User blog:Y g 2/Hexxel Notation

So it's safe to say that I really don't understand a lot of googology. I came up with this notation to practice the whole "iterating a function and then diagonalizing it" thing, but I have no idea how fast it grows or if I made any critical errors. I probably did, which explains why it looks so confusing; or I might have accidentally invented someone else's notation, actually. No idea.



Anyone care to walk me through, please?



Warning: contains unicode hexagons, which might cause problems on some browsers.



...

STAGE ONE
At this stage the dots don't actually do anything, other than separate the numbers after a hex from the "b" at the end.

...

a• ⬡ •b = a ↑↑↑↑ b

a• ⬡ 2•b = a ↑↑↑↑↑↑ ...{a ↑↑↑↑ b arrows}... ↑↑↑↑↑↑ b

a• ⬡ 3•b = a ↑↑↑↑↑↑ ...(a ↑↑↑↑↑↑ ...(a ↑↑↑↑ b arrows)... ↑↑↑↑↑↑ b arrows)... ↑↑↑↑↑↑ b

....

(thus 3• ⬡ 64•3 = Graham's number)

...

a• ⬡⬡ •b = continues the monohex pattern, but nested a• ⬡ •b times

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬡⬡ 2•b = continues the monohex pattern, but nested a• ⬡ 2•b times

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬡⬡ 3•b = continues the monohex pattern, but nested a• ⬡ 3•b times

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬡⬡⬡ •b = continues the monohex pattern, but nested a• ⬡⬡ •b times

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬡2 •b = a• ⬡⬡⬡ ...(a• ⬡ •b hexes)... ⬡⬡⬡ •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬡3 •b = a• ⬡⬡⬡ ...(a• ⬡2 •b hexes)... ⬡⬡⬡ •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">...

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬢ •b = a• ⬡ (a• ⬡ •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬢ 2•b = a• ⬡ (a• ⬡2 •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬢ 3•b = a• ⬡ (a• ⬡3 •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬢⬡ •b = a• ⬡ (a• ⬡⬡ •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬢⬢ •b = a• ⬡ (a• ⬢ •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬢⬢⬢ •b = a• ⬡ (a• ⬢⬢ •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬢ 2•b = a• ⬢⬢⬢ ...(a• ⬢ •b hexes)... ⬢⬢⬢ •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a• ⬢ 3•b = a• ⬢⬢⬢ ...(a• ⬢2 •b hexes)... ⬢⬢⬢ •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">...

STAGE TWO
<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">Brackets get introduced here. The idea was to diagonalize over white hexagons and black hexagons.

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">...

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡ }1•b = a• ⬡ •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡ }2•b = a• ⬢ •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">...

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡ }3•b = a• ⬢ (a• ⬢ •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡ }4•b = a•{ ⬡ }3 {{sub|(a•{}} ⬡ {{sub|}3•b)}} •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡ }5•b = a•{ ⬡ }4 {{sub|(a•{}} ⬡ {{sub|}4•b)}} •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡⬡ }•b = a•{ ⬡ (a• ⬢ •b)}•b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡⬡ }2•b = a•{ ⬡ (a• ⬢ (a• ⬢ •b) •b)}•b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡⬡⬡ }•b = a•{ ⬡⬡ (a• ⬢ •b)}•b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡ }2•b = a•{ ⬡⬡⬡ ...(a•{ ⬡} •b hexes)... ⬡⬡⬡} •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬡ }3•b = a•{ ⬡⬡⬡ ...(a•{ ⬡ 2 } •b hexes)... ⬡⬡⬡} •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬢ }•b = a•{ ⬡ } (a• ⬢ •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬢ 2}•b = a•{ ⬡ } (a• ⬢2 •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬢ 3}•b = a•{ ⬡ } (a• ⬢3 •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬢⬡ }•b = a•{ ⬡ } (a• ⬢⬢ •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬢⬢ }•b = a•{ ⬡ } {{sub|(a•{}} ⬢ {{sub|}•b)}} •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬢⬢⬢ }•b = a•{ ⬡ } {{sub|(a•{}} ⬢⬢ {{sub|}•b)}} •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬢ 2}•b = a•{ ⬢⬢⬢ ...(a•{ ⬢ }•b hexes)... ⬢⬢⬢ }•b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•{ ⬢ 3}•b = a•{ ⬢⬢⬢ ...(a•{ ⬢ 2}•b hexes)... ⬢⬢⬢ }•b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;min-height:14px;">...

STAGE THREE
<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;min-height:14px;">Still working on this one. The dots actually do something now, but I don't think that this is the most efficient way to increase the function's growth rate.

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;min-height:14px;">...

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•• ⬡ ••b = a•{ ⬢ } (a• ⬢ •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•• ⬡ 2••b = a•{ ⬢ } (a• ⬢2 •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•• ⬡ 3••b = a•{ ⬢ } (a• ⬢3 •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•• ⬡⬡ ••b = a•{ ⬢ } (a• ⬢(a•⬢•b) •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•• ⬡⬡⬡ ••b = a•{ ⬢ } (a• ⬢(a•⬢(a•⬢•b)•b) •b) •b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•• ⬡ 2••b = as above, with a•{ ⬢} •b layers of subscript

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•• ⬡ 3••b = as above, with a•{ ⬢ 2 } •b layers of subscript

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a•• ⬢ ••b = a••{ ⬡ } (a•• ⬡ ••b) ••b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a••• ⬡ •••b = a••{ ⬢ } (a•• ⬢ ••b) ••b

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">

<p style="margin-top:0px;margin-bottom:0px;font-size:12px;line-height:normal;font-family:Helvetica;">a: ⬡ :b = a•••...(a•{ ⬢ }•b dots)...•••{ ⬢ } {{sub|(a•••...(a•{}} ⬢ {{sub|}•b dots)...•••}} ⬢ {{sub|•••...(a•{}} ⬢ {{sub|}•b dots)...b)}} •••...(a•{ ⬢ }•b dots)...•••b