User:SurrealWarrior/Draft of Jonathan Bower's Array Notation

Array Notation was invented by Jonathan Bowers as a way to represent large numbers. Array notation creates huge numbers with relativly small entries. Bowers uses his array notation to define his Infinity Scrapers.

Definitions
Array - structured set of entries (sometimes represented as A)

Entry - a position in the array that holds a positive integer value (1 is default)

Base - first entry (value represented by b)

Prime entry - second entry (value represented by p)

Pilot - first non-1 entry after prime entry (can be as early as 3rd entry)

Copilot - entry right before pilot (if pilot is the first entry on row, then there's no copilot)

Structures - structures are sub-arrays of the following types: entry (X^0), row (X^1), plane (X^2), realm (X^3), flune (X^4),..,X^n,..,dimensional group (X^X)=X^^2,..,n-D array of dimensional groups (X^(X+n)),..dimensional group of dimensional groups (X^(2X)),...X^(2X+n),...m-D array of dim.group of dim.group of.....dim.groups n times (X^(nX+m)),...dimensional gang (X^(X^2)),...X^(polynomial),...X^(X^X)=X^^3,....X^(X^X+polynomial),........X^^4,....X^^n,...X^^X,......X^^^X,....X^^^^````^^^X,...etc.

Previous Structures - previous entries are entries before the pilot but are on the same row, previous rows are rows before the row the pilot is on, but are in the same plane, previous n-spaces are n-spaces in the same n+1-space that are before the pilot's n-space, etc.

Prime Block - the prime block of an X^n structure is the first p^n block of entries in that structure, - this generalizes to all structures, turning the X's to p's, where p is the prime. - so if p=10, and the structure is X^(3X^5+2X^4+3X+6), the prime block is the first 10^(3*10^5+2*10^4+3*10+6) set of entries.

Airplane - a set of entries that include the pilot, all previous entries, and the prime block of all previous structures.

Passengers - any entry in the airplane that is not the pilot or copilot. This will always include the base.

Allowance
Default entries (entries that have a 1 value) can be chopped off as long as none of the entries change their positions.

Rules

 * 1) The prime rule - Condition: default prime (p = 1) - Result: A=b.
 * 2) The initial rule - Condition: no pilot (has two entries) - Result: A=b^p (in old version of array notation, it was A=b+p)
 * 3) The catastrophic rule - Condition: first two rules don't apply - Result: Array changes in the following way:
 * pilot - decreases by 1
 * copilot (if exists) - becomes original array, but with the prime entry reduced by 1
 * passengers take on base value
 * remainder of array remains unchanged

Linear
Array consists of one row, ex: {3,5,6,1,3,4,2} - these are the smallest and simplest arrays - even though the arrays are small, the value of the arrays can be so huge that not even Conway's Chained Arrow notation can keep up (an array with only 5 entries will clobber Chained Arrow!!). The positions on the array can be represented by a positive integer, i.e. the 17th position.

Dimensional
Arrays that have a multidimensional structure - these are the second smallest arrays. These arrays consist of rows, planes, realms, flunes (4-spaces), and various n-spaces. The rules above can easily handle dimensional arrays - although the result nearly reaches infinity. The positions in the array can be described with a linear array, i.e. (4,5,3,1,2) represents the 4th entry on 5th row on 3rd plane on 1st realm on 2nd flune. Dimensional arrays can be represented as follows: {4,2,5,7 (2) 5,6,1,2 (2) 5,4 (1) 6 (3)(3)(2) 2,5,6 (1) 1,1,1,7 (2) 6,8,5,2 (5) 7,5,6,8}. Here (2) represents going to the next second dimension (or going onto next plane), (3) is next realm, (5) is next 5-space. (Note: Theoretically, you should replace the commas in the above array with "(0)", but, in practice it is inconvenient.)

Tetrational
These arrays go way beyond dimensional arrays, and take on structures that require tetrational spaces. They consist not only of rows, planes, realms, etc, but also dimensional groups, rows of groups, planes of groups, groups of groups, groups of groups of groups of groups, gangs, rows of gangs, groups of gangs, gangs of gangs, realms of groups of groups of gangs of gangs of gangs, etc, etc, etc, superdimensional groups, trimensional groups, etc,etc. These are the third smallest arrays (and the largest type that I have fully grasped how they work). Tetrational arrays consist of super dimensional arrays, trimensional, quadramensional, etc in the same way dimensional arrays consists of planar, realmic, flunic, etc arrays.

Positions of superdimensional arrays can be described by a dimensional array, i.e. (5,6,3 (1) 3, 5 (1) 6,7 (2) 5) represents the 5th entry of the 6th row of the 3rd plane of the 3rd dimensional group, of the 5th row of groups, of the 6th group of groups, of the 7th row of group of groups of the 5th dimensional gang. To represent a 3^(3^2) array, let A represent the 3^3 dimensional group {3,3,3 (1) 3,3,3 (1) 3,3,3 (2) 3,3,3 (1) 3,3,3 (1) 3,3,3 (2) 3,3,3 (1) 3,3,3 (1) 3,3,3}, now a 3^(3^2) array of 3's would be {A,A,A (1,1) A,A,A (1,1) A,A,A (2,1) A,A,A (1,1) A,A,A (1,1) A,A,A (2,1) A,A,A (1,1) A,A,A (1,1) A,A,A} To get a 3^(3^3) array let A be equal to the group of groups mentioned above (the 3^(3^2) array that is) and let (1,1) and (2,1) change into (1,2) and (2,2) respectively.

