What is a Random Walk?
The easiest way to describe a random walk is to build a few, and watch how they develop. We'll build three walks, to see the different ways in which they can develop. These sample walks are computer-generated, which means a computer made all of the random decisions. The main walks on this site are processed by a computer, but the random decisions are made by people - the participants on this site.
The Starting PointAll random walks begin the same way, with a single point. The starting point in all of these images is a large green dot.
Up or Down?After a starting point is established, we face our first decisioin: Do we go up or down? In the first walk shown here, the first step starts off with a move upward. The second two walks begin by moving down from the starting point. After deciding what direction to move, we also choose how far to move in that direction.
Left or Right?After making the vertical part of the first move, we decide whether to move left or right, and how far to move in that direction. Wherever we end up, we place a point. Including the starting point, this is the second point. The first two walks have made a move to the right, while the third walk makes no horizontal move at all. This allows for a purely vertical move.
RepetitionFrom this point forward, the same steps are repeated over and over again. Here, we again choose whether to move up or down in each walk, and how far to go in that direction. The first walk heads up, while the third walk moves down. The middle walk makes no vertical move at all, which will result in a purely horizontal move.
Third PointAll three walks have moved to the left. The first walk moved up and left, the second walk moved straight left, and the third walk moved down and left.
After 5 PointsWe can see some interesting things after 5 points. The first walk is heading straight up, but going back and forth along the way. The second walk has returned to its starting point! This does not mean a pattern is about to repeat itself; the walk could still go anywhere from here. The third walk continues to head down and to the left.
After 10 PointsThe scale on these walks has changed, in order to fit them into a reasonable space.
After 25 PointsIt might be more correct to say "after 25 moves". Twenty-five moves have been made, but if a point was reached twice, it only shows up once in the drawing.
After 100 PointsThe first and third walks are continuing to expand, while the middle walk has kept a tight spacing between its points.
After 1000 PointsThe large-scale structure of each walk has begun to emerge.
Completed Walk - 5000 PointsThe finished walks. Although all walks start with the same conditions and follow the same rules, each walk is unique. By the time we have placed 5000 points, there are more possible end results than there have been seconds in the Universe. You could have begun making these walks, once each second from the moment the Universe began, and you would still be making a unique random walk each second until today. Click on each walk for a larger image.
What's next?See the gallery of sample walks, or the section on visual design for colored walks. Better yet, place the next point in the current walk!