Uploaded August 2020 | Updated September 2026, 1 week ago
An explorer staggers randomly throught the world. Will they always eventually return close to the starting place?
It is well-known that in the Euclidean plane they will return with probability 1, but in higher Euclidean dimensions, they are likely to never return: en.wikipedia.org/wiki/Random_walk
In hyperbolic geometry (even two-dimensional) it is even worse. Because of the exponential growth, more directions take our explorer away from the starting point, than bring them back. So while in Euclidean space, we are roughly as likely to go closer or further from the starting point, and the average distance after time t is √t, in hyperbolic space we tend to go away (at a roughly constant speed). Even if the explorers reproduce from time to time, it is likely that none of the descendants will ever return! (This depends on how fast they reproduce, relative to the curvature of the world: intuitively, even though the population grows exponentially with time, the space to explore grows even faster.)
In this visualization we show such a branching random walk. We animate the straight line from the starting point to the final point of a chosen explorer, with constant speed. (If we always centered on the explorer, the video would be very shaky.)
More fun facts:
* we are using the Poincaré disk model, which is conformal. A random walk in a conformal projection will look like an Euclidean random walk. However, the timing is different (reaching the edge of the disk takes infinite time, while in Euclidean plane we would just continue).
* Brown Island in HyperRogue is roughly based on these ideas.
* the moving explorer tends to stay close to the camera (moving at constant speed), although in most realizations of this visualization, the distance is larger (at some point of time, the explorer is lagging behind the camera and hardly visible).
An explorer staggers randomly throught the world. Will they always eventually return close to the starting place?
It is well-known that in the Euclidean plane they will return with probability 1, but in higher Euclidean dimensions, they are likely to never return: en.wikipedia.org/wiki/Random_walk
In hyperbolic geometry (even two-dimensional) it is even worse. Because of the exponential growth, more directions take our explorer away from the starting point, than bring them back. So while in Euclidean space, we are roughly as likely to go closer or further from the starting point, and the average distance after time t is √t, in hyperbolic space we tend to go away (at a roughly constant speed). Even if the explorers reproduce from time to time, it is likely that none of the descendants will ever return! (This depends on how fast they reproduce, relative to the curvature of the world: intuitively, even though the population grows exponentially with time, the space to explore grows even faster.)
In this visualization we show such a branching random walk. We animate the straight line from the starting point to the final point of a chosen explorer, with constant speed. (If we always centered on the explorer, the video would be very shaky.)
More fun facts:
* we are using the Poincaré disk model, which is conformal. A random walk in a conformal projection will look like an Euclidean random walk. However, the timing is different (reaching the edge of the disk takes infinite time, while in Euclidean plane we would just continue).
* Brown Island in HyperRogue is roughly based on these ideas.
* the moving explorer tends to stay close to the camera (moving at constant speed), although in most realizations of this visualization, the distance is larger (at some point of time, the explorer is lagging behind the camera and hardly visible).
