  For an arbitrary normed space $\mathcal X$ over a field $\mathbb F \in \{
\mathbb R, \mathbb C \}$, we define the directed graph $\Gamma(\mathcal X)$
induced by Birkhoff-James orthogonality on the projective space $\mathbb
P(\mathcal X)$, and also its nonprojective counterpart $\Gamma_0(\mathcal X)$.
We show that, in finite-dimensional normed spaces, $\Gamma(\mathcal X)$ carries
all the information about the dimension, smooth points, and norm's maximal
faces. It also allows to determine whether the norm is a supremum norm or not,
and thus classifies finite-dimensional abelian $C^\ast$-algebras among other
normed spaces. We further establish the necessary and sufficient conditions
under which the graph $\Gamma_0(\mathcal{R})$ of a (real or complex) Radon
plane $\mathcal{R}$ is isomorphic to the graph $\Gamma_0(\mathbb F^2,
\|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of
such nonsmooth Radon planes.
