Poly-cell notation

Poly-cell notation is a notation invented by Sbiis Saibian. This is an early version of Hyper-E Notation, which is more compactly and extendable.

Stack notation
Stack notation is an extension of scientific notation. It is a triadic operator with "base" (written as square), "replicator" (witten as bar), and "determinant" (written as triangle):



In ASCII form:

[10][10]

This means "base b repeated r times with d at the top", for example:

[2][2]<3> = 22 3 [2][3]<3> = 22 2 3   [10][10] = 1010 ... 10 303   (with 10 tens)

Some named numbers can be expressed in this form:

Googol = [10][1] or [10][2]<2> Googolplex = [10][2] or [10][3]<2> Decker = [10][10]<1> or [10][9]<10> Giggol = [10][100]<1> or [10][99]<10> Grangol = [10][100] (E100#100 in Hyper-E) Tritri = [3][7625597484987]<1> or [3][7625597484986]<3>

This notation has tetrational growth.

Diamond notation
Diamond notation is an extension of stack notation. It based in fact that stack notation might be embedded to itself arbitrary number of times:

[10][10] [10][[10][10] ] [10][[10][[10][10] ] ]

And diamond notation has following form:



In ASCII form:

/303 | 4\ \10 |10/

To comprehend how it works consider a few examples:

/303 | 1\ \10 |10/ = [10][10]

/303 | 2\ \10 |10/ = [10][[10][10] ]

/303 | 3\ \10 |10/ = [10][[10][[10][10] ] ]

/303 | 4\ \10 |10/ = [10][[10][[10][[10][10] ] ] ]

In otherwords, 2nd replicator defines a number of "replicator embeddings" in stack notation.

This notation has pentational growth.

5-cells and beyond
5-cell notation can determine number of "2nd replicator embeddings" in diamond notation and has hexational growth:



6-cell notation can determine number of "3rd replicator embeddings" in 5-cell notation and has heptational growth.

7-cell notation can determine number of "4th replicator embeddings" in 6-cell notation and has octational growth.

And in general:

n-cell notation can determine number of "(n-3)-th replicator embeddings" in (n-1)-cell notation and has (n+1)-ational growth.

And clearly picture of poly-cell notations (poly-cell hierarchy):