift all ambiguities without sounding overly pedantic,
"¶" is also routinely called
"curly d", "rounded d" or "curved d". 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
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.

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 doctor
Imhotep
(27th century BC).

The Rod of Asclepius, symbol of medicine,
is a single snake entwined around a stick.
Originally, the symbol may have depicted the treatment of
dracunculiasis
(very common in the Ancient World) in which the long parasitic worm was
traditionally extracted through the patient's skin
by wrapping it around a stick over a period of days or weeks
(because a faster procedure might break the worm).

Any symbol involving a snake would seem natural for medicine:
The snake is a symbol of renewed life out of old shedded skin,
not to mention the perpetual renewal of life evoked by the
ouroboros
symbol (a snake feeding on its own tail).
A snake around a walking stick is also an ancient symbol of supernatural powers
which can triumph over death, like medicine can
(biblically, the symbol of Moses' divine mission was his ability to change his
walking stick into a snake).

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

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

Ophiuchus is the only zodiacal constellation which has not given
its name to one of the 12 signs of the zodiac associated with
the 12 traditional equal subdivisions of the solar year, which form the
calendar used by astrologers.
However, some
modern
astrologers are advocating a reformed system
with uneven zodiacal signs, where Ophiuchus has found its place...

Astrological belief systems are not proper subjects
for scientific investigation.
Nevertheless, we must point out that it's a plain error to associate Ophiuchus
with the caduceus symbol
(two snakes around a winged staff)
since that symbol of Hermes
(messenger of the gods) is associated with
commerce, not medicine.

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".

(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 seerTiresias,
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 elementmercury 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.

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.

(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 are all circles
[no other counterexamples are known].

The design was once the symbol of the alliance between
the Visconti, Sforza and Borromeo families.
It's been named after the Borromeo family who has perused the three-ring symbol,
with several
other interlacing patterns!
The three rings are found among the many symbols featured on the
Borromeo
coat of arms (they are not nearly as prominent as one would expect,
you may need a closer look).

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

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

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

Borromean rings are but the simplest example of
Brunnian links.

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

The Chinese Taiji symbol (Tai-Chi, or taijitu)
predates the Song dynasty (960-1279). Known in the West as the
Yin-Yang symbol,
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).
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.

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:

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 scientific meaning...

The oldest known Tai-Chi symbol
was carved in the stone of a Korean Buddhist temple in AD 682.
A stylized version of the Ying-Yang symbol
(Eum-Yang to Koreans) appears on the modern [South] Korean Flag
(T'aeGuk-Ki)
which was first used in 1882, by the diplomat Young-Hyo Park on a mission to Japan.
The flag was banned during the Japanese occupation of Korea, from 1910 to 1945.

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).

(2012-08-11) Dangerous Bend Symbol
(Bourbaki, Knuth)
Announces a delicate point, possibly difficult or counterintuitive.

Certains passages sont destinés à prémunir le lecteur contre des
erreurs graves, où il risquerait de tomber; ces passages sont
signalés en marge par le signe
☡ ("_{ }tournant dangereux "). ^{ }Nicolas Bourbaki
(1935 - ¥)

A single warning sign may also indicate a
hazardous discussion of minute details, to skip on first reading.

A triple sign warns against possible crackpottery.

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 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!