now, what kind of imaginary mario could exist that is both 1) distinct from both mario and wario, but in a way where reapplying the process that obtained that mario gives you wario, and 2) has only one brother-mario with properties of both a luigi and a waluigi?

i posit: sonic the hedgehog

these establish two important properties of marios in this particular imaginary space:

first, the imaginary mario 𝑰(x) is in some sense halfway between the base mario x and the wario 𝑾(x) of that mario, but in a rotational kind of halfway and not a scalar kind of halfway

second, the luigi 𝑳(x) of an imaginary mario is its own wario 𝑾(x). this suggests that imaginary marios only need to have one corresponding brother, and that brother has both luigi-like and waluigi-like properties

but we're not here to model canonical marios, we are here to invent new ones.

let's suppose the existence of an imagination function 𝑰(x) capable of producing imaginary marios. a simple definition of this function would be:

𝑰(x) = 𝑖 * x

using this definition, we can again explore some identities:

𝑰(𝑰(x)) = 𝑖 * 𝑖 * x = -x = 𝑾(x)
𝑳(𝑰(x)) = 1 / (𝑖 * x) = 𝑖 / (𝑖 * 𝑖 * x) = - 𝑖 / x = 𝑾(𝑳(𝑰(x)))