Editing Knuth's up-arrow notation (section)

===Writing out up-arrow notation in terms of powers===
Attempting to write <math>a \uparrow \uparrow b</math> using the familiar superscript notation gives a [[tetration|power tower]].

:For example: <math>a \uparrow \uparrow 4 = a \uparrow (a \uparrow (a \uparrow a)) = a^{a^{a^a}}</math>

If <math>b</math> is a variable (or is too large), the power tower might be written using dots and a note indicating the height of the tower.

:<math>a \uparrow \uparrow b = {} \underbrace{a^{a^{.^{.^{.{a}}}}}}_{b}</math>

Continuing with this notation, <math>a \uparrow \uparrow \uparrow b</math> could be written with a stack of such power towers, each describing the size of the one above it.

:<math>a \uparrow \uparrow \uparrow 4 = a \uparrow \uparrow (a \uparrow \uparrow (a \uparrow \uparrow a)) = 
  \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{a} }}</math>

Again, if <math>b</math> is a variable or is too large, the stack might be written using dots and a note indicating its height.

:<math>a \uparrow \uparrow \uparrow b = 
  \left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\} b</math>

Furthermore, <math>a \uparrow \uparrow \uparrow \uparrow b</math> might be written using several columns of such stacks of power towers, each column describing the number of power towers in the stack to its left:

:<math>a \uparrow \uparrow \uparrow \uparrow 4 = a \uparrow \uparrow \uparrow (a \uparrow \uparrow \uparrow (a \uparrow \uparrow \uparrow a)) = 
  \left.\left.\left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
                     \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
                     \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
                     a</math>

And more generally:

:<math>a \uparrow \uparrow \uparrow \uparrow b = 
  \underbrace{
    \left.\left.\left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
                       \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
                       \cdots \right\}
                       a
  }_{b}</math>

This might be carried out indefinitely to represent <math>a \uparrow^n b</math> as iterated exponentiation of iterated exponentiation for any <math>a</math>, <math>n</math>, and <math>b</math> (although it clearly becomes rather cumbersome).

====Using tetration====

The Rudy Rucker notation <math>^{b}a</math> for [[tetration]] allows us to make these diagrams slightly simpler while still employing a geometric representation (we could call these ''tetration towers'').

: <math> a \uparrow \uparrow b = { }^{b}a</math>

: <math> a \uparrow \uparrow \uparrow b = \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{b}</math>

: <math> a \uparrow \uparrow \uparrow \uparrow b = 
   \left. \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \underbrace{\vdots}_{a} }} \right\} b</math>

Finally, as an example, the fourth Ackermann number <math>4 \uparrow^4 4</math> could be represented as:
: <math>\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{4} }} = 
        \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ ^{^{^{4}4}4}4 }}</math>