Solution (source code)

= Solution

Treat the specified particles only as rigid geometric shadowing objects; no biological preparation or interpretation is needed. Their projected shapes are rounded-ended rods with straight central length $100-30=70\,\mathrm{nm}$ and cap radius $15\,\mathrm{nm}$. \b[Normal, collimated evaporation deposits metal in the gaps between projected particle footprints], leaving isolated small islands after mask removal, rather than a metal replica of each rod.

One concrete dense parallel packing has center translations
$$
\mathbf u=(70+15\sqrt3,15)\,\mathrm{nm},\qquad
\mathbf v=(0,30)\,\mathrm{nm}.
$$
Adjacent rounded ends then touch at triangular arrangements. The open regions are small curvilinear triangular gaps near those ends. This is a specified geometrical packing for the sketch; the words close-packed do not uniquely prescribe the domain orientation or registry of all real sheets.

\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2018/paper-1-mask.png]
{title=Original geometric calculation of rounded-rod shadow masks, deposited islands and the effect of shifting a second layer; no source illustration is reproduced}

Particle size and shape distributions change the gap size and spacing. Misorientation, missing particles, boundaries between ordered domains, overlaps, incomplete coverage and drying-induced cracks interrupt long-range order. Substrate roughness and uneven adhesion distort the arrangement. A finite source angle produces <deposition shadowing> and penumbra; surface diffusion, island nucleation and increasing film thickness can broaden or merge deposited regions. These are generic geometry and materials effects, not assumptions about a biological assembly mechanism.