Another Way of Thinking About Hyper Operators (Tetration and Pentation)

The common definition of hyper operators refers to the previous hyper operator. For example, 3 ↑↑ 3 equals 3 ↑ 3 ↑ 3, and generally a↑^bc = a↑^b-1 a ↑^b-1 a ...↑^b-1 a with c a's overall.

This isn't particularly intuitive, given that it is difficult to imagine tetration and the numbers it generates (e.g.  10↑2 is a number too large to represent using digital notation even if every single particle in the universe was used to write it down).

However, they can also be expressed using power towers, and low-quality ASCII art. Firstly, observe that:
For Tetration:
              10 <-
             10   /
           ...   / 10
10       10     /
  10 = 10     <-

10 tetrated to ten is equal to a power tower of 10s. Generally, a tetrated to b is equal to a power tower of a with a height of b. Let's continue with pentation:
For Pentation:
             10 <-         <-    10 <-
            10   /          /   10   /
           ...  /- . . .   /-  ...  /- 10
          10   /          /   10   /
10↑↑10 = 10  <-         <-   10  <-
         ^                              ^
         |------------------------------|
                       10

If this is difficult to read, then notice that it is a power tower of 10s, with a number of tens equal to a power tower of tens... and so on.

Another to think of this is to imagine the sequence:

• 10 (a power tower of tens with a height of 1)
• ¹⁰10 (a power tower of tens with a height of 10)
• ¹⁰¹⁰10 (a power tower of tens with a height of ¹⁰10)

And so on, until you reach the tenth term in this sequence. That number is 10 pentated to 10.

You can actually continue in the diagrammatic method forever - for example, 10 hexated to 10 is equal to the above diagram for pentation, but the "10" in the length of the power-tower-chain is replaced with ten copies of the pentation sequence. And so on.

--Thomas