 Solution

A decahedral nanoparticle is a multiply twinned nanoparticle: five fcc-like tetrahedral domains share a common fivefold axis and meet on coherent twin planes. The angle of an ideal fcc tetrahedral wedge is $\arccos (1/3)=70.53^{\circ }$, so five wedges leave about $7.36^{\circ }$ of angular deficit. Elastic distortion and defects accommodate closure. The external shape can be a pentagonal bipyramid or a truncated variant; it is not one unstrained, periodically repeated fcc crystal.

An icosahedral nanoparticle comprises twenty tetrahedral domains arranged around a common centre, with twenty approximately triangular outer facets, twelve vertices and thirty edges. Five domains meet around each radial fivefold direction. Coherent twinning exposes many low-energy close-packed facets, but fivefold symmetry cannot tile ordinary three-dimensional space periodically without strain. The competition between favourable surface/twin energies and volume-dependent elastic strain explains why such morphologies are especially plausible for small particles.

These labels describe the observed domain geometry; cobalt can adopt different crystal phases, so identifying fcc-like twinning also requires diffraction rather than assuming every cobalt particle is bulk fcc.