  Maniplexes are coloured graphs that generalise maps on surfaces and abstract
polytopes. Each maniplex uniquely defines a partially ordered set that encodes
information about its structure. When this poset is an abstract polytope, we
say that the associated maniplex is polytopal. Maniplexes that have two
properties, called faithfulness and thinness, are completely determined by
their associated poset, which is often an abstract polytope. We show that all
faithful thin maniplexes of rank three are polytopal. So far only one example,
of rank four, of a thin maniplex that is not polytopal was known. We construct
the first infinite family of maniplexes that are faithful and thin but are
non-polytopal for all ranks greater than three.
