12th

12 (twelve) is the natural number following 11 and preceding 13. Twelve is a superior highly composite number, divisible by 2, 3, 4, and 6.

It is the number of years required for an orbital period of Jupiter. It is central to many systems of timekeeping, including the Western calendar and units of time of day and frequently appears in the world's major religions.

Name

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. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance.

Derived from Old English, ' and ' are first attested in the 10th-century Lindisfarne Gospels' Book of John. It has cognates in every Germanic language (e.g. German ''zwölf''), whose Proto-Germanic ancestor has been reconstructed as ', from ' ("two") and suffix ' or ' of uncertain meaning. It is sometimes compared with the Lithuanian ', although ' is used as the suffix for all numbers from 11 to 19 (analogous to "-teen"). Every other Indo-European language instead uses a form of "two"+" ten", such as the Latin '. 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,''Oxford English Dictionary'', 1st ed. "eleven, ''adj.'' and ''n.''" Oxford University Press (Oxford), 1891. the earliest forms of twelve are often considered to be connected with Proto-Germanic ' or ' ("to leave"), with the implicit meaning that "two is left" after having already counted to ten.''Oxford English Dictionary'', 1st ed. "twelve, ''adj.'' and ''n.''" Oxford University Press (Oxford), 1916. The Lithuanian suffix is also considered to share a similar development. The suffix ' has also been connected with reconstructions of the Proto-Germanic for ten.

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''.

Written representation

In prose writing, ''twelve'', being the last single-syllable numeral, is sometimes taken as the last number to be written as a word, and 13 the first to be written using digits.
This is not a binding rule, and in English language tradition, it is sometimes recommended to spell out numbers up to and including either ''nine'', ''ten'' or ''twelve'', or even ''ninety-nine'' or ''one hundred''. Another system spells out all numbers written in one or two words (''sixteen'', ''twenty-seven'', ''fifteen thousand'', but ''372'' or ''15,001'').
In German orthography, there used to be the widely followed (but unofficial) rule of spelling out numbers up to twelve (''zwölf''). The Duden (the German standard dictionary) mentions this rule as outdated.

Mathematical properties

Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one being twenty-four.

Twelve is 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. Twelve is the smallest of two known sublime numbers, numbers that has a perfect number of divisors, and the sum of its divisors is also a perfect number. Since there is a subset of 12's proper divisors that add up to 12 (all of them but with 4 excluded), 12 is a semiperfect number.

If an odd perfect number is of the form 12''k'' + 1, it has at least twelve distinct prime factors.

A twelve-sided polygon is a dodecagon. A twelve- faced polyhedron is a dodecahedron. Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices. Twelve is a pentagonal number. The densest three-dimensional lattice sphere packing has each sphere touching 12 others, and this is almost certainly true for ''any'' arrangement of spheres (the Kepler conjecture). Twelve is also the kissing number in three dimensions.

Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant Δ(''q'') whose Fourier coefficients are given by the Ramanujan -function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function. This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function at −1 i.e. ζ(−1) = −, the fact that the abelianization of SL(2,Z) has twelve elements, and even the properties of lattice polygons.

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

There are 12 Latin squares of size 3 × 3.

There are 12 two-dimensional regular tilings when enantiomorphic forms are counted separately.

The duodecimal system (12₁₀ welve= 10₁₂), which is the use of 12 as a division factor for many ancient and medieval weights and measures, including hours, probably originates from Mesopotamia.

In base thirteen and higher bases (such as hexadecimal), twelve is represented as C. In base 10.

12 is a Harshad number in all bases except octal.

12 is a pronic number and a Pell number.

12 is the negative reciprocal of the Ramanujan summation of the infinite series 1 + 2 + 3 + 4 + ⋯. Although the series is divergent, methods such as Ramanujan summation can assign finite values to divergent series.

12 is the superfactorial of 3.

List of basic calculations

In nature

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.

Religion

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

Ancient Greek religion

*The Twelve Olympians are the principal gods of the pantheon, they were preceded by twelve Titans, and Hercules carries out twelve labours.
*Several sets of twelve cities are identified in history as a dodecapolis, the most familiar being the Etruscan League. In ancient Rome, the twelve lictors carried fasces of twelve rods.

Judaism and Christianity

* The significance is especially pronounced in the Hebrew Bible

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