  In this article we consider an open conjecture about coherently labelling a
polyhedron in three dimensions. We exhibit all the forty eight possible
coherent labellings of a tetrahedron. We also exhibit that some simplicial
polyhedra like bipyramids, Kleetopes, gyroelongated bipyramids are coherently
labellable. Also we prove that pyramids over $n$-gons for $n\geq 4$, which are
not simplicial polyhedra, are coherently labellable. We prove that among
platonic solids, the cube and the dodecahedron are not coherently labellable,
even though, the tetrahedron, the octahedron and the icosahedron are coherently
labellable. Unlike the case of a tetrahedron, in general for a polyhedron, we
show that a coherent labelling need not induce a coherent labelling at a
vertex. We prove the main conjecture in the affirmative for a certain class of
polyhedra which are constructible from tetrahedra through certain types of edge
and face vanishing tetrahedron attachments. As a consequence we conclude that a
cube cannot be obtained from only these type of tetrahedron attachments. We
also give an obstruction criterion for a polyhedron to be not coherently
labellable and consequentially show that any polyhedron obtained from a pyramid
with its apex chopped off is not coherently labellable. Finally with the
suggestion of the affirmative results we prove the main theorem that any
simplicial polyhedron is coherently labellable.
