12 (number)

Twelve is the largest number with a single-syllable name in English. Early Germanic numbers have been theorized to have been non-decimal: evidence includes the unusual phrasing of eleven and twelve, the former use of "hundred" to refer to groups of 120, and the presence of glosses such as "tentywise" or "ten-count" in medieval texts showing that writers could not presume their readers would normally understand them that way.[1][2][3] Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance.

Derived from Old English, twelf and tuelf are first attested in the 10th-century Lindisfarne Gospels' Book of John.[note 1][5] It has cognates in every Germanic language (e.g. German zwölf), whose Proto-Germanic ancestor has been reconstructed as *twaliƀi..., from *twa ("two") and suffix *-lif- or *-liƀ- of uncertain meaning.[5] It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19 (analogous to "-teen").[5] Every other Indo-European language instead uses a form of "two"+"ten", such as the Latin duōdecim.[5] The usual ordinal form is "twelfth" but "dozenth" or "duodecimal" (from the Latin word) is also used in some contexts, particularly base-12 numeration. Similarly, a group of twelve things is usually a "dozen" but may also be referred to as a "dodecad" or "duodecad". The adjective referring to a group of twelve is "duodecuple".

As with eleven,[6] the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan ("to leave"), with the implicit meaning that "two is left" after having already counted to ten.[5] The Lithuanian suffix is also considered to share a similar development.[5] The suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten.[6][7]

As mentioned above, 12 has its own name in Germanic languages such as English (dozen), Dutch (dozijn), German (Dutzend), and Swedish (dussin), all derived from Old French dozaine. It is a compound number in many other languages, e.g. Italian dodici (but in Spanish and Portuguese, 16, and in French, 17 is the first compound number), Japanese 十二 jūni.

Twelve is the sixth composite number, and the smallest number with exactly six divisors, its divisors being 1, 2, 3, 4, 6 and 12. It is a pronic number, and the smallest abundant number, since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 16) is greater than itself.[9][10] It is the second semiperfect number, since there is a subset of the proper divisors of 12 that add up to itself.[11] If an odd perfect number is of the form 12k + 1, it will have at least twelve distinct prime factors. Twelve is the fifth highly composite number, the next one being twenty-four and the previous six (uniquely, twice twelve and half of twelve, respectively).[12] 12, as a number with a perfect number of divisors (six), has a sum of divisors that yields the second perfect number, σ(12) = 28.[13] In turn, 28 is the arithmetic mean of the twelve divisors of the fourth harmonic divisor number, 140 (like 6, and 28), which generate an integer harmonic mean of 5.[14][15][16] It is the smallest of two known sublime numbers, which are numbers that have a perfect number of divisors whose sum is also perfect.[17]

Twelve is the number of divisors of 60 and 90, the second and third unitary perfect numbers (6 is the first). It is also the number of distinct prime factors that belong to the fifth unitary perfect number, the largest known,

2¹⁸ × 3 × 5⁴ × 7 × 11 × 13 × 19 × 37 × 79 × 109 × 157 × 313 = 146,361,946,186,458,562,560,000.[18][19]

12 is the fifth Pell number, preceded by 0, 1, 2, and 5, as well as the third pentagonal number.[20][21] 12 is the superfactorial of 3.[22] It is also a Harshad number in all bases except octal.

There are 12 Latin squares of size 3 × 3, where symbols appear exactly once in each row and exactly once in each column.[23]

There are twelve Jacobian elliptic functions and twelve cubic distance-transitive graphs.

A twelve-sided polygon is a dodecagon. In its regular form, it is the largest polygon that can uniformly tile the plane alongside other regular polygons, as with the truncated hexagonal tiling or the truncated trihexagonal tiling. There are 12 regular and semiregular tilings when enantiomorphic forms of the snub hexagonal tiling are counted separately.[24]

A regular dodecahedron has twelve pentagonal faces. Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices. The rhombic dodecahedron has twelve rhombic faces and is able to tessellate three-dimensional space. Its dual polyhedron, the cuboctahedron, has 12 vertices with radial equilateral symmetry, and is one of two quasiregular polyhedra.

The densest three-dimensional lattice sphere packing has each sphere touching twelve other spheres, and this is almost certainly true for any arrangement of spheres (the Kepler conjecture). Twelve is also the kissing number in three dimensions.

There are twelve complex apeirotopes in dimensions five and higher, which include van Oss polytopes in the form of complex ${\displaystyle n}$-orthoplexes.[25] There are also twelve paracompact hyperbolic Coxeter groups of uniform polytopes in five-dimensional space.

Bring's curve is a Riemann surface of genus four, with a domain that is a regular hyperbolic 20-sided icosagon.[26] By the Gauss-Bonnet theorem, the area of this fundamental polygon is equal to ${\displaystyle 12\pi }$.

Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant ${\displaystyle \Delta (q)}$ whose Fourier coefficients are given by the Ramanujan ${\displaystyle \tau }$-function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function:

${\displaystyle \Delta (\tau )=(2\pi )^{12}\eta ^{24}(\tau )}$

This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the fact that the abelianization of SL(2,Z) has twelve elements, to the value of the Riemann zeta function at −1 being −+1/12.

As the negative reciprocal of the Ramanujan summation of the infinite series 1 + 2 + 3 + 4 + ⋯ at ζ(−1),

${\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}}}$

Although the series is divergent, methods such as Ramanujan summation can assign finite values to divergent series.

Notably, twelve is the number of full lunations in a solar year, hence the number of months in a solar calendar, as well as the number of signs in the Western and the Chinese zodiac. Twelve is also the number of years for an orbital period of Jupiter.

