Irrational numbers have decimal expansions that neither terminate nor become periodic. So we can get unlimited “random walk” steps from an irrational numbers.

With the following code, the first 10000 digits of Sqrt[2] is presented as a random walk by converting it in base 4. 0, 1, 2 and 3 digit in base 4 represent 4 directions. Starting point is the green dot, the red one is the ending point.

x = N[Sqrt[2], 10000];

walk = First@ RealDigits[x, 4];

rn = FoldList[Plus, {0, 0}, {{0, 1}, {1, 0}, {0, -1}, {-1, 0}}[[# + 1]] & /@ walk];

Graphics[{Line[rn], PointSize[Large], Green, Point[First@rn], Red, Point[Last@rn]}]

We can also display it with ArrayPlot by constructing a sparse array.

(* shift the moves to {1,1} *)

minx = Min[rn[[All, 1]]];

miny = Min[rn[[All, 2]]];

m = # + {-minx + 1, -miny + 1} & /@ rn;

(* sparse array *)

tt = Tally[m];

cd = SparseArray[tt[[All, 1]] -> tt[[All, 2]]];

cd = Transpose[cd];

ArrayPlot[cd, ColorFunction -> "Rainbow", DataReversed -> True, ColorRules -> {0 -> White}]

Let’s visualize Sqrt[2], e, Pi in their first 50000 digits. It seems there are some similarities among these images.

Sqrt[2]

Pi

e

**Update:**

DrMajorBob said... in comments:

Here's a 3D version:

x = N[Sqrt[2], 10000];

walk = First@RealDigits[x, 6];

rn = FoldList[ Plus, {0, 0, 0}, {{0, 1, 0}, {1, 0, 0}, {0, -1, 0}, {-1, 0, 0}, {0, 0, 1}, {0, 0, -1}}[[# + 1]] & /@ walk ];

Graphics3D[{Line[rn], PointSize[Large], Green, Point[First@rn], Red, Point[Last@rn]}]

You can also try non-irrational number, e.g. 121/5^10

Thanks.

**Update 2:**

Visualize genome sequence. I know nothing about it, it probably totally meaningless.

GenePlot[g_] := Module[{cs, rn}, walk = Characters[GenomeData [g, "FullSequence"]] /. {"A" -> 1, "T" -> 2, "G" -> 3, "C" -> 4};

rn = FoldList[Plus, {0, 0}, {{0, 1}, {1, 0}, {0, -1}, {-1, 0}}[[#]] & /@ walk];

Graphics[{Line[rn], PointSize[Large], Green, Point[First@rn], Red, Point[Last@rn]}, Frame -> True, FrameTicks -> None] ]

GenePlot["DDK4"]

GenePlot["DKK3"]