# Create parameterizable map between $\mathbb{Z}$ and $\mathbb{Z}^2$

I would like to create a parameterizable map between $\mathbb{Z}$ and $\mathbb{Z}^2$. This map I'll call $M$ and the parameter I'll call $k$. $k \in \mathbb{Z}$ (but if there is a better space for $k$, I'm open to suggestions)

• $M(z, k) \rightarrow (x, y)$
• $M^{-1}(x, y, k) \rightarrow z$

The requirements for $M$:

• $M$ and $M^{-1}$ are defined for all values in their domains
• For any $(x,y)$ there exists at least one set of values for $z$ and $k$ such that $M(z, k) = (x, y)$

## My project

(for those who care about why I want to do this)

I'm a computer programmer, and I'm working on a side project (for fun). My idea is that I want to build a model of an "entity" inside of an "environment".

Entity

• Input: $z \in \mathbb{Z}$, $c \in \mathbb{R}, [0,1]$
• Output: $z' = z + \Delta z$

$\Delta z$ is calculated based on an internal algorithm that isn't important to the question.

Environment

• Internal State: $x, y \in \mathbb{Z}$
• Input (Output From Entity): $z'$
• Output (Input To Entity): $M^{-1}(M(z',k),k) = z$ and $c = e^{-|M(z',k)-p*|}$