The number twelve carries religious, mythological and magical symbolism, generally representing perfection, entirety, or cosmic order in traditions since antiquity.[27]

• The number of twelve jurors in jury trials is depicted by Aeschylus in the Eumenides. In the play, the innovation is brought about by the goddess Athena, who summons twelve citizens to sit as jury.
• In English Common Law, the tradition of twelve jurors harks back to the 10th-century law code introduced by Aethelred the Unready.

• The lunar year is 12 lunar months. Adding 11 or 12 days completes the solar year.[31]
• Most calendar systems – solar or lunar – have twelve months in a year.
• The Chinese use a 12-year cycle for time-reckoning called Earthly Branches.
• There are twelve hours in a half day, numbered one to twelve for both the ante meridiem (a.m.) and the post meridiem (p.m.). 12:00 p.m. is midday or noon, and 12:00 a.m. is midnight.
• The basic units of time (60 seconds, 60 minutes, 24 hours) are evenly divisible by twelve into smaller units.

• The atomic number of magnesium in the periodic table.
• The Standard Model identifies twelve types of elementary fermions as well as twelve types of elementary gauge bosons.
• The human body has twelve cranial nerves.
• The duodenum (from Latin: duodecim, lit. 'twelve') is the first part of the small intestine, that is about twelve inches (30 cm) long. More precisely, this section of the intestine was measured not in inches but in fingerwidths. In fact, in German the name of the duodenum is Zwölffingerdarm, in Dutch the name is twaalfvingerige darm and in Bulgarian the name is дванадесетопръстник, all meaning "twelve-finger bowel".
• Force 12 on the Beaufort wind force scale corresponds to the maximum wind speed of a hurricane.
• The human body has twelve thoracic vertebrae.

• In both soccer and American football, the number 12 can be a symbolic reference to the fans because of the support they give to the 11 players on the field. Texas A&M University reserves the number 12 jersey for a walk-on player who represents the original "12th Man", a fan who was asked to play when the team's reserves were low in a college American football game in 1922. Similarly, Bayern Munich, Hammarby, Feyenoord, Atlético Mineiro, Flamengo, Seattle Seahawks, Portsmouth and Cork City do not allow field players to wear the number 12 on their jersey because it is reserved for their supporters. It also serves as the jersey number for some the National Football League's best and most well-known quarterback, Tom Brady.
• In Canadian football, 12 is the maximum number of players that can be on the field of play for each team at any time.
• In cricket, another sport with eleven players per team, teams may select a "12th man", who may replace an injured player for the purpose of fielding (but not batting or bowling).
• In women's lacrosse, each team has 12 players on the field at any given time, except in penalty situations.
• In rugby league, one of the starting second-row forwards wears the number 12 jersey in most competitions. An exception is in the Super League, which uses static squad numbering.
• In rugby union, one of the starting centres, most often but not always the inside centre, wears the 12 shirt.
• In an NBA game, a quarter lasts 12 minutes.

• ASCII and Unicode code point for form feed.
• The number of function keys on most PC keyboards (F1 through F12).
• The number of keys in any standard digital telephone (1 through 9, 0, * and #).
• Microsoft's Rich Text Format specification assigns numbers congruent to 12 mod 256 to variants of the French language.

12 stars are featured on the Flag of Europe.

• There are 12 troy ounces in a troy pound (used for precious metals).
• Twelve of something is called a dozen.
• In the former British currency system, there were twelve pence in a shilling.[33]
• In Greek mythology, the number of Labours of Heracles was increased from ten to make twelve.
• In English, twelve is the number of greatest magnitude that has just one syllable and the last one to contain a single syllable.

The numerical range on the analogue clock ends at 12.

• 12 is the last number featured on the analogue clock, and also the starting point of the transition from A.M. to P.M. hours or vice-versa.
• There are twelve months within a year, with the last one being December.
• The level of grades in which one must attend school typically ends at 12 (although some jurisdictions may include a thirteenth grade depending on the country).
• Twelve hours form half a day, and twelve hours away from another lead to the same time but with a different period (ex. Twelve hours away from 6:00AM leads to 6:00PM).
• There are normally twelve pairs of ribs in the human body.
• The Twelve Tables or Leges Duodecim Tabularum, more informally simply Duodecim Tabulae, was the ancient legislation underlying Roman law.
• In the United States, twelve people are appointed to sit on a jury for felony trials in all but four states, and in federal and Washington, D.C. courts. The number of jurors gave the title to the play (and subsequent films) Twelve Angry Men.
• Twelve men have walked on Earth's moon.
• The United States is divided into twelve Federal Reserve Districts (Boston, New York, Philadelphia, Cleveland, Richmond, Atlanta, Chicago, St. Louis, Minneapolis, Kansas City, Dallas, and San Francisco); American paper currency has serial numbers beginning with one of twelve different letters, A through L, representing the Federal Reserve Bank from which the currency originated.
• According to UFO conspiracy theory, Majestic 12 is a secret committee, allegedly set up by U.S. President Harry S. Truman to investigate the Roswell UFO incident and cover up future extraterrestrial contact.
• 12 is the number of the French department Aveyron.
• King Arthur's Round Table had 12 knights plus King Arthur himself.
• 12 inches in a foot.
• Alcoholics Anonymous has 12 steps, 12 traditions and 12 concepts for world service.
• Wilhelm Heinrich Schüßler developed a list of 12 biochemical cell salts, also known as tissue salts.

• The Twelfth

• Media related to 12 (number) at Wikimedia Commons