Here is some information on dimensional squares:
A dimensional square, like cube sets, should be treated as a set, where the set is defined by [math]\partial_{n}[/math]. More specifically, the partial sets would be defined by , where a represents the two parameters [math]a_{f}[/math] and [math]u_{f}[/math], while b represents the two parameters [math]b_{f}[/math] and [math]v_{f}[/math]. The partial set notation for the first two parameters is and the notation for their inverse is .
There is a way to analyze the parameters of a dimensional, which is to divide the two partial sets of the dimensional square.
If the parameters were the Collatz parameters, the following would occur.
This can be become a function, which will describe the properties of the first partial square compared to the second partial square, or the inverse. It will also describe the cube set that encompasses these two partial squares.
[math]\Upsilon_{\mathbb{W}}(a,b,u,v)=\frac{\partial_{a\bigsqcup u}}{\partial_{b\bigsqcup v}}[/math]
The following would be a representation of a function from the above example of the Collatz parameters.
[math]\Upsilon_{\mathbb{W}}(\frac{x}{2},\frac{x-1}{3},3x+1,2x)=\frac{\partial_{a\bigsqcup u}}{\partial_{b\bigsqcup v}}[/math]
[math]\Upsilon_{\mu }(x)=\frac{-2x-1}{2x^{2}-2x}[/math]
This function would output the following graph.
In this graph, the parabola closest to the top represents the first partial set of the set of parameters. The lower parabola represents the inverse parameters. This function will apply to all cube sets.[math]\Upsilon_{\mathbb{W}}(x)=\Upsilon_{\Im}(x)[/math]
This shows the commonality of cube sets. This sets a basic blueprint for all number systems that exist.