Four is, of course, equal to 3+1, or S(3) using the successor function. It is a number still small enough that even four of something in a row can be easily and immediately recognized, so it’s definitely in the primary range of numbers. However it’s large enough that words for four don’t usually get special names - an exception is "quarter" which is still based on four. Four is sometimes considered to be the limit of numbers you can recognize at once, though I personally prefer making that limit be 6.

Properties of the number 4

Four is yet another number with zillions of mathematical properties, which shouldn't be surprising with such a small number. For example, four is:

- the first composite number

- the second square number (2^2)

- the first non-trival perfect power

- the second busy beaver number, preceded by 1 and followed by 6

- the largest number of spatial dimensions to contain more than three convex regular figures (infinitely many regular polygons for two dimensions, five Platonic solids for three, six similar 4-dimensional solids for four, and all higher have only three)

- the first integer that isn't a Fibonacci number

- the period of the sequence of integer powers of i (the imaginary unit)

- the smallest number of colors that you can use to color any 2-D map without the same colors touching (the four-color theorem)

- the last digit of pi (not!)

Some of these properties are unique to the number 4, but some are further discussed in later entries on this list - perfect powers at 16, squares at 25, composites at 18, and the busy beaver function at 107 and 4098.

Four in our world

While a set of three has a special feel of symmetry to it, a set of four has a special feel of completeness to it - things are very often grouped in fours. For example there are:

- the four seasons (spring, summer, fall, winter, picture right)

- four alchemic elements (fire, earth, air, water

- the four cardinal directions (north, south, east, west)

- four corners of a square

- four bases in DNA and RNA

- four legs on many animals

- four human blood types

- the Fantastic Four superheroes

- the four Teenage Mutant Ninja Turtles

- the Beatles were a band of four people

- 4 suits in a card deck (see also 52)

- four presidents on Mount Rushmore

- etc, etc.

In the Chinese language 4 is considered unlucky because the word for 4 sounds similar to the word for death. The only difference is the tone of the word - four in Chinese uses the falling tone, and death uses the falling-and-then-rising tone. There are other similar examples of Chinese numerology, which interestingly relates to the language itself, such as:

"three" sounds like "birth"

"five" sounds like "not"

"six" sounds like "flow"

"eight" sounds like "wealth"

"nine" sounds like "long lasting"

Numbers can even be combined in meaning - for example 54 can be lucky because it sounds like "no death". Not all Chinese numerology relates to the language though. For example, 5's luck is associated with the five Chinese classical elements, and 7 for reasons I'm not sure of represents togetherness.

The two main prefixes for four are tetra- (Greek, such as tetrahedron, tetration, and the famous game Tetris) and quad- (Latin, such as quadrillion, quadruple, quadrilateral). Both are very commonly used.

Four in googology

In googology, four shows up again and again when working with large number functions of twos because 2+2, 2*2, 2^2, 2^^2, 2^^^2, 2^^^...^^^2 with as many ^s as we please are all equal to four. Therefore larger arrays that start with 2 will all degenerate to 2^^...^^2 with however many ^s, which equals four. For example, even a giant-looking array like {2,500,64,3333,293467934,12904732948546209834652} still evaluates to just 4, because it starts with a two.

Other than that four isn't particularly significant in googology, though it does appear in googolisms' names like googolquadriplex and tetratri.

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