Steinhaus–Moser notation


In mathematics, Steinhaus–Moser notation is a notation for expressing certain large numbers. It is an extension of Hugo Steinhaus's polygon notation, devised by Leo Moser.

Definitions

etc.: written in an -sided polygon is equivalent with "the number inside nested -sided polygons". In a series of nested polygons, they are associated inward. The number inside two triangles is equivalent to inside one triangle, which is equivalent to raised to the power of.
Steinhaus defined only the triangle, the square, and the circle, which is equivalent to the pentagon defined above.

Special values

Steinhaus defined:
Moser's number is the number represented by "2 in a megagon". Megagon is here the name of a polygon with "mega" sides.
Alternative notations:
A mega, ②, is already a very large number, since ② =
square = square =
square =
square =
square =
square =
triangle) =
triangle) ~
triangle) =
...
Using the other notation:
mega = M = M
With the function we have mega = where the superscript denotes a functional power, not a numerical power.
We have :
Similarly:
etc.
Thus:
Rounding more crudely, we get mega ≈, using Knuth's up-arrow notation.
After the first few steps the value of is each time approximately equal to. In fact, it is even approximately equal to . Using base 10 powers we get:
...
It has been proven that in Conway chained arrow notation,
and, in Knuth's up-arrow notation,
Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number: