The two rays with the dotted lines hit the center points accurately and set the basis for the 12 cu The other rays were drawn with dashed lines, since they are not close hits. The dashed and dotted lines will be used in the remainder of the circumferential charts to distinguish between good hits and fair hits.

The approximate points indicated by the dashed lines on this chart seem to detract from its quality, but before you jump to any hasty conclusions, let us take another look at the phenomenon of longitudinal glide reflection. By careful inspection, you will see that #58 and #88 are on opposite sides of the ray line. Likewise, #26 and #42 lie on opposite sides of their common ray. And the same can be said for #12 and the obscure #14, and for the two obscure points #7 and #8. These dashed lines that are approximate hits, really represent average positions between two or more points. This then gives added significance to this symmetry pattern since it also uses some average positions. The preceding charts were examples of simple symmetry, whereas now, I have introduced a clear cut example of complexity which involves glide symmetry. You can now go back to the previous charts and find many other examples of glide reflection which I have not pointed out specifically.

### 155

This chart demonstrates two different rings. The inner ring is based on the fundamental units of 11 cu and 6 cu, while 12 cu and 18 cu are multiples of 6 cu. The outer ring, made up of the points #46, #37, #60, and #79, has been drawn to show that it has the fundamental of 11 cu, which is a continuation of 11 cu in the inner ring.

### 156

All the angles are divisible by the main symmetry element of 4. There is a balance left and right of bilateral symmetry around the 16 cu between #72 and #39.

### 157

The main symmetry element is 13 cu, followed by 5 cu and its multiples, plus one 14 cu. The hard core of this chart was established with #79, #80, #76, #40, and #51. The rest of the rays were drawn as a matter of interest. Point #48 seems to fit.

### 158

Here, again, we have bilateral symmetry with the 4 cu between #76 and #63 as the center piece.

### 159

This pattern involves the symmetry elements of 8 cu and 10 cu.