I was playing with a mapping $f:[0,1] \to \text{Sierpinski's Triangle}$, and I'd like to know if there's a name for this kind of thing. [It's a lot like playing the chaos game in reverse order.]
The process:
- Label the vertices of the Sierpinski Triangle (ST) with the digits $0$, $1$ and $2$.
- Begin with an arbitrary point $P$ in the largest empty triangle of ST (from this construction).
- Take a real number $x$ between $0$ and $1$, and write its infinite ternary expansion. [If a number terminates after finitely many digits, continue it with infinitely many zeros.]
- Start reading the digits of the expansion from left-to-right, and for each digit, move point $P$ into the next-smallest empty triangle "in the direction" of the vertex with that number. ["In the direction" here just means down-left, down-right or up.]
- Take the limiting location of point $P$ as $f(x)$.
Example: Let $x=0.4 = 0.\overline{1012}_3$
Point $P$ starts in the center triangle. The first digit 1 moves $P$ into the center triangle that is down to the right (since vertex $1$ is down to the right). The next digit 0 moves $P$ down to the left (since vertex $0$ is down to the left). The next digit 1 moves $P$ down to the right, then 2 moves it up, etc. See the GeoGebra animation below:
Similarly, the number $0.5 = 0.\overline{1}_3$ would be at vertex 1, and $\frac{1}{3} = 0.\overline{3} = 0.1\overline{0}_3$ would be midway between vertices 0 and 1.
More Info. It seems every $x$ should have a unique point on the fractal, unless we allow it to be written in two equivalent ways where one ends in an infinite number of $2$'s or $1$'s. [Notice that writing $\frac{1}{3}=0.0\overline{2}_3$ also puts $\frac{1}{3}$ midway between vertices 0 and 2.] Similarly, appending an infinite number of $0$'s at the end of an expansion means that a point on the fractal won't have a unique number mapping to it. [The number $\frac{1}{6} = 0.0\overline{1}_3$ is also at the midpoint between vertices 0 and 1.]
Here's a GeoGebra manipulative showing the function from $x=0$ to $x=1$ in steps of $0.001$. [Press the play button at bottom left.] Even in huge steps of $0.001$, it's easy to see lots of discontinuities where a ternary number is getting, say, a large sequence of $2$'s -- the "next" number mapped suddenly jumps to a $1$ in the previous digit.
Again, I thought this was silly and fun, but I figured it might have been looked at in the past and have a name.