Tetration stacks exponents into towers that explode almost immediately
Multiplication is repeated addition and exponentiation is repeated multiplication. Repeat exponentiation and you get tetration, where a number is raised to itself again and again. Stack just four 2s this way and the tower already works out to 65,536, which is why the operation is used to write down enormous numbers.
The calculation runs from the top of the tower downward. For four 2s, start with 2 to the 2, which is 4; then 2 to the 4 gives 16; then 2 to the 16 gives 65,536. The number of copies stacked is called the height and the repeated number is the base, so this result is read as the fourth tetration of 2. The top-down direction is essential, and mixing it up is a common source of confusion with related expressions called iterated exponentials.
Tetration sits in a ladder of operations called hyperoperations. Counting up by one is treated as the zeroth rung, then come addition, multiplication and exponentiation, with tetration fourth and pentation next. Each rung iterates the one below, but tetration is the first that is not an elementary function. It has two inverse operations, the super-root and the super-logarithm, playing roles like ordinary roots and logarithms.
The name comes from Reuben Goodstein, who blended the prefix tetra, meaning four, with iteration in a 1947 paper on transfinite ordinals, and Rudy Rucker's book Infinity and the Mind helped popularise it. Rivals never caught on as widely. Superexponentiation appeared in Ed Nelson's 1986 book Predicative Arithmetic and in a 1987 paper by Bromer, while power tower is still used informally.
In its original form the height must be a whole number; raising four to itself half a time makes no obvious sense. A recursive definition, in which zero height gives 1 and each step raises the base to the previous result, opens the door to extending tetration to real, complex and even ordinal heights, and many of those extensions remain active research.
Source: Tetration
