Tetration: A concept so negligible even my math teacher has never heard of it. Up until now, you probably have never heard of this arithmetic operation. You’ve probably scrolled the internet trying to figure out what it meant because the concept is never used in a traditional educational setting. Does it serve a purpose, or should we have just stopped at exponents?
To put it simply, tetration is the act of repeating the exponentiation of a base number to a certain degree, or:
For example, if you were to have 32, or 2 tetrated to 3, that would be equivalent to 2^2^2, or 16.
Another common notation for this operator is the arrow notation: x ↑↑ y, where x is the base and y is the degree of repetition.
The idea behind tetration was that it represents an extension of exponentiation, just as exponentiation is an extension of multiplication, and multiplication is an extension of addition. Just as multiplication is just a x n = a + a + a (n-times) and exponentiation is just a^n = a x a x a (n-times), tetration is simply na = a^a^a (n-times). The term tetration is also a type of iterated exponentiation.
But the real question is: does it serve as useful or not? Short answer, no. Long answer, maybe.
One of the main qualities of this operator is that the numbers inflate incredibly quickly. One moment you’re starting with the expression 32, equal to 16, and the next moment you find yourself with the expression 52, equal to 2^65536 or 2 x 10^19,728 (for context, there are estimated to be only 10^78 and 10^82 atoms in the entire universe). Therefore, the physical applications of tetration are quite limited.
One of the coolest things about tetration (besides its name) is its fractal properties. For those who are unfamiliar, fractals are patterns resulting from iterating (or repeating) a process, shape, or mathematical equation again and again, creating a feedback loop where the solution of one iterated process becomes the input for the next. This is then graphed onto the complex plane, making some pretty cool visual patterns.
This specific fractal is denoted as the power tower fractal.
On a basic level, the outreach of tetration is quite limited as a computational tool. To put it in the words of a Redditor, “pretty much the only times it ever comes up in math discussions are when people post about it on Reddit”. One of the most practical applications of the operator I found on the internet was to represent Graham’s number, basically an upper limit on the answer to a problem that arose in Ramsey Theory, a branch of combinatorics.
Although, one of the reasons it may appear as a dead end is mainly due to its chaotic nature. Tetration has been studied since the era of Euler, however, very few breakthroughs have been made, except in research specifically studying the convergence of infinite tetration, because of the lack of computers. The erratic behavior of this seemingly useless operator is incredibly difficult, almost nonsensical, to track and research without using a device that can store and manipulate large amounts of data. Additionally, most people can’t even agree on the basic definition of tetration past integer-based values. 32 may be easy to evaluate, but what about e2? How do you repeat the exponentiation of 2 e times? Once the tetration of non-integer heights is defined, it may be possible that new applications of the operation will be discovered. And the journey of arithmetic succession doesn’t even end there! After that, we have pentation, then hexation, then heptation, and so on. Good luck trying to find any realistic applications of those!