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 4

*pi*/

*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*= 2

*n*/(

*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*= 2

*n*/(

*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*= 2

*n*/(

*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*= 2

*n*/(

*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 2

*n*: 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.