[Ansteorra] Centurion Cloaks
brandonsmcd at yahoo.com
Tue Dec 6 10:29:58 PST 2011
We in the Order of the Centurions of the Sable Star of Ansteorra have, as long as I have been a member, regarded the original 6 as an entity. They are "THE" 6. No more precedence is needed. Interestingly the #6 has many unique features. See below.
Count Lochlan Dunn
CSSA XCVI (96)
Six is the second smallest composite number, its proper divisors being 1, 2 and 3.
Since six equals the sum of these proper divisors, six is the smallest perfect number, Granville number, and -perfect number. As a perfect number, 6 is related to the Mersenne prime 3, since 21(22 - 1) = 6. (The next perfect number is 28.) It is the only even perfect number that is not the sum of successive odd cubes. Being perfect, six is the root of the 6-aliquot tree, and is itself the aliquot sum of only one number; the square number, 25. Unrelated to 6 being a perfect number, a Golomb ruler of length 6 is a "perfect ruler." Six is a congruent number.
Six is the first discrete biprime (2.3) and the first member of the (2.q) discrete biprime family.
Six is the only number that is both the sum and the product of three consecutive positive numbers.
Six is a unitary perfect number, a harmonic divisor number and a highly composite number. The next highly composite number is 12.
5 and 6 form a Ruth-Aaron pair under either definition.
The smallest non-abelian group is the symmetric group S3 which has 3! = 6 elements.
S6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of n isomorphic objects on an n-set A, invariant under all permutations of A, but not naturally in 1-1 correspondence with the elements of A. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for n=6.
6 similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.
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