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Twelve is a composite number, the smallest number with exactly six divisors, its proper divisors being 1, 2, 3, 4 and 6. Twelve is also a highly composite number, the next one being 24.
 The duodecimal system (1210 [twelve] = 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. As Schoolhouse Rock explains in its song "Little Twelvetoes", if humankind had been born with twelve fingers, they would have counted and multiplied using the duodecimal system. (There is no need to have twelve fingers though, since one will easily arrive at the duodecimal system by simply counting the phalanx bones of fingers with the same hand's thumb, a practice in use with some people until the present day.)
Twelve is a 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.
A twelve-sided polygon is a dodecagon. A twelve-sided polyhedron is a dodecahedron. 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.
In base thirteen and higher bases (such as hexadecimal), twelve is represented as C.
Twelve is superabundant, sparsely totient, a Harshad number and a Pell number.
Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant ?(q) whose Fourier coefficients are given by the Ramanujan t-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 function at -1 i.e. ?(-1)=-1/12, the fact that the abelianization of SL(2,Z) has twelve elements, and even the properties of lattice polygons.
Geometry
Using a compass, construct a simple circle.
Taking care not to change the radius setting, draw a second circle with center anywhere on the perimeter of the first.
Now go to the two points where the second circle crosses the first, and draw two more circles from these centers.
Continue this process to draw the final three circles, working your way around the perimeter of the first.
Finally connect the center of the original circle with all other points of intersection, using a straightedge.
You now have twelve radial lines emanating from the center, separated by twelve 30-degree sectors, all constructed from seven equal-sized circles. - Demo User
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