Chaos, fractals, and Arcadia

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Chaos, Fractals, and Arcadia

### The Sierpinski Hexagon

As another example of Thomasina's algorithm, start with six points
arranged at the vertices of a hexagon. Number them from one
to six and erase the colors on your die. Beginning with an initial
seed in the hexagon, roll the die and then move the point two-thirds of the
distance toward the appropriate vertex. Then iterate this
procedure. As before, do not record the first 15 or so iterations, but
plot the rest. The result is again anything but a random mess: It is
the Sierpinski hexagon. See Figure 3.

Fig. 3. The Sierpinski hexagon.

Note that these two fractal images are not quite the lifelike images
that Thomasina promised us, but there is a hint of what is to come.
Look at the innermost boundary of the Sierpinski hexagon. Note how it
resembles a snowflake. Indeed, this innermost boundary is the
Koch snowflake curve, another very famous fractal.
This particular algorithm can also be expressed in terms of linear
contractions of a given square. Start with a square and place six
points on the square so that they form the boundary of a
hexagon with equal sides.
Geometry exercise: Where should these points be placed if
vertex number one lies in the middle of the left side of the square?
At each iteration, contract all points in the original square toward
the appropriate vertex. The image is, in each case, a square exactly
one-third the size of the original and located as shown in Figure 4.

Fig. 4. The six contractions to produce the
Sierpinski hexagon.

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