TITLE: Does $\pi$ span an infinite grid?
QUESTION [8 upvotes]: This is similar to a random walk using $\pi$ as the input.
Consider taking steps on an infinite grid based on the digits of $\pi$ (though we could use any number).  If we express the number in base $8$, then each digit $0$ to $7$ represents a step to one of the $8$ neighbors including diagonals (base 4 would be an alternative with up/down left/right).  One possible mapping of these steps could be from the current position (or pixel) $P$ to one of these neighbors as
$$
\begin{array}  \\ \hline
0 & 1 & 2 \\
7 & P & 3  \\
6 & 5 & 4  \\ \hline
\end{array}
$$
Considering $\pi$ in base 8 we have 
$$
\pi_8 = 3.1103755 \dots
$$ 
so the first step would be one cell to the right (3).  The second and third steps both move one cell up (1).  The fourth step moves up and to the left, $etc$.  
The first 1022 steps (starting at green position ending at black) looks like this:


While discussing this game a friend of mine claimed that as we continued steps toward infinity, $\pi$ would fill the plane.  Clearly a bold claim, and I am suspicious.  Is it provable?  If it doesn't work in base $8$, could it work in the base $4$ case where we move up/down or left right?

REPLY [7 votes]: Equiprobability in base $8$ is not  necessary, as you could tour the plane with e.g. digits $1,4,6$. 
And normality is not sufficient: consider a sequence that is random except that when the walker would otherwise hit $[1,0]$ you change the digit.   Although this will require changing infinitely many digits (because a random walker in the plane will hit $[1,0]$ infinitely many times), the fraction of digits to be changed goes to $0$ (asymptotically, the probability of the random walker being at $[1,0]$ at time $t$ decreases like $1/t$), so that these changes do not affect normality in any base.