How To Polygon

Chapter 3: Morphing the grids: the zig-zag-zog
Any grid or set of repeating tiles can be modified into a new set by certain changes to the lines. If the same changes are applied all positions of the same type, the new set will also be repeating tiles, each one the same.
For instance:
Changing each line into a zig-zag will produce a wide variety of new tiles, depending on the angles you use in the zigs and the zags.
For uniformity, the middle line in the zig-zag should cross the original grid line at its midpoint, but it can be drawn at any chosen angle in the grid.
Below, the angle is that of a diagonal of a 1 x 2 rectangle. (26.6°) This results in a tile that is made up of five regular square-agons. This same pattern could also be generated additively, by the piling up of squares.
Successive zig-zagging creates a fracturing of the tiles and successively finer filigree at the perimeters, thus making tiles with the character of fractals.

Here's a strange looking grid . . . the eye wants to see waves in it. It is actually just a field of large and small squares, trapezoids and rhombuses. Applying a zig-zag process (p. 16) yields the ziggy-funky pattern below:

Ziggy-funky:
Applying zig-zag methods to the rhombus and square tiling.
Note that this can be seen as just rotating half the “plus” tiles in a plus grid.
Another way of seeing this is as a rotate of the four lines meeting at each of the corners of a grid. Disconnect at the midpoints and rotate them to whatever angle is desired, and then re-connect the endpoints.
In the above, whole lines are rotated in opposite directions at corners diagonal from each other. Connecting the remaining corners results in irregular five-gon tiles in a curious pattern. . .four of these cluster into elongated six-gons. (as in p 11). It's a 4-way symmetry, but the twist here is 15°, so that 120° angles are formed which makes possible the arrangement of these same tiles (plus six-agons) into a pattern with 6-way symmetry. (see image, page 3)

Stepped parallels:
Another way to create tiles by modifications of grid lines is by making a series of stepped parallels, running in different directions.
When they cross each other, various tiles will form.
Notice that one of the tiles appears as a zig-like form (red).

Chapter 4
What to do with Five-agons?
OK, so only regular 3-, 4-, and 6-agons will tile by themselves. But can we do anything with regular 5-agons? Not really, because there are gaps to deal with.
Pairs can be placed abutted and then at right angles to each other, (a handy method for fitting shapes) but there are 4-point stars between them. Don't fear the gaps!
Sometimes shapes from the leftover spaces are beautiful themselves, and can be useful in forming your favorite patterns. The extras in one movie might be the stars in the next.
In the above, the 5-agons meet in pairs back-to-back. But if you begin by arranging them nose-to-nose, you get square and triangular spaces between them, so there are no concave gaps. Fish appear in these strange wavy grid lines:


In another pattern with five-gons, 3-, 4-, and 5- become 6-, 8-, and 5-agons:
The above might seem unlikely if you had not seen it with your own eyes. . . Eight-gons mixed with five-gons . . and six-gons? . . . “Impossible!” you shout.
Yes, it defies common sense... (the secret is that not all of the polygons are completely regular). Seeing it with the polygons centered at crossing lines from another grid may make it seem a little more forgivable. (Also pages 29 and 35)

Rotational Symmetry and A-Periodic Patterns:
You can also make patterns that work beautifully in rotational symmetry. The above is made with just 2 regular polygons: five-agons and ten-agons. But there are irregular six-agons, in there, too. These are formed at “pinch-points” where five-agons, overlap. This pattern can cover an unlimited plane. An example of a kind of “a-periodic” tiling, it cannot be overlaid upon itself when shifted except by rotation around a unique center.
Here is what can be done when such a pattern is further subdivided:
Chapter 5: Looping Rules:
Notice that there are ten 5-agons that fit snugly around a 10-agon. These form a kind of “loop” which works according to the following rule:
Any regular polygon of N sides can form loops of N or N/2 polygons when N is even, or a loop of 2N polygons when N is an odd number.
So, if N=14, loops of 14 or 7 fourteen-agons can be made, but if N is 7, one can only make a loop of fourteen. (see also looping fractals, p 21).
6 six-agons, 8 eight-agons, 10 ten-agons, 12 twelve-agons, and ten five-agons

Penrose: For some interesting five-way a-periodic tiles, check out Roger Penrose*. He developed many interesting patterns and theories of geometry including 2 ways to cover the plane, each with just two tiles: the kite & dart, and the 2 rhombuses.
In the above example, we fill in a 10-way looping symmetry with “fat” tiles (green and blue), and “skinny” tiles (pink and orange). More transformations (such as the zig-zags) can be applied to this pattern, but it's nice to see that complex patterns can be made from the fewest numbers of simplest tiles.