Positions of trimensional arrays can be described by a superdimensional array, positions of a quadramensional array can be described by a trimensional array, etc.

Pentational
Next in line are pentational arrays, these require pentational spaces. Trying to understand how these work might give someone a migrane. One would need to keep up with tetration levels which can also get out of hand. There will be tetrational groups that show up here, X^^n arrays are more tidy examples of tetrational groups. a X^X^X^X^X array is a tetrational one, quintamensional to be exact (keep in mind that powers are solved in reverse) - for pentational arrays, the powers will sort into groups, i.e. an X^X^X^X^{X^X^X^X^X}^{X^X^X^X^X}^{{X^X^X^X^X}^{X^X^X^X^X}^{X^X^X^X^X}^{X^X^X^X^X}^{X^X^X^X^X}} array, the {} are not to be solved like ordinary parenthesis, but are used to group up the exponents into tetrational blocks - the number of entries in this array can be found out by removing the {} and solving - ie. X^^39 in this case - and this is barely scratching the beginning of pentationals.

Beyond Pentationals
Hexational, heptational, expandal, multiexpandal, explosal, powerexplosal arrays, etc. arrays so huge that the space its in needs to be represented by array notation (linear, dimensional, tetrational, and so forth) - how to work with these? - Only God knows - but they should form some massive arrays - and utterly unspeakable numbers when solved.

The Next Legion
Let & represent "array of", so 3 & 3 = {3,3,3} = tritri = 3 to the power of itself 3^27 times - now consider 3&3&3 (solved frontwards) this is a {3,3,3} array of 3's which is a 3^^^3 array of 3's - now imagine a new array notation such that the second rule changes to this:

Condition - no pilot, result - v(A)=b&b&b&b&.....&b&b -- p times - welcome to the next level of array notation.

Another way to represent this is to simply add an entry with value 2 in a second "legion", where the first legion can take on any of the previous arrays - {3,3 / 2} = 3&3&3, the big boowa = {3,3,3 / 2} = 3&3&3&3&3&............&3 - where there are 3&3&3.......&3 3's arrayed to each other - where there are 3&3&3.....3 3's arrayed to each other - where there are 3&3&3.....&3 3's arrayed to each other - ... - ... - ... - - - - this is said 3&3&3&3&......&3 times - where there are 3&3&3 3's arrayed to each other here before getting to the end (which is "where there are 3 3's arrayed to each other")!!! - keep in mind that in {3,3,3 / 2} 3,3,3 are in the first legion and 2 is in the second legion - this 2 represents "second level of array notation".

One can consider {b,p / 2} as if the pilot is in the 2nd legion, and in this case the airplane is defined to be a b&b&b...&b p times - array in the first legion - of course you can go to the 75th level of array notation if you like - {b,p / 75} = {b&b&b....&b - p times - array / 74}. Note that {A / 1} = A, so 1's are still default.

Linear Legion Arrays
{A1 / A2 / A3 / A4 / ..... An} - here each of the An's are arrays (can be linear, dimensional, tetrational, pentational, etc - up to but not including legion arrays) - nuf said!!!

Dimensional Legions, Tetrational Legions, Legion Legions, and beyond
One can take arrays further to include things like the following:

{A / A / A / A (/2) A / A / A (/3,4,5,2) A } - this is a tetrational legion array.

{3,3 // 2} - a legion legion array - this is equal to 3&&3&&3, where 3&&3 = {A / A / A} where A = 3&3&3 array (in other words 3&&3 is a 3 legion array of 3's), 3&&3&&3 = 3&&3 legion-array of 3's. - note that a mere 3^3 legion array of 3's would be {A / A / A (/1) A / A / A (/1) A / A / A (/2) A / A / A (/1) A / A / A (/1) A / A / A (/2) A / A / A (/1) A / A / A (/1) A / A / A} - which is nothing compared to a 3&&3 legion array of 3's

{3,3 /// 3} - a legion legion legion array - which equals 3&&&3&&&3, where 3&&&3 = {A // A // A} where A = 3&&3&&3 array of 3's

{3,3 //////////////// 2 } - use your imagination.

Linear
{2,2,3} =(Rule 3)

{2,{2,1,3},2} =(Rule 1)

{2,2,2} =(Rule 3)

{2,{2,1,2},1} =(Rule 1)

{2,2,1} =(Allowence)

{2,2} =(Rule 2)

2^2 = 4

In general: {a,b,c} = hyper(a,c+2,b) (see Hyper operator)

Links

 * Large Numbers at MROB
 * Jonathans's Home Page
 * Array Notation Page