User blog:XXxKennyDashxXx/A NEW NOTATION! HBAN.

I am working on a new notation called "Hyperbracket Array Notation", or HBAN for short. It starts out almost identical to BEAF, but then shoots out into its own direction! It will become an actual array notation after a while. So, without further ado, here it is!

Hyperbracket Array Notation (HBAN) Definition
Rule 1: &[n] = n^^n

Rule 2: &[a][b][c][d]...[y][z] = &[a][b][c][d]...[&[a][b][c]...[......][z-1][z-1]...[z-1] with a nests.

That's all! For growth rates and numbers defined by me with HBAN and extensions, check out the bottom of the page! HOWEVER, there is more! Let's extend this to...

Extended Hyperbracket Array Notation (EHBAN) Definition
Let @ be the rest of the array.

Rule 1: &{1}[n] = &[n][n]...[n][n] with n entries.

Rule 2: &{n}[m] = &{n-1}{n-1}{n-1}...{n-1}{n-1}{n-1}[m] with m {n-1}s.

Rule 3: &{a,b,c...y,z}[n] = &{a,b,c...&{a,b,c... ...y,z},z-1},z-1}...} with n nests.

Rule 4: &{a&b}[n] = &{a,a,a...a,a,a}[n] with b entries.

Here, it's still the same as BEAF. Also define a &{a,b(1)c}[n], and so on to match with the HBAN array of operator variation.

Rule 5: &{a&&&...&&&b}[n] with m &s = &{a&&&...&&&a&&&...&&&a}[n] with m-1 &s and b & groups.

Rule 6: &{a^b}[n] = &{a&&&...&&&a}[n] with b &s.

Next, define it with the same rules of Extended Cascading-E Notation, up until &{a&^^^...^^^&b}[n]

And that concludes the EHBAN rules. BEAF legions only match up with &{a&&b}[n], or the number of single &s.

Remember, this notation is a W.I.P. Definitions may be slightly ill. Next, we have:

Geometrical Hyperbracket Array Notation (GHBAN) Definition (informal)
From this point on, currently (will change) rules are not official.

&{n^^^...^^^n}[n] with m ^s = {n,m(●) 2}

Next, there's {n,m( ●) 3} which is {(n,n( ●) 2)( ●) 2)...} with m nests.

Continuing, there's {n,n( ●) (n,n( ●) n,n...)...)} with m nests, which is {n,m( ● ●) <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">2}

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">Using the Hyper-E rules again, we have: {a,b( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●^ <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●)2} = {a,a( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">● <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">● <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●... <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">● <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">● <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●)a} with b  <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●s.

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">...

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">Then, {a,a( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●^^^...^^^ <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●)a} with b ^s = {a,b( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●_2)2}

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">Do the entire process again.

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">{a,a( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●_2^^^...^^^ <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●_2)a} with b ^s = {a,b( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●_3)2}

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">{a,a( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●_{a,a( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●_...)a} with b nests = {a,b( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">♦)2}

Do the same process again.

Next...

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">{a,a( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">♦_{a,a( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">♦_...)a} with b nests = {a,b( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">▲)2}

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">And so on.

That's the end of GHBAN! Next comes...

Alphabetical Hyperbracket Array Notation (AHBAN) Definition (informal)
This extension uses Greek letters to diagonalize over lower-class Greek letters! So here's a very informal definition of it!

We start with: {a,b( <span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Ξ)2} = {a,a(b-th symbol)a} where the sequence of symbols is { <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●, <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">♦, <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">▲...} It's easy to remember after the first two, because it's also the number of sides that the symbol has. After the triangle, it would be a sqaure, and then a pentagon, and so on. Examples:

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">{3,1( <span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Ξ)2} = {3,3( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●)3}

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">{3,2( <span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Ξ <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">)2} = (3,3( <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">♦)3}

{3,10( <span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Ξ)2} = {3,3(some kind of filled-in decagon)3}

<span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Do the same thing until...

<span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">{a,a( <span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Ξ^^^...^^^ <span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Ξ)a} which would be {a,a( <span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Ξ_2)a}.

<span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Next, {a,a( <span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Ξ_...a)a} with b subscript nests = {a,b( <span style="color:rgb(0,0,0);font-family:sans-serif;line-height:22.399999618530273px;">Ξ <span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">●)2}

<span style="color:rgb(51,51,51);font-family:'OpenSans','HelveticaNeue',Helvetica,Arial,sans-serif;line-height:20px;">Under construction, being worked on RIGHT NOW! Seriously! This is just a save so that I don't accidently delete the whole thing!