Uniform rings of regular polygons.
Version of Thursday 9 July 2020.
Dave Barber's other pages.
Below are some examples of uniform rings of regular polygons, where "uniform" will be defined later. In figure 1, the image in the upper left is the only possible ring of pentagons. The other images are the four possible rings of dodecagons.
For polygons of many sides, diagrams in the plain style of figure 1 are not very helpful, because the polygons will look very much like circles. Hence figure 2 adds a numeral indicating the how many sides each polygon has, and draws partial spokes to emphasize its corners. Also, the sides on the interior of the ring are drawn doubled. This duplication makes the difference among the dodecagonal rings shown here easier to discern.
figure 2




1 inner side
 4 inner sides




3 inner sides
 2 inner sides
 1 inner side

The rings examined in this report are uniform in the sense that all polygons are equal and regular; and all polygons have the same number of inner sides. As a consequence, the centers of the polygons form a regular polygon in its own right.
By way of contrast, figure 3 shows a ring that is NOT uniform, because some polygons have more inner sides than others:
If n is prime, only one ring will be possible for an ngon; if n is composite, rings of different sizes will likely be possible. This observation can be made more precise, as follows.
A polygon ring can be characterized by the total number of sides (t) in each polygon, along with how many of them are inner sides (i). These two quantities lead to the basis for categorizing the rings in this report, which is the core (c), defined thus:
c = t − 2i − 2
The ring will succeed if and only if 2t is a multiple of c.
Some limitations:
 t ≥ 3 because these are polygons
 i ≥ 0
 if t is even, then 2i ≤ t − 4
 if t is odd, then 2i ≤ t − 3
 c ≥ 1
Figure 4 displays the only rings where the polygons have zero inner sides:
figure 5 core = 1



The table below gives numbers for several sequences of rings; the count is simply the number of polygons in the ring. The table also has links to sample images in the format of figure 5 above, where some rings are shown only in part to keep the image's file size from becoming excessive.
core = 1 images
 n  total sides  inner sides  count


1  3  0  6
 2  5  1  10
 3  7  2  14
 4  9  3  18
 n  2n + 1  n − 1  4n + 2


core = 2 images
 n  total sides  inner sides  count


1  4  0  4
 2  6  1  6
 3  8  2  8
 4  10  3  10
 n  2n + 2  n − 1  2n + 2


core = 3 images
 n  total sides  inner sides  count


1  9  2  6
 2  15  5  10
 3  21  8  14
 4  27  11  18
 n  6n + 3  3n − 1  4n + 2


core = 4 images
 n  total sides  inner sides  count


1  6  0  3
 2  8  1  4
 3  10  2  5
 4  12  3  6
 n  2n + 4  n − 1  n + 2


core = 5 images
 n  total sides  inner sides  count


1  15  4  6
 2  25  9  10
 3  35  14  14
 4  45  19  18
 n  10n + 5  5n − 1  4n + 2


core = 6 images
 n  total sides  inner sides  count


1  12  2  4
 2  18  5  6
 3  24  8  8
 4  30  11  10
 n  6n + 6  3n − 1  2n + 2


core = 7 images
 n  total sides  inner sides  count


1  21  6  6
 2  35  13  10
 3  49  20  14
 4  63  27  18
 n  14n + 7  7n − 1  4n + 2


core = 8 images
 n  total sides  inner sides  count


1  12  1  3
 2  16  3  4
 3  20  5  5
 4  24  7  6
 n  4n + 8  2n − 1  n + 2



Here is a further generalization, where n is any positive integer:
core an odd number


core  total sides  inner sides  count


1  2n + 1  n − 1  4n + 2
 3  6n + 3  3n − 1  4n + 2
 5  10n + 5  5n − 1  4n + 2
 7  14n + 7  7n − 1  4n + 2

c
 2cn + c
 cn − 1
 4n + 2


core an odd multiple of 2


core  total sides  inner sides  count


2  2n + 2  n − 1  2n + 2
 6  6n + 6  3n − 1  2n + 2
 10  10n + 10  5n − 1  2n + 2
 14  14n + 14  7n − 1  2n + 2

c
 cn + c
 cn ÷ 2 − 1
 2n + 2


core a multiple of 4


core  total sides  inner sides  count


4  2n + 4  n − 1  n + 2
 8  4n + 8  2n − 1  n + 2
 12  6n + 12  3n − 1  n + 2
 16  8n + 16  4n − 1  n + 2

c
 cn ÷ 2 + c
 cn ÷ 4 − 1
 n + 2

