= Solution
For a <graphene> honeycomb with C-C <bond length> $b$, a primitive cell has two atoms and area $3\sqrt3b^2/2$. Its areal atomic density is
$$
n_s=\frac{4}{3\sqrt3b^2}=39.3\,\mathrm{nm^{-2}}\quad(b=0.14\,\mathrm{nm}).
$$
For the <nanotube atom-count estimate>, treat each shell as a <graphene> cylinder. The atoms per axial length are
$$
\boxed{\frac{N_C}{\ell}=2\pi n_s\sum_{j=0}^{n-1}(r_{\rm in}+jd_{002})=2\pi n_s\left[nr_{\rm in}+\frac{d_{002}n(n-1)}2\right].}
$$
Using the fringe spacing as an internal ruler gives representative estimates $n\simeq16$, $r_{\rm in}\simeq2.2\,\mathrm{nm}$ and $r_{\rm out}\simeq7.45\,\mathrm{nm}$, consistent with $15$ radial spacings of $0.35\,\mathrm{nm}$. Then
$$
\frac{N_C}{\ell}\simeq2\pi(39.3)[16(2.2)+0.35(16)(15)/2]\simeq\boxed{1.9\times10^4\text{ atoms per nm}}.
$$
The meaningful estimate is \b[of order $2\times10^4$ atoms per nm], not an exact count. It assumes nearly complete cylindrical shells, uses projected dimensions in a reasonably straight segment, and neglects defects, corrugation and extra external carbon. Image uncertainty in wall count and diameter changes the estimate by tens of percent; no external scale bar is available in this figure.
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