![]() ![]() ![]() There are six basic trigonometric functions. It is a triangular number and so is its square ( 36). Because 6 is the product of a power of 2 (namely 2 1) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Ī six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. 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.Ħ is the largest of the four all-Harshad numbers. ![]() This makes 6 the answer to the two-dimensional kissing number problem. Six 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 category has a non-trivial functor to itself only for n = 6. 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. 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 one-to-one correspondence with the elements of A. 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. ![]() S 6, with 720 = 6 ! elements, is the only finite symmetric group which has an outer automorphism. The smallest non- abelian group is the symmetric group S 3 which has 3! = 6 elements. It is also the smallest Granville number, or S. Since 6 equals the sum of its proper divisors, it is a perfect number 6 is the smallest of the perfect numbers. Six is the smallest positive integer which is neither a square number nor a prime number it is the second smallest composite number, behind 4 its proper divisors are 1, 2 and 3. ![]()
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