= Solution
Read the displayed axes as horizontal $[001]$ and vertical $[010]$; the viewing direction is therefore consistent with $[100]$. <crystal facets> edge-on in this projection have normals in the $yz$ plane. The flat upper/lower boundaries are approximately horizontal, the end face vertical, and the bevels approximately $45^\circ$ to both.
A probable low-index assignment is \b[upper face $(010)$, end face $(001)$ and lower-right bevel $(0\bar11)$], with the upper-right bevel $(011)$ if it is also indexed. Thus the flat faces belong to $\{100\}$ and the diagonal faces to $\{110\}$.
For example,
$$
\cos\phi_{(010),(001)}=0,\qquad
\cos\phi_{(001),(011)}=\frac1{\sqrt2},
$$
giving $90^\circ$ and $45^\circ$ between normals. The upper $(010)$ normal and lower bevel $(0\bar11)$ normal instead make $135^\circ$; their acute unoriented plane angle is $45^\circ$. Adjacent polygon corners between a flat edge and bevel have internal angles near $135^\circ$.
\b[The distinction between a normal angle, an acute plane angle and an internal polygon angle avoids a supplementary-angle error.] This is a probable projected indexation, not a unique full three-dimensional <crystal facet> reconstruction. The associated work uses tomography precisely to recover information absent from one projection.<Three-Dimensional Elemental Mapping at the Atomic Scale in Bimetallic Nanocrystals by Bart Goris and others>[2013, study scope: three-dimensional structure and composition from electron-tomography data]{ref} The original facet-normal diagram in the growth-and-LED figure shows the $45^\circ$ relation independently of the source micrograph.
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