This result collects three fundamental symmetries of Brownian motion: invariance under orthogonal transformations (rotation), invariance under parabolic rescaling, and the simple Markov property. These are direct consequences of the defining properties of Brownian motion — Gaussian increments, independence, and stationarity — and they underpin most of the deeper theory. Rotational invariance says that applying any orthogonal matrix $U$ to Brownian motion yields another Brownian motion. This is a consequence of the fact that orthogonal transformations preserve the multivariate Gaussian distribution: if $B_t - B_s \sim \mathcal{N}(0, (t-s)I_d)$, then $U(B_t - B_s) \sim \mathcal{N}(0, (t-s)UU^\top) = \mathcal{N}(0, (t-s)I_d)$. The scaling property $\lambda^{-1/2} B_{\lambda t} \stackrel{d}{=} B_t$ reflects the self-similar structure of Brownian motion and is used in the proof of the [Recurrence-Transience Dichotomy](/theorems/1185), in the analysis of the zero set of Brownian motion, and in [Donsker's Invariance Principle](/theorems/1189). The simple Markov property — that $(B_{t+s} - B_s)_{t \geq 0}$ is a Brownian motion independent of $\mathcal{F}_s^B$ — is the probabilistic expression of the memoryless nature of Brownian motion. It is the deterministic-time version of the [Strong Markov Property](/theorems/1180) and serves as its foundation: the strong Markov property is proved by approximating a stopping time from above by dyadic times and applying the simple Markov property at each approximation. The Markov property also connects Brownian motion to PDE theory through the semigroup $P_t f(x) = \mathbb{E}_x[f(B_t)]$, whose generator is $\frac{1}{2}\Delta$.