Small Numbers  (6)

The smallest composite squarefree number (), and the third triangular number (). It is the also smallest perfect number, since . The number 6 arises in combinatorics as the binomial coefficient , which appears in Pascal's triangle and counts the 2-subsets of a set with 4 elements. It is also equal to  (3 factorial), the number of permutations of three objects, and the order of the symmetric group  (which is the smallest non-Abelian group).

Six is indicated by the Latin prefix sex-, as in sextic, or by the Greek prefix hexa- (-), as in hexagon, hexagram, or hexahedron.

The six-fold symmetry is typical of crystals such as snowflakes. A mathematical and physical treatment can be found in Kepler (Halleux 1975), Descartes (1637), Weyl (1952), and Chandrasekharan (1986).

REFERENCES:

Chandrasekharan, K. Hermann Weyl (1885-1985): Centenary Lectures. Berlin: Springer-Verlag, 1986.

Descartes, R. Discours de la méthode: Les météores. Leyden, Netherlands, 1637.

Kepler, J. Étrenne ou la Neige sexangulaire. Translated from Latin by R. Halleux. Paris, France: J. Vrin Éditions du CNRS, 1975.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 67-69, 1986.

Weyl, H. Symmetry. Princeton, NJ: Princeton University Press, 1952.