  We consider uncountable almost disjoint families of subsets of $\mathbb N$,
the Johnson-Lindenstrauss Banach spaces $(\mathcal X_{\mathcal A}, \|\
\|_\infty)$ induced by them, and their natural equivalent renormings $(\mathcal
X_{\mathcal A}, \|\ \|_{\infty, 2})$. We introduce a partial order $\mathbb
P_{\mathcal A}$ and characterize some geometric properties of the spheres of
$(\mathcal X_{\mathcal A}, \|\ \|_{\infty})$ and of $(\mathcal X_{\mathcal A},
\|\ \|_{\infty, 2})$ in terms of combinatorial properties of $\mathbb
P_{\mathcal A}$. Exploiting the extreme behavior of some known and some new
almost disjoint families among others we show the existence of Banach spaces
where the unit spheres display surprising geometry:
  1) There is a Banach space of density continuum whose unit sphere is the
union of countably many sets of diameters strictly less than $1$.
  2) It is consistent that for every $\rho>0$ there is a nonseparable Banach
space, where for every $\delta>0$ there is $\varepsilon>0$ such that every
uncountable $(1-\varepsilon)$-separated set of elements of the unit sphere
contains two elements distant by less than $1$ and two elements distant at
least by $2-\rho-\delta$.
  It should be noted that for every $\varepsilon>0$ every nonseparable Banach
space has a plenty of uncountable $(1-\varepsilon)$-separated sets by the Riesz
Lemma.
  We also obtain a consistent dichotomy for the spaces of the form $(\mathcal
X_{\mathcal A}, \|\ \|_{\infty, 2})$: The Open Coloring Axiom implies that the
unit sphere of every Banach space of the form $(\mathcal X_{\mathcal A}, \|\
\|_{\infty, 2})$ either is the union of countably many sets of diameter
strictly less than $1$ or it contains an uncountable
$(2-\varepsilon)$-separated set for every $\varepsilon>0$.
