The fractals included in this collection were chosen because of their beauty or because of something interesting about them. There is a small description about each fractal following the applet. As with the other fractal applets, be careful about taking them too far.
| Koch Curve |
The Koch Curve fractal, one of the classic ones, is also one of the
easiest ones to implement. This is already covered in the Koch
Curve specific page.
| Koch Snowflake Curve |
The Koch Snowflake Curve fractal is essentially a three sided Koch Curve.
One of the interesting things about this one is that by rescaling the lines
at every iteration (by a factor of 3), the overall size remains constant
while the length of all the lines added together grows towards infinity.
| Hilbert Curve |
The Hilbert Curve, proposed by David Hilbert, is called a space-filling fractal, because as you can see, it continuously attempts to fill in the empty area within. At the scale that this applet draws it, it is pretty well filled in starting with the 7th iteration. Of course, as with many fractals, if you could zoom in, you would see that you could continue filling it in forever.
This fractal introduces something new regarding the L-systems. When we introduced the L-Systems, we mentioned some of the symbols that are used to describe the axioms and production rules. Remember we said that you could use this codes to describe someone how to draw the shapes. Select the Hilbert Curve fractal, reset it, and advance it to the first iteration. Repeat this a couple of times. To be able to do this we need something new in addition to what was described. Can you tell why it wouldn't work with the few symbols that were described?.
| Sierpinski Triangle |
The Sierpinski Triangle fractal implemented here uses the L-System method
instead of the collection of triangles that is used in the Sierpinski
Triangle page. One advantage of the other method is that when you are
dealing with triangles (instead of only lines that don't know anything
about the geometrical figures being represented) it is easy to assign colors
to each triangle. Additionally, in the L-System implementation one does
not start with the large triangle, but as you move to higher iterations,
its shape becomes apparent.
| Dragon Curve |
The Dragon Curve fractal implemented here uses the L-System method instead
of the collection of points that is used in the Dragon
Curve page. One of the differences that this one has is that it gets
rescaled at every iteration (by a factor of square root of 2). The other
difference is how the colors work. In this case they are completely random
and they change every 200 lines. By looking at the colors you can see the
real small area that 200 lines can occupy. You can also get a feel
for the amazing number of lines that are used in the high iteration numbers
by looking at the amount of different colors.
| Stairs |
The Stairs fractal is one that I came up with. It is the square version
of the Koch Curve which as it grows, becomes an interesting pyramid with
lots of tiny steps and interesting decorations or tunnels into the pyramid.
Maybe I should call it the pyramid rather than the stairs fractal. If this
is a known fractal please let me know what it is and I'll change it here.
| Peano Curve |
The Peano Curve fractal is another example of a space-filling fractal.
In this case it uses squares or tiles to fill in a shape that becomes a
rhombus. It is interesting to watch it as it fills in the rhombus, especially
when it reaches iterations 4 or 5. If you don't have a computer that can
get it to these iterations, try to find one!
| Peano-Gosper Curve |
The Peano-Gosper Curve is yet another space-filling fractal but with a very interesting initial shape. To make it more interesting this fractal is drawn using the colors in the normal rainbow color sequence:
| red | orange | yellow | green | blue | indigo | violet |
| Low frequency | High frequency |
It does this by transforming the colors as slowly as possible covering the entire rainbow spectrum of light as evenly as possible. The smoothness obviously depends on the number of lines for the particular iteration.
| Bush One |
The first Bush fractal introduces something new regarding the L-systems. As with the Hilbert Curve above, this fractal needs something new. Select the Bush One fractal, reset it, and advance it to the first iteration. Repeat this a couple of times. To be able to do this fractal we need additional symbols with new meanings.
The other thing to mention about this fractal is the use of colors.
The attempt is being made to make the bush look as real as possible. It
starts with brown, then a sequence of greens that gets lighter and lighter,
and then a few different colors that look like flowers. Unless you have
a pretty powerful computer, you may not be able to see the flowers, but
nevertheless, it is a very nice fractal.
| Bush Two |
The second Bush fractal, which uses the colors the same way as the first
bush, shows the flowers very nicely. They first appear with the 7th
iteration and get better with the 8th and
9th.
| Bush Three |
Best for Last! This last bush fractal, which uses the same color scheme as the other bushes, is simply spectacular. It looks extremely real, 3D, and it is great to see it as it gets painted. It is hard to believe that it is a fractal unless you see it grow step by step as with this applet. If your computer is not powerful enough to enjoy this fractal, I recommend that you try it on one.