User blog:Dhacorrea/Steinhaus-Moser: some 2's in step-by-step.

Steinhaus-Moser Notation is a well know notation created by Hugo Steinhaus, and extended by Leo Moser.[1]

My intention in this post is to show the resolution of some 2's inside polygons of Steinhaus-Moser notation. It´s well know that the first polygon is the triangle, and any number \(n\) inside a triangle is converted to the \(n\) power of \(n\) or \(n^{n}\).

Here we represent a value \(n\) inside a triangle as \(n[3]_{1}\), according  the  general notation  proposed by Susan Stepney. The notation extends to: ​Let´s see how fast the notation escalate the results from one \((k)gon\) to the next consecutive  \((k+1)gon\).
 * ​\(n\) inside a square or \(n[4]_{1}\) = \(n\) inside \(n\) triangles = \(n[3]_{n}\);
 * \(n\) inside a pentagon or \(n[5]_{1}\) = \(n\) inside \(n\) squares = \(n[4]_{n}\);
 * \(n\) inside a hexagon or \(n[6]_{1}\) = \(n\) inside \(n\) pentagons = \(n[5]_{n}\);
 * \(n\) inside a \((k)gon\) or \(n[k]_{1}\) = \(n\) inside \(n\) \((k-1)gon\) = \(n[k-1]_{n}\).
 * \(n\) inside a \((k)gon\) or \(n[k]_{1}\) = \(n\) inside \(n\) \((k-1)gon\) = \(n[k-1]_{n}\).

\(\text{Solving 2[3]:}\)

\(2[3] = 2[3]_{1} = 2^{2} = 4.\)

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\(\text{Solving 2[4]:}\)

\(2[4] = 2[4]_{1} = 2[3]_{2} = 2[3][3]_{1} = 4[3]_{1} = 4^{4} = 256.\)

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\(\text{Solving 2[5] or Mega:}\)

\(2[5] = 2[5]_{1} = 2[4]_{2} = 2[4][4]_{1} = 256[4]_{1} = 256[3]_{256},\)

\(256[3]_{256} = 256[3]_{1}[3]_{255} = 256^{256}[3]_{255},\)

\(256^{256} = 3.2317006071311007...\times10^{616} = 10^{616.50943111983348779773...},\)

\(256[3]_{256} = 256[3]_{2}[3]_{254},\)

\(256[3]_{2} = 256[3]_{1}[3] = (256[3]_{1})^{(256[3]_{1})},\)

\(256[3]_{2} = (256[3]_{1})^{(256[3]_{1})},\)