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The [[word]] "'''twelve'''" is the largest [[number]] with a single-[http://en.wikipedia.org/wiki/Morpheme morpheme] name in [[English]]. Etymology suggests that "twelve" (similar to "eleven") arises from the Germanic [http://en.wikipedia.org/wiki/Compound_(linguistics) compound] twalif "two-leftover", so a [[literal]] [[translation]] would yield "two remaining [after having [[ten]] taken]". This compound [[meaning]] may have been [[transparent]] to speakers of [https://nordan.daynal.org/wiki/index.php?title=English#ca._600-1100.09THE_OLD_ENGLISH.2C_OR_ANGLO-SAXON_PERIOD Old English], but the modern form "twelve" is quite opaque. Only the remaining tw- hints that twelve and two are related.

The [[word]] "'''twelve'''" is the largest [[number]] with a single-[https://en.wikipedia.org/wiki/Morpheme morpheme] name in [[English]]. Etymology suggests that "twelve" (similar to "eleven") arises from the Germanic [https://en.wikipedia.org/wiki/Compound_(linguistics) compound] twalif "two-leftover", so a [[literal]] [[translation]] would yield "two remaining [after having [[ten]] taken]". This compound [[meaning]] may have been [[transparent]] to speakers of [https://nordan.daynal.org/wiki/index.php?title=English#ca._600-1100.09THE_OLD_ENGLISH.2C_OR_ANGLO-SAXON_PERIOD Old English], but the modern form "twelve" is quite opaque. Only the remaining tw- hints that twelve and two are related.

A group of twelve things is called a ''Duodecad''. The ordinal adjective is ''duodenary'', twelfth. The adjective referring to a [[group]] consisting of twelve [[things]] is ''duodecuple''.

A group of twelve things is called a ''Duodecad''. The ordinal adjective is ''duodenary'', twelfth. The adjective referring to a [[group]] consisting of twelve [[things]] is ''duodecuple''.

The [[number]] twelve is often used as a sales unit in trade, and is often referred to as a [http://en.wikipedia.org/wiki/Dozen dozen]. Twelve dozen are known as a [http://en.wikipedia.org/wiki/Gross_(unit) gross]. (Note that there are thirteen items in a [http://en.wikipedia.org/wiki/Baker%27s_dozen baker's dozen].)

The [[number]] twelve is often used as a sales unit in trade, and is often referred to as a [https://en.wikipedia.org/wiki/Dozen dozen]. Twelve dozen are known as a [https://en.wikipedia.org/wiki/Gross_(unit) gross]. (Note that there are thirteen items in a [https://en.wikipedia.org/wiki/Baker%27s_dozen baker's dozen].)

As shown below, the number twelve is frequently cited in the [http://en.wikipedia.org/wiki/Abrahamic_religions Abrahamic religions] and is also central to [http://en.wikipedia.org/wiki/Gregorian_calendar Western calendar] and [[units]] of [[time]].

As shown below, the number twelve is frequently cited in the [https://en.wikipedia.org/wiki/Abrahamic_religions Abrahamic religions] and is also central to [https://en.wikipedia.org/wiki/Gregorian_calendar Western calendar] and [[units]] of [[time]].

==In mathematics==

==In mathematics==

Twelve is a [http://en.wikipedia.org/wiki/Composite_number composite number], the smallest number with exactly six [http://en.wikipedia.org/wiki/Divisor divisors], its proper divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one being 24. It is the first composite number of the form p2q; a square-prime, and also the first member of the (p2) family in this form. 12 has an [http://en.wikipedia.org/wiki/Aliquot_sum aliquot sum] of 16 (133% in [[abundance]]). Accordingly, 12 is the first [http://en.wikipedia.org/wiki/Abundant_number abundant number] (in fact a [http://en.wikipedia.org/wiki/Superabundant_number superabundant number]) and demonstrates an 8 member aliquot sequence; {12,16,15,9,4,3,1,0} 12 is the 3rd composite number in the 3-aliquot tree. The only number which has 12 as its aliquot sum is the [http://en.wikipedia.org/wiki/Square_number square] 121. Only 2 other square primes are abundant (18 and 20).

