Poly-cell notation

Poly-cell notation is a notation invented by Sbiis Saibian when he was in 2nd grade.A 2nd Graders Close Encounter with the Infinite 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 .
;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:
* = d. When r is zero, there is no b, so there is only d.
* = 
In other words, = with r copies of b (including the base b). This is equivalent to E(b)d#r in hyper-E notation.
Examples:
* = = = 256
* = = 1.1579208924... × 
* = million
* = googol
* = centillion
* or = trialogue
* = googolplex
* = decker
* = with 10 copies of 10
* = giggol
* = grangol
* = 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 = 
===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.
== Sources ==
== See also ==
*Extensible-E System
*Steinhaus-Moser notation
*Array notation
Category:Notations
Category:Functions
Category:Functions by Sbiis Saibian