Three views of the same object: a sample path, the random-walk scaling limit that builds it (Donsker), and the quadratic variation that makes Itô calculus different.
Each curve is one realization of $W_t$ on $[0,10]$. Increments are Gaussian with variance equal to the time gap, so the path wanders with standard deviation $\sqrt{t}$. Press New paths to resample.
A simple symmetric random walk $S_i$ ($\pm1$ steps) is rescaled to $S_{\lfloor nt\rfloor}/\sqrt{n}$ and plotted over $[0,1]$. As $n$ grows the staircase smooths into a continuous Brownian path.
For one path on $[0,1]$, the sum of squared increments $\sum(\Delta W)^2$ over a partition of $m$ equal intervals. For a smooth curve this would collapse to $0$; for Brownian motion it converges to $t=1$.