Random

http://www.wired.com/2016/08/mathematicians-building-unified-theory-geometric-randomness/

My own assumption about a random process on a random surface would be
that it moves twice as fast. Think about it: program a random number generator
between +2 and -2 to move your bug starting at 0. The fact that the space on which it
moves is randomly distorted as well might push Buggy to +4 on the first turn, a feat
not otherwise possible. That's Buggy on a thread, or in 1 dimension.

Let's change things up, and move Buggy to a checkerboard space, thus in 2 dimensions.
He still moves forward or back with a maximum of 2, but each move can be considered
two moves, one on each axis, with the constraint that 2 is the maximum on any
one axis (here, where the other axis is a 0 move). Now make the board random
within the 2 rule. Buggy has sped up by 2 here as well, with the added consequence
that the history of his movement could paint a skewed picture, and indeed probably will.
Because as observers of randomness, we bring expectations about how much order we will
tolerate.