Twelve is a [https://en.wikipedia.org/wiki/Composite_number composite number], the smallest number with exactly six [https://en.wikipedia.org/wiki/Divisor divisors], its proper divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one being 24. It is the first composite number of the form p2q; a square-prime, and also the first member of the (p2) family in this form. 12 has an [https://en.wikipedia.org/wiki/Aliquot_sum aliquot sum] of 16 (133% in [[abundance]]). Accordingly, 12 is the first [https://en.wikipedia.org/wiki/Abundant_number abundant number] (in fact a [https://en.wikipedia.org/wiki/Superabundant_number superabundant number]) and demonstrates an 8 member aliquot sequence; {12,16,15,9,4,3,1,0} 12 is the 3rd composite number in the 3-aliquot tree. The only number which has 12 as its aliquot sum is the [https://en.wikipedia.org/wiki/Square_number square] 121. Only 2 other square primes are abundant (18 and 20).

Twelve is a [http://en.wikipedia.org/wiki/Sublime_number sublime number], a [[number]] that has a [http://en.wikipedia.org/wiki/Perfect_number 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 [http://en.wikipedia.org/wiki/Semiperfect_number semiperfect number].

Twelve is a [https://en.wikipedia.org/wiki/Sublime_number sublime number], a [[number]] that has a [https://en.wikipedia.org/wiki/Perfect_number 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 [https://en.wikipedia.org/wiki/Semiperfect_number semiperfect number].

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

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

Twelve is a [http://en.wikipedia.org/wiki/Superfactorial superfactorial], being the product of the first three factorials. Twelve being the product of three and four, the first four positive integers show up in the equation 12 = 3 × 4, which can be continued with the equation 56 = 7 × 8.

Twelve is a [https://en.wikipedia.org/wiki/Superfactorial superfactorial], being the product of the first three factorials. Twelve being the product of three and four, the first four positive integers show up in the equation 12 = 3 × 4, which can be continued with the equation 56 = 7 × 8.

Twelve is the ninth [http://en.wikipedia.org/wiki/Perrin_number Perrin number], preceded in the sequence by 5, 7, 10, and also appears in the [http://en.wikipedia.org/wiki/Padovan_sequence Padovan sequence], preceded by the terms 5, 7, 9 (it is the sum of the first two of these). It is the fourth [http://en.wikipedia.org/wiki/Pell_number Pell number], preceded in the sequence by 2 and 5 (it is the sum of the former plus twice the latter).

Twelve is the ninth [https://en.wikipedia.org/wiki/Perrin_number Perrin number], preceded in the sequence by 5, 7, 10, and also appears in the [https://en.wikipedia.org/wiki/Padovan_sequence Padovan sequence], preceded by the terms 5, 7, 9 (it is the sum of the first two of these). It is the fourth [https://en.wikipedia.org/wiki/Pell_number Pell number], preceded in the sequence by 2 and 5 (it is the sum of the former plus twice the latter).

A twelve-sided polygon is a [http://en.wikipedia.org/wiki/Dodecagon dodecagon]. A twelve-faced polyhedron is a [http://en.wikipedia.org/wiki/Dodecahedron dodecahedron]. Regular cubes and [http://en.wikipedia.org/wiki/Octahedron octahedrons] both have 12 edges, while regular [http://en.wikipedia.org/wiki/Icosahedron icosahedrons] have 12 vertices. Twelve is a [http://en.wikipedia.org/wiki/Pentagonal_number pentagonal number]. The densest three-dimensional [http://en.wikipedia.org/wiki/Lattice_(group) lattice] [http://en.wikipedia.org/wiki/Sphere_packing sphere packing] has each sphere touching 12 others, and this is almost certainly true for any arrangement of [[spheres]] (the [http://en.wikipedia.org/wiki/Kepler_conjecture Kepler conjecture]). Twelve is also the [http://en.wikipedia.org/wiki/Kissing_number kissing number] in [[three]] [[dimensions]].

A twelve-sided polygon is a [https://en.wikipedia.org/wiki/Dodecagon dodecagon]. A twelve-faced polyhedron is a [https://en.wikipedia.org/wiki/Dodecahedron dodecahedron]. Regular cubes and [https://en.wikipedia.org/wiki/Octahedron octahedrons] both have 12 edges, while regular [https://en.wikipedia.org/wiki/Icosahedron icosahedrons] have 12 vertices. Twelve is a [https://en.wikipedia.org/wiki/Pentagonal_number pentagonal number]. The densest three-dimensional [https://en.wikipedia.org/wiki/Lattice_(group) lattice] [https://en.wikipedia.org/wiki/Sphere_packing sphere packing] has each sphere touching 12 others, and this is almost certainly true for any arrangement of [[spheres]] (the [https://en.wikipedia.org/wiki/Kepler_conjecture Kepler conjecture]). Twelve is also the [https://en.wikipedia.org/wiki/Kissing_number kissing number] in [[three]] [[dimensions]].

