Brownian motion, or random walk, can be regarded as the trace of some cumulative normal random numbers.
The location of the next step is current location + random Gaussian numbers, i.e.,
$$x_{k+1} = x_{k} + rnorm(1)$$
$$y_{k + 1} = y_{k} + rnorm(1)$$
where \((x, y)\) stands for the location of a point.
library(animation)
ani.options(interval = 0.05, nmax = 150)
brownian.motion(pch = 21, cex = 5, col = "red", bg = "yellow", main = "Demonstration of Brownian Motion")