User blog:Kaptain Kerbab/A slightly better Array Notation

(Insert witty array name that may or may not be added later)

Rules

1) x•1 = x^x

2) x•2 = ((...((x•1)•1)...)•1)•1 with x brackets (equivalent to x^^x)

3) x•(n+1) = ((...((x•n)•n)...)•n)•n with x brackets

Generally x•n is roughly x{n}x in Extended Array Notation

Next:

4) x•x•1 = x•(x•(...x•(x•(x•x))...)) with x brackets

For a chain of length n+1 it converts into x chains of length each one incrementing the n th link.

E.g.

3•3•2 = 3•(3•(3•3)) = 3•7625597484987 ≈ 1.26*10^3638334640024 and

3•3•3•3• = 3•3•(3•3•(3•3•(3•3•3))) Quite large in everyday terms.

Now to diagonalize over those chains we can define:

5) x••1 = x•x•x•...x•x with x x's

6) and x••(n+1) = ((...((x••n)••n)...)••n)••n

Generally x••...••(n+1) with m number of •'s becomes (...((x••...••n)••...••n)...)••...••n)••...••n with x brackets and x••...••1 with m+1 •'s becomes x••...••x••...••x ... x••...••x with x number of ••...•• (with n •'s)

Another Diagonalization ! :D

x◊1 = x••...••x with x •'s

This symbol follows the rules 1 to 6.

We can keep adding symbols until the heat death of the observable universe but there is a better way of expressing this. Now we can define linear arrays.

7) [a, b, c, d] = a...b with c number of 's. "d" denotes the "level" of the symbol . Each symbol with a level n+1 can be reduced to another symbol with a level of n using the first six rules.

Examples

[3, 3, 3, 3] = 3∆∆∆3 (where ∆ is a symbol with a level of 3)

= 3∆∆(3∆∆(3∆∆3))

This is the end of the "linear array" and if I have enough time (and motivation) I may extend it.