You can also make up other tilings with these rings of pentagons - to get the one below to work you have to sneak in some dented or overlapping pentagons.
But which regular n-gons can form rings like this? You obviously can't do it with a square.
And some regular n-gons, like heptagons, nonagons, decagons, and hendecagons (11-gons) don't work either.
All the angles of the regular n-gon are (n-2)pi/n - so the angles of the polygon in the center would have to be 4pi/n, but for that interior polygon to be a regular polygon itself, there must be some k for which the angle is also (k-2)pi/k. Equating these two values and solving for k gives k = 2n/(n-4). If we look for n that give integer values for k, then we have the n-gons that can form this sort of ring.
Which tells us that only the pentagon, hexagon, octagon, and dodecagon can form a ring around another regular n-gon (the regular decagon, hexagon, quadrilateral, and triangle, respectively). Coincidentally, these are the same polygons that can form a dented pinwheel, as described here.
But what if we skip over another edge (so 2 are skipped over) while forming the ring? We end up getting a star instead of a polygon in the center, and the smallest regular polygon this works for is the heptagon:
With a little bit of work, you may believe that this will work for n that give integer values for k = 2n/(n-6), and this turns out that those n values correspond to the regular heptagon, octagon, nonagon, decagon, dodecagon and octadecagon (18-gon).
But we can go further, and skip over another edge (3 now) when forming the ring of polygons. The center is no longer a star, but a bumpy gear-like polygon, and the smallest regular polygon that can do this is the nonagon:
What other polygons can form this third kind of ring where 3 edges are skipped? Our function is now k = 2n/(n-8), and we get integer values for n = 9, 10, 12, 16, and 24.
Our hendecagons still won't form a ring when skipping 3 edges, but will once we start skipping 4.
If we skip m edges when putting the ring together, we can find the number of regular n-gons that will form the ring using the formula k = 2n/(n-2(m+1)), and will only get closed rings when k takes on integer values.
From this relationship we can find out a few things about these rings. For example, for any odd n, where n is 5 or more, we can form a ring by skipping (n-3)/2 edges and have a ring of 2n: for regular pentagons, we skip 1 edge and get a ring of 10, for heptagons we skip 2 edges and get a ring of 14, and for regular hendecagons, we skip 4 edges to get a ring of 22. Another observation: the eminently factorable 12 allows the dodecagon to form rings of 3, 4, 6, or 12.
I was lead to this while playing with regular heptagon, having fun making rings (and rings of rings) like the ones below.