Here are a few pictures and speculations to add to the the great ones from @j.c.

Here are the pentagons which arise from starting with the central green one and doing the reflections which do not overlap any previous ones. 

[![enter image description here][1]][1]

These are just the outer frontiers of the $C_i.$ (Specifically, this shows the frontiers for $i \le 7$ and half the frontier for $i=8.)  It seems clear (though I do not claim a proof) that for a target point inside one of the pentagons one does best to roll out to it using just these pentagons. For a target point in one of the rhombi one should roll out to a neighboring pentagon and at the last step reflect over a previously unused (for reflections) edge. One can use one of the bordering pentagons closest to the center for the closer half of the rhombus and a small part of the further half.  


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Here is a central portion of the same thing with lines connecting the centers of pentagons which share an edge. The resulting hexagons tile the plane with a $5$-fold rotational symmetry.

[![enter image description here][2]][2]

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Finally, here is a picture of just these hexagons going out further. For each of the $5$ orientations, the hexagons of that orientation fall into $2$ connected components. One orientation is highlighted in blue.

[![enter image description here][3]][3]


  [1]: https://i.sstatic.net/1eDAj.png
  [2]: https://i.sstatic.net/Yk9mG.png
  [3]: https://i.sstatic.net/XAyrX.png