1 Horror Vacui 2 a Very Patient Turtle Who Draws Lines 3 a Taxonomy of Fractology 4 Gallery of Specimens 
Root 2 Family  Root 3 Family  Root 4 Square Grid Family  Root 4 Triangle Grid Family 
Root 5 Family  No Root 6!  Root 7 Family  Root 8 Family 
Root 9 Square Grid Family  Root 9 Triangle Grid Family  Root 10 Family  Root 12 Family 
Root 13 Square Grid Family  Root 13 Triangle Grid Family  Root 16 Square Grid Family  Root 16 Triangle Grid Family 
Root 17 and Beyond...  5 My Brain Fillith Over  6 References  7 acknowledgements 
For the sake of completeness I will start with the simplest specimen of all: a nonfractal curve consisting of a straight line. Its fractal dimension is 1, and its interval length is 1. I will use this as an introduction to the diagrams used throughout the book. In this diagram, the header bar at the top shows the name of the fractal at the left (although many don't have names). To the right of that are the interval length (expressed as a square root) and the fractal dimension. Below the header bar at the left is some information about the genetics of the fractal generator. This includes the grid type (not relevant in the case of this single line), and the number of segments. Below that is a list of numbers that specify the segments in the generator. Each line segment in the generator is specified using four numbers. The first two numbers specify its displacement within the grid. In this example, the line extends one unit in the x direction and 0 units in the y direction, and so the numbers are 1 and 0. The third and fourth numbers describe the segment's flippings. I will explain that next. 
Segment Flippings
Remember the four flipped variations of the turtle I showed you earlier? These four kinds of flippings are represented in the third and fourth numbers. So: 1, 1 means no flipping; 1, 1, means it is flipped in x; 1, 1 means it is flipped in y; and 1, 1 means it is flipped in both x and y. Now let's look at an Lshaped generator with no flippings. It creates a fractal known as the Levy Ccurve: 
This in an interesting fractal curve  in a gnarly kind of way. Now, consider what happens when we try a few different flippings among these two segments. Take note of the subtle difference in flippings here: 

Quite different results, eh? There are in fact 16 different possible ways to flip these two segments (since each of the two segment can be flipped four ways: 4^{2} = 16). Here are the fractal curves that result from all possible flippings: 

In the graph, I have labeled the rows A, B, C, D, and the columns 1, 2, 3, 4. Notice the diagonal symmetry mirrored along the axis that stretches from topleft to lowerright (A1, B2, C3, D4). also notice the four boxes arranged along the opposite diagonal (A4, B3, C2, D1). They specify the only wellbehaved fractal curves of this family. and they happen to be gridfillers. You can see that the wellbehaved fractal curves come in two forms (which I will introduce shortly). Two of them are simply flipped versions of the other two, and so we conclude that there are really just two planefilling curves of this family, which I call the root2 family.
The four flippings in the upperleft corner all result in the Levy Ccurve. and the four curves in the lower right corner all result in Cesaro's Sweep, which is a double density gridfiller, meaning, it is everywhere selftouching along its edges. Here is a diagram showing the fractalization of the Lshaped generator to create Cesaro's Sweep: 

The fractal curves located at A3, B4, C1, and D2 are quite misbehaved: they cross over themselves and they leave lots of holes in the process. This is not to say that they are uninteresting. In fact, as a nod to all the misbehaved fractal curves in the world (which is most of them) I shall offer a portrait of the fractal curve at C1...here.
The fractal curves that selfcross or selftouch can be considered as creatures that have reinforced regions in their bodies. The density of the fabric of their flesh is uneven  some spots are thick  other spots have holes. although they may lack the aesthetic elegance of planefilling curves, they often do exhibit some interesting forms of selfsimilarity, and they evoke familiar forms in nature. 

End of chapter. 
1 Horror Vacui 2 a Very Patient Turtle Who Draws Lines 3 a Taxonomy of Fractology 4 Gallery of Specimens 
Root 2 Family  Root 3 Family  Root 4 Square Grid Family  Root 4 Triangle Grid Family 
Root 5 Family  No Root 6!  Root 7 Family  Root 8 Family 
Root 9 Square Grid Family  Root 9 Triangle Grid Family  Root 10 Family  Root 12 Family 
Root 13 Square Grid Family  Root 13 Triangle Grid Family  Root 16 Square Grid Family  Root 16 Triangle Grid Family 
Root 17 and Beyond...  5 My Brain Fillith Over  6 References  7 acknowledgements 