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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

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> = bb...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> = 223 = 28 = 256
• [2][3]<3> = 2223 = 1.1579208924... × 1077
• [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> = 1010...10303 with 10 copies of 10
• [10][100]<1> = giggol
• [10][100]<100> = grangol
• [3][327]<1> = tritri

Stack notation exhibits tetrational growth with respect to the replicator.

## Diamond notation

Diamond notation is an extension to stack notation created by nesting stack notation within itself. A second replicator r2 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, r2, r, b.

In ASCII, this is written as

/d|r2\
\r|b/.

Inline, this is written "a d, r2, r, b diamond."

It is recursively defined by the following:

• d, 0, r, b diamond = r
• d, r2 + 1, r, b diamond = [b][d, r2, r, b diamond]<d>

### Examples

/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

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.