version 1.4

January 1997

It was profoundly disturbing to mathematicians to discover that 1-dimensional curves exist that fill 2-dimensional space. A famous example is Heighway's Dragon curve. It was originally formed by paperfolding. Take a long strip of paper laid out in front of you from left to right. Fold the right end over the left end. Next fold the right end of the result over the left end, fold the right end of the result over the left end, and so on. It is claimed that you can't do this more than seven times because it would be like trying to fold a 128 page book in half. Now open each fold to 90 degrees. The result looks like a scaly dragon. Here is an L-system to draw this curve.

Hiway { Angle 4 Axiom fx x=fx+fy y=fx-fy f= }

x represents the first half of the curve and y represents the second half.
After FRACTINT draws the first half, no matter what orientation FRACTINT is in
when it finishes drawing the first half, it must turn left 90 degrees because
that is the middle fold, the very first fold created in the folding process.
The second half y is exactly the same as the first half x but it is traced
backwards. This is why the replacement string for x is fx+fy and the
replacement string for y is fx-fy, x backwards.
This L-system is somewhat simpler and more efficient than Adrian Mariano's
Dragon L-system in
fractint.l. Adrian's L-system has the feature that the
beginning and end of the curve form the endpoints of a horizontal line at all
orders, while the above L-system has these endpoints turning left 45 degrees
with each increase in order.

While studying Heighway's dragon, Davis and Knuth invented their own spacefilling curve called the Terdragon. It's L-system is:

Terdragon { Angle 3 ; Only need turns of 120=360/3 degrees Axiom f ; Draw one Terdragon f=f+f-f ; Replace with the Terdragon, turn left, another Terdragon, turn right, another ; Terdragon }

The above spacefilling curves were invented long after the mathematical crisis engendered by spacefilling curves. One of the original crisis-makers was Peano's spacefilling curve. Here is an L-system to render Peano's curve.

Peano1 { ; Adrian Mariano ; from The Fractal Geometry of Nature by Mandelbrot Angle 4 Axiom F F=F-F+F+F+F-F-F-F+F }

I've changed the axiom line of Peano1 in the file fractint.l so that only
Peano's original curve shows. Note that FRACTINT ignores text following a
**";"** and also ignores case; **"F"** is the same
as **"f"** to FRACTINT.

Another recently discovered spacefilling curve was invented by Dekking. I include it here because it exemplifies the case where a figure is made up of parts dissimilar from each other.

Dekking { ; Dekking's Church ; Advances in Math, vol. 44, 1982, pp. 78-104 Angle 4 Axiom C32WC06ZC15YC06X F= W=FW+F-XFW-F+Z X=++F--Y-F+Z++F--Y-F+Z Y=++F--Y+F-X Z=FW+F-X }

Note that each part, W, X, Y, Z, has a different replacement string. Note
also that each + or - that precedes an f is followed by its opposite. This is
because FRACTINT does not allow replacing + and - by nothing (maybe in version
19.6?). The strings **"c32"**, **"c06"**, and
**"c15"** tell Fractint to color the following lines in the color
in the color palette in the position indicated by the number following the
**"c"**.

This curve looks like a church on which a mural is painted reminiscent of
the painting of God giving life to Adam.

Earlier I described a tiling involving dissecting two isosceles triangles. This tiling was chosen in such a way that I could create a spacefilling curve based on the tiling. Here is the resulting L-system.

Pentive3 { Angle 10 Axiom q f= p=--fr++++fs--fu q=ft++fr----fs++ r=++fp----fq++ft s=fu--fp++++fq-- t=+fu--fp+ u=-fq++ft- }

All of the spacefilling curves described so far revisit certain points, but there are many spacefilling curves which never revisit any points. An example is the Cross. It is named so because there is an underlying tiling of crosses.

Cross { Angle 4 Axiom FX X=FX+FX+FXFY-FY- Y=+FX+FXFY-FY-FY F= }

The symbol X tells FRACTINT how the Cross is made up. It says draw X, turn
left, draw X, turn left, draw X, draw Y (i.e., draw X backwards), turn right,
draw Y, turn right. The F's precede each X and Y because, without them
FRACTINT would create long strings of X's and Y's with no drawing commands.
The replacement string for Y must do the same thing that the replacement
string for X does only backwards. The text line **"F="** is
there to tell FRACTINT to replace each F by nothing so that F's used at one
order are not involved in the next order.

Note also that the replacement strings for X and Y each have the same number of +'s as -'s. This is to insure that when FRACTINT is drawing the Cross it is oriented the same way after drawing X or Y as it was when it began drawing X or Y. This is the safe thing to do if you don't understand how the orientation may affect the outcome of your L-system.

There is another famous spacefilling curve that does not revisit any points due to Hilbert. It's discussion is relegated to the final section which follows the next section.

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noel@triumf.ca