Twelve is the smallest weight for which a [http://en.wikipedia.org/wiki/Cusp_form cusp form] exists. This cusp form is the discriminant Δ(q) whose Fourier coefficients are given by the [http://en.wikipedia.org/wiki/Ramanujan Ramanujan] τ-function and which is (up to a constant multiplier) the 24th power of the [http://en.wikipedia.org/wiki/Dedekind_eta_function 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 [http://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function] function at -1 i.e. ζ(-1)=-1/12, the fact that the abelianization of SL(2,Z) has twelve elements, and even the properties of lattice polygons.

Twelve is the smallest weight for which a [https://en.wikipedia.org/wiki/Cusp_form cusp form] exists. This cusp form is the discriminant Δ(q) whose Fourier coefficients are given by the [https://en.wikipedia.org/wiki/Ramanujan Ramanujan] τ-function and which is (up to a constant multiplier) the 24th power of the [https://en.wikipedia.org/wiki/Dedekind_eta_function 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 [https://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function] function at -1 i.e. ζ(-1)=-1/12, the fact that the abelianization of SL(2,Z) has twelve elements, and even the properties of lattice polygons.

There are twelve [http://en.wikipedia.org/wiki/Jacobian_elliptic_function Jacobian elliptic functions] and twelve cubic [http://en.wikipedia.org/wiki/Distance-transitive_graph distance-transitive graphs].

There are twelve [https://en.wikipedia.org/wiki/Jacobian_elliptic_function Jacobian elliptic functions] and twelve cubic [https://en.wikipedia.org/wiki/Distance-transitive_graph distance-transitive graphs].

The [http://en.wikipedia.org/wiki/Duodecimal duodecimal system] (1210 [twelve] = 1012), which is the use of 12 as a division factor for many ancient and [http://en.wikipedia.org/wiki/Medieval_weights_and_measures medieval weights and measures], including hours, probably originates from [http://en.wikipedia.org/wiki/Mesopotamia Mesopotamia].

The [https://en.wikipedia.org/wiki/Duodecimal duodecimal system] (1210 [twelve] = 1012), which is the use of 12 as a division factor for many ancient and [https://en.wikipedia.org/wiki/Medieval_weights_and_measures medieval weights and measures], including hours, probably originates from [https://en.wikipedia.org/wiki/Mesopotamia Mesopotamia].

In [http://en.wikipedia.org/wiki/Base_thirteen base thirteen] and higher bases (such as [http://en.wikipedia.org/wiki/Hexadecimal hexadecimal]), twelve is represented as C. In [http://en.wikipedia.org/wiki/Base_10 base 10], the number 12 is a [http://en.wikipedia.org/wiki/Harshad_number Harshad number].

In [https://en.wikipedia.org/wiki/Base_thirteen base thirteen] and higher bases (such as [https://en.wikipedia.org/wiki/Hexadecimal hexadecimal]), twelve is represented as C. In [https://en.wikipedia.org/wiki/Base_10 base 10], the number 12 is a [https://en.wikipedia.org/wiki/Harshad_number Harshad number].

==Color Theory==

==Color Theory==

There are twelve basic [http://en.wikipedia.org/wiki/Hue hues] in the [http://en.wikipedia.org/wiki/Color_wheel color wheel]; 3 primary colors (red, yellow, blue), 3 secondary colors (orange, green & purple) and 6 tertiary colors (names for these vary, but are intermediates between the primaries and secondaries).[http://en.wikipedia.org/wiki/12_%28number%29]

There are twelve basic [https://en.wikipedia.org/wiki/Hue hues] in the [https://en.wikipedia.org/wiki/Color_wheel color wheel]; 3 primary colors (red, yellow, blue), 3 secondary colors (orange, green & purple) and 6 tertiary colors (names for these vary, but are intermediates between the primaries and secondaries).[https://en.wikipedia.org/wiki/12_%28number%29]

[[Category: Mathematics]]

[[Category: Mathematics]]

[[Category: History]]

[[Category: History]]