2'' Squares

In mathematics, Steinhaus–Moser notation is a means of expressing certain extremely large numbers. It is an extension of Steinhaus's polygon notation. more...

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n inside n triangles, which are all nested."

n inside n squares, which are all nested."

etc.: n written in an (m+1)-sided polygon is equivalent with "the number n inside n m-sided polygons, which are all nested."

Steinhaus only defined the triangle, the square, and a circle Steinhaus defined:

"mega" is the number equivalent to 2 in a circle: ②; "megiston" is the number equivalent to 10 in a circle: ⑩;

Moser's number is the number represented by "2 in a megagon", where a "megagon" is a polygon with "mega" sides.

Alternative notations:

use the functions square(x) and triangle(x); let M(n,m,p) be the number represented by the number n in m nested p-sided polygons; then the rules are:

M(n,1,3) = nⁿ; M(n,1,p + 1) = M(n,n,p);

mega = M(2,1,5); moser =  Mega

Note that ② is already a very large number, since ② = square(square(2)) = square(triangle(triangle(2))) = square(triangle(2²)) = square(triangle(4)) = square(4⁴) = square(256) = triangle(triangle(triangle(...triangle(256)...))) = triangle(triangle(triangle(...triangle(256²⁵⁶)...))) = triangle(triangle(triangle(...triangle(3.2 × 10⁶¹⁶)...))) = ...

Using the other notation:

mega = M(2,1,5) = M(256,256,3)

With the function f(x) = x^x we have mega = f²⁵⁶(256) = f²⁵⁸(2) where the superscript denotes a functional power, not a numerical power.

We have (note the convention that powers are evaluated from right to left):

M(256,2,3) = Similarly:

M(256,4,3) ≈

etc.

Thus:

mega = f(n) = 256ⁿ.;

Read more at Wikipedia.org