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Gallery of Specimens
  A chapter from Brainfilling Curves by Jeffrey Ventrella








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 non-fractal 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 L-shaped generator with no flippings. It creates a fractal known as the Levy C-curve:


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: 42 = 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 top-left to lower-right (A1, B2, C3, D4). also notice the four boxes arranged along the opposite diagonal (A4, B3, C2, D1). They specify the only well-behaved fractal curves of this family. and they happen to be gridfillers. You can see that the well-behaved 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 plane-filling curves of this family, which I call the root2 family.

The four flippings in the upper-left corner all result in the Levy C-curve. 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 self-touching along its edges. Here is a diagram showing the fractalization of the L-shaped 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 self-cross or self-touch 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 plane-filling curves, they often do exhibit some interesting forms of self-similarity, 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