  We construct Fano threefolds with very ample anti-canonical bundle and Picard
rank greater than one from cracked polytopes - polytopes whose intersection
with a complete fan forms a set of unimodular polytopes - using Laurent
inversion; a method developed jointly with Coates-Kasprzyk. We also give
constructions of rank one Fano threefolds from cracked polytopes, following
work of Christophersen-Ilten and Galkin. We explore the problem of classifying
polytopes cracked along a given fan in three dimensions, and classify the
unimodular polytopes which can occur as 'pieces' of a cracked polytope.
