By planar I mean there is no $K_{3,3}$ minor of $K_5$ minor. Also, I am only considering the $\mathbb{R}^2$ surface, not a torus not any other surfaces.

I know that to construct such graph, For $k \le 6$, it can be done easily by tesselation of regular $k$-gon and/or dual operation of graph. However, this construction does not work for $k \ge 7$ as regular $k$-gon cannot be "tightly packed together" (a little ambiguous here, I don't know how to explain what I am thinking, sorry about that) in $\mathbb{R}^2$.

Can anyone give me an idea on how to construct such infinite $k$-connected graph. A friend of mine gave me an idea of building a tree rooted at $v$ and extend it indefinitely such that some list of properties hold (I cannot remember all of them as there are several, but basically properties such as every vertex has to have degree at least $k$, etc)

P/S: I talked to my professor about this, and was told by the professor that any $k \ge 6$ has to be an infinite graph. But is there an example of 4-connected planar graph and 5-connected planar graph that is finite?