  Following work by the first author and Banchoff, we investigate
triangulations of real and complex projective spaces of real and complex
dimension $k$ that are adapted to the decomposition into "zones of influence"
around the points $[1,0,\ldots,0],$ $\ldots,$ $[0,\ldots,0,1]$ in homogeneous
coordinates. The boundary of such a "zone of influence" must admit a simplicial
version of the Hopf decomposition of a sphere into "solid tori" of various
dimensions. We present such {\em Hopf triangulations} of $S^{2k-1}$ for $k \leq
4$, and give candidate triangulations for arbitrary $k$.
  In the complex case, a crucial role of this construction is the central
$k$-torus as the intersection of all "zones of influence". Candidate
triangulations of the $k$-torus with $2^{k+1}-1$, $k\geq 1$, vertices --
possibly the minimum numbers -- are well known. They admit an involution acting
like complex conjugation and an automorphism of order $k+1$ realising the
cyclic shift of coordinate directions in $\mathbb{C}P^k$. For $k=2$, this can
be extended to what we call a {\em perfect equilibrium triangulation} of
$\mathbb{C}P^2$, previously described in the literature. We prove that this is
no longer possible for $k=3$, and no perfect equilibrium triangulation of
$\mathbb{C}P^3$ exists. In the real case, the central torus is replaced by its
fixed-point set under complex conjugation: the vertices of a $k$-dimensional
cube. We revisit known equilibrium triangulations of $\mathbb{R}P^k$ for $k\leq
2$, and describe new equilibrium triangulations of $\mathbb{R}P^3$ and
$\mathbb{R}P^4$.
  Finally, we discuss the most symmetric and vertex-minimal triangulation of
$\mathbb{R}P^4$ and present a tight polyhedral embedding of $\mathbb{R}P^3$
into 6-space. No such embedding was known before.
