**Poly-cell notation** is a notation invented by Sbiis Saibian when he was in 2nd grade.^{[1]} It is an early ancestor of the more powerful Hyper-E notation, which operates on a similar basis.

## Stack notation Edit

Stack notation is an extension of scientific notation. It is a triadic operator consisting of three arguments: a *base*, a *replicator*, and a *determinant*. The replicator is a nonnegative integer; the base and the determinant can be any real numbers, but they must be nonnegative integers in further extensions to the notation.

The base is drawn inside a square, followed by the replicator in a rectangle, and the determinant inside a triangle, as shown in the image. In plaintext, it is represented by [*b*][*r*]<*d*>.

- Definition

- [b] is a square.
- [r](in the second) is a rectangle.
- <d> is a triangle.

The notation is defined recursively with the following rules:

- [
*b*][0]<*d*> =*d*. When*r*is zero, there is no*b*, so there is only*d*. - [
*b*][*r*+ 1]<*d*> =*b*^{[b][r]<d>}

In other words, [*b*][*r*]<*d*> = *b*^{b...bd} with *r* copies of *b* (including the base *b*). This is equivalent to E(b)d#r in hyper-E notation.

Examples:

- [2][2]<3> = 2
^{23}= 2^{8}= 256 - [2][3]<3> = 2
^{223}= 1.1579208924... × 10^{77} - [10][1]<6> = million
- [10][1]<100> = googol
- [10][1]<303> = centillion
- [10][2]<10> or [10][3]<1> = trialogue
- [10][2]<100> = googolplex
- [10][10]<1> = decker
- [10][10]<303> = 10
^{10...10303}with 10 copies of 10 - [10][100]<1> = giggol
- [10][100]<100> = grangol
- [3][3
^{27}]<1> = tritri

Stack notation exhibits tetrational growth with respect to the replicator.

## Diamond notation Edit

Diamond notation is an extension to stack notation created by nesting stack notation within itself. A second replicator *r*_{2} is introduced to indicate the depth of this stack. For the notation, a square is rotated 45 degrees partitioned into four cells in which the four arguments are placed in reading order: *d*, *r*_{2}, *r*, *b*.

In ASCII, this is written as

- /
*d*|*r*_{2}\ - \
*r*|*b*/.

Inline, this is written "a *d*, *r*_{2}, *r*, *b* diamond."

It is recursively defined by the following:

*d*, 0,*r*,*b*diamond =*r**d*,*r*_{2}+ 1,*r*,*b*diamond = [*b*][*d*,*r*_{2},*r*,*b*diamond]<*d*>

### ExamplesEdit

/303| 1\

\10 |10/ = [10][10]<303>

/303| 2\

\10 |10/ = [10][ [10][10]<303> ]<303>

/303| 3\

\10 |10/ = [10][ [10][[10][10]<303>]<303> ]<303>

This notation exhibits pentational growth with respect to the second replicator.

## 5-cells and beyond Edit

In 5-cell notation, the third replicator determines the number of "2nd replicator embeddings" in diamond notation and has hexational growth.

In 6-cell notation, the fourth replicator determines the number of "3rd replicator embeddings" in 5-cell notation and has heptational 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.