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Exploding Dimension Realm

The following numbers describe steps in computing a notation.

0*.

{0, …} =>

Relating to Knuth’s Arrow Notation.

x = ( (cardinality of …) - 1) / 2

x is initially drawn up as the number just left of centre.

Nx+1 = (Nx ^Nx+1 Nx+2)

Each step, the centre point is redrawn as each central bracket is solved for:

3 ^3 3 *3

3 ^(3 ^3 3) 3 *5

3 ^(3 ^(3 ^3 3) 3) 3 *7

3 ^(3 ^(3 ^(3 ^3 3) 3) 3) 3 *9

...

1.

{..., {N1, N2, …, Nn-1,1}, …} => {..., {N2, …, {N1, N2, …, Nn-1}}, …}

2. {..., {N1, N2, …, Nn}, …} => {..., {N2, …, {N1, N2, …, Nn-1}}, …}

3.

{..., {N1, Nn}, ...} => {..., N1, Nn, ...}

4.

{..., {N}, ...}, {N} => N

5.

{..., {1, N2, …, D, …, Nn}, …} => {..., 1, N2, …, …, Nn, …}

6*.

{D, …, D, …, Nn} => {D-1, …, …, Nn}


D describes the main array’s dimension, and is carried forth. It only matters decrementally, and for remerging.


Examples:

{1, 3, 3, 3} => {1, 3, 3, {1, 3, 3, 2}} (2) => {1, 3, 3, {1, 3, 3, {1, 3, 3, 1}}} (2) => {1, 3, 3, {1, 3, 3, {1, 3, 3}}} (1) => {1, 3, 3, {1, 3, 3, {1, 3, {1, 3, 2}}}} (2) => {1, 3, 3, {1, 3, 3, {1, 3, {1, 3, {1, 3, 1}}}}} (2) => {1, 3, 3, {1, 3, 3, {1, 3, {1, 3, {1, 3}}}}} => {1, 3, 3, {1, 3, 3, {1, 3, {1, 3, 1, 3}}}} (3) => {1, 3, 3, {1, 3, 3, {1, 3, 1, 3, 3}}} (5) => {1, 3, 3, {1, 3, 3, 1, 3, 3, 3}} (5) => {1, 3, 3, 1, 3, 3, 3, 3, 3} (5) => {0, 3, 3, 3, 3, 3, 3, 3} (6) => 3 ^(3 ^(3 ^3 3) 3) 3 (0)

Generally, arrays with N-items equal to D are very regressive. Equally, arrays with smaller ranges tend to be regressive. Higher Ds usually make larger numbers.

{2, 5, 5} => {2, 5, {2, 5, 4}} => {2, 5, {2, 5, {2, 5, 3}}} => {2, 5, {2, 5, {2, 5, {2, 5, 2}}}} => {2, 5, {2, 5, {2, 5, {2, 5, {2, 5, 1}}}}} => {2, 5, {2, 5, {2, 5, {2, 5, {2, 5}}}}} => {2, 5, {2, 5, {2, 5, {2, 5, 2, 5}}}} => {2, 5, {2, 5, {2, 5, 2, 5, 2, 5}}} => {2, 5, {2, 5, 2, 5, 2, 5, 2, 5}} => {2, 5, 2, 5, 2, 5, 2, 5, 2, 5} => {2, 5, 5, 5, 5, 5} => {1, 5, 5, 5, 5, 5} => {1, 5, 5, 5, 5, {1, 5, 5, 5, 5, 4}} => ...

{2, 8, 5, 9} => {2, 8, 5, {2, 8, 5, 8}} => {2, 8, 5, {2, 8, 5, {2, 8, 5, 7}} => … => {2, …, 2, ...} => {1, …, …} => … => {1, …, 1, ...} = {0, …, ...}


Using another array for the value of D is especially powerful.

{{9, 6, 3, 9}, ...} => … => {{9, 6, 3, 9}, …, {9, 6, 3, 9}, ...} => {{9, 6, 3, 9}, …, ...} => {{9, 6, 3, 9} - 1, …} => …

Nesting arrays within each other, esecially into the place of D, creates even more powerful structures.

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