User blog:Allam948736/Allam's Extended Steinhaus-Moser Notation

Steinhaus-Moser notation can be essentially thought of as a fast iteration hierarchy where f_0(n) = n^n, as is stated on the article here. Based on this, I defined my own extension of Steinhaus-Moser notation, where n in a circle is f_w(n) in the aforementioned hierarchy, or n in an n+3-gon. I denote n in a circle as n[w], because a circle can be thought of as a polygon with infinitely many (or omega) sides. Below are the first few values of n[w].

1[w] = 1[4] = 1 2[w] = 2[5] (mega) 3[w] = 3[6] (grand megision) 4[w] = 4[7] 5[w] = 5[8]

To continue, I moved up to 3-dimensional shapes, and defined n in a tetrahedron (or n[3, 3]) as n in n circles. In the hierarchy that Steinhaus-Moser notation forms, this would be f_w+1(n). 2[3, 3] is comparable to and "slightly" greater than Moser's number (being equal to mega[mega + 3]), and 64[3, 3] is greater than Graham's number.

I next defined n in a cube as n in n tetrahedrons (which would be f_w+2(n) in the hierarchy formed by Steinhaus-Moser notation). 2[4, 3] is already much greater than Graham's number and even G(G(1)), and it is a number that I refer to as the cube mega, from the shape that the 2 would be in and the number mega, and used Aarex's naming scheme for numbers defined with the regular Steinhaus-Moser notation to name 3[4, 3], 4[4, 3], 5[4, 3], etc. I then continued with dodecahedrons (which have pentagonal faces) and icosahedrons, and then defined n in a sphere (or n[w, 3]) as n[n+2, 3]. This has a growth rate of f_2w in the fast-growing hierarchy.

I then continued with 4-dimensional shapes, 5-dimensional shapes, 6-dimensional shapes, etc. Although it is impossible to actually visualize these, they can at least be denoted as n[m, k], where k is the number of dimensions. I then continued to an infinite number of dimensions, defining n[m, w] as n[m, n+1], and n[w, w] (that would be n in an infinite-dimensional sphere, as crazy as that sounds) as n[w, n+1] or n[n+2, n+1]. This achieves a growth rate of f_w^2 in the fast-growing hierarchy.

But I didn't stop there. I next defined n[3, 2, 2] as [w, w] applied to n n times, n[4, 2, 2] as [3, 2, 2] applied to n n times, and so on, until n[w, 2, 2] which is n[n+2, 2, 2], and continued with n[3, 3, 2] being [w, 2, 2] applied to n n times. In general, each time you increase the first number in the array by 1, you apply the previous operator the input number of times (that is, n[a+1, b, c] is [a, b, c] applied to n n times if a > 3). This is true for any number of arguments.

So let's skip to n[3, w, 2], which is equal to n[3, n+1, 2]. In general, n[a, w, b] is equal to n[a, n+1, b]. At n[w, w, 2] (which is n[w, n+1, 2]), we reach the end of the operators with 2 as the third argument in the array. This achieves the growth rate of f_(w^2)*2(n) in the fast-growing hierarchy, and to continue, I simply defined n[3, 2, 3] as [w, w, 2] applied to n n times. Eventually, we reach n[w, w, w], which has a growth rate of f_w^3(n) in the fast-growing hierarchy, and is equal to n[w, w, n], which is n[w, n+1, n], or n[n+2, n+1, n]

We can just add more arguments to the array after that. In general, n[3, 2, 2, 2, ..., 2, 2, 2] w/ m 2s is equal to [w, w, w, ..., w, w, w] w/ m omegas applied to n n times, and n[w, w, w,..., w, w, w] w/ m omegas is equal to n[w, w, w, ..., w, w, n] w/ m-1 omegas which is in turn equal to n[n+2, n+1, n, n, ..., n, n] w/ m-2 copies of n. This achieves a growth rate of f_w^w in the fast-growing hierarchy, on par with Bowers' linear arrays.