What I mean is a set similar to the menger sponge/carpet, but expanding, instead of intricate. Iteration $0$, $\ddot m_0$ is the unit cube/square from $0$ to $1$, and the following iterations are described by the following:
$\begin{array}[llclclcl]\\ \ddot m_0 & \ni (x,y) & \iff & 0 < x < 1 & \land & 0 < y < 1 \\ \ddot m_{n + 1} & \ni (x,y) & \iff & 0 < x < 3^{n + 1} & \land & 0 < y < 3^{n + 1} & \land &(x \mathbin{\text{mod}} 3^n, y \mathbin{\text{mod}} 3^n) \in \ddot m_n\\ \end{array}$
Does the set $\ddot m_\infty = \lim_{n \to \infty} \ddot m_n$ exist, have any special properties or is in any other way relevant? Most importantly, however, is it a fractal, and, if it's a known set, what is its name?