Frontier of the Undefined

Without division it does not exist

With x, 0, and any operations, it is a line of infinite length

If you add y, it becomes a square with infinite lengthed sides

By mixing variables, this frontier can become a any shape, as far as I can imagine, and still represents a map of the limit nodes where the observer knows, and can represent mathematically, that math cannot define the result.

I'm realizing that the juncture, at least for equations with only x, only x and y, and only x, y, and z- with a single instance of zero in each case would have a "point of maximum instances of failure" at the center of the shape (line, square, cube- at zero). This moves from 2 extremes with only x (infinity, negative infinity, along the line), to 4 extremes in the square, to 8 extremes for the cube. It also represents 2 failures for x and 0 inputs- x/0 and 0/x when x=0- whereas the remainder of the failure points only represent 1 failure each (x/0). When you combine x, y, and 0 though, the central failure point of x and y equaling zero represents x/y, y/x, xy/0, 0/xy, (x+y)/0, (x-y)/0, etc. Most, but not all of these failure arrangements at the central point would produce a shape of its own within the cube. Some arrangements, like 0/(x+y), would form shapes along the values of x=-y, while others like x/(y+0) would form a line, where x is any value but y must remain 0.

I am also wondering how valid it would be to represent the frontier limit as a circle or sphere, rather than a square or cube. If each value would presumably "expand" at the same rate (picturing an actual "ticking" of time in expressing the limit as each value approaches infinity), then this evenness might be accurately represented with a radius of infinite length, rather than sides of infinite length.

I am now realizing that the manager of mathematics can be used here to express a specific location or shape that mathematics cannot directly access. In this way something can be known in one sense quite precisely, but cannot be accessed directly by the one communicated with, despite a communication method having been clearly established. This is effectively, from a pure logic perspective, the Double Faraday Cage, as presumably this would be true from every acting perspective, and perhaps every possible (possible acting?) perspective. This would mean that this effect cannot be undone as a whole, as the shape of nothing would necessarily persist even if everything reached a static/predictable state. A new pattern would be forged after such a state, but the more precisely this is done from every possible logical perspective, the less one can change the other, as these frontier lines of logic would become more precisely defined and absolute. Presumably there would be a limit to this process for an individual perspective, from the perspective of another.

What would be the shape of the undefined at the limit of all variables included in the base equation? Would this limit be infinite or fictionally infinite based on a limiting factor within the foundation of logic in reality?