# 6

Six

6

I like to consider the number 6 the limit of numbers we can recognize at once. For example:

o o o

o o o

This layout instantly allows us to perceive six because they are arranged in the optimal way, but our brains are even worse at recognizing sixness that they are at fiveness, and six needs to be virtually always seen as three and three.

But since 6 seems to be a boundary in that respect, 6 is the base of Robert Munafo's idea of classes - six is the boundary between class 0 (1-6) and class 1 (6-1,000,000) numbers. Class 0 numbers are described as numbers that can be immediately recognized in the mind, while class 1 numbers are numbers that can't be immediately recognized but can be physically perceived directly. With that idea, one, two, three, four, five, and six are class 0, while 7, 27, 100, 2048, and are all class 1 numbers. For more on classes, see the entry for a million.

Six is an interesting number as well, with several notable properties. For example, it is:

- the smallest perfect number (6's divisors, 1, 2, and 3, add up to 6)

- the factorial of three (3! = 3*2*1 = 6)

- the second primorial (2*3, product of the first two primes)

- the only semiprime that isn't a deficient number

- the smallest number x such that all multiples of x are abundant

- the number of basic trigonometric functions (sine, cosine, tangent, cotangent, secant. cosecant)

- the number of faces on a cube

- the largest number of sides of a regular polygon that can tesselate a plane

- the third busy beaver number, preceded by 4 and followed by 13

- and more

For more on some of thse properties, see factorials at 24, primorials at 30, semiprimes at 14, abundant vs deficient numbers at 12, and perfect numbers at 496.

The prefixes for six are sex- (Latin, such as sextillion and sextuple) and hexa- (Greek, such as hexagon and hexahedron). "Hexa" is much more common than "sex" for 6 - you know why :)