Humans and other animals build up spatial knowledge of the environment on the basis
of visual information and path integration. We compare three hypotheses about the
geometry of this knowledge of navigation space: (a) 'cognitive map' with metric Euclidean
structure and a consistent coordinate system, (b) 'topological graph' or network of
paths between places, and (c) 'labelled graph' incorporating local metric information
about path lengths and junction angles. In two experiments, participants walked in
a non-Euclidean environment, a virtual hedge maze containing two 'wormholes' that
visually rotated and teleported them between locations. During training, they learned
the metric locations of eight target objects from a 'home' location, which were visible
individually. During testing, shorter wormhole routes to a target were preferred,
and novel shortcuts were directional, contrary to the topological hypothesis. Shortcuts
were strongly biased by the wormholes, with mean constant errors of 37° and 41° (45°
expected), revealing violations of the metric postulates in spatial knowledge. In
addition, shortcuts to targets near wormholes shifted relative to flanking targets,
revealing 'rips' (86% of cases), 'folds' (91%), and ordinal reversals (66%) in spatial
knowledge. Moreover, participants were completely unaware of these geometric inconsistencies,
reflecting a surprising insensitivity to Euclidean structure. The probability of the
shortcut data under the Euclidean map model and labelled graph model indicated decisive
support for the latter (BFGM>100). We conclude that knowledge of navigation space
is best characterized by a labelled graph, in which local metric information is approximate,
geometrically inconsistent, and not embedded in a common coordinate system. This class
of 'cognitive graph' models supports route finding, novel detours, and rough shortcuts,
and has the potential to unify a range of data on spatial navigation.