  Fully 3-dimensional computations of flow through a long pipe demand a huge
number of degrees of freedom, making it very expensive to explore parameter
space and difficult to isolate the structure of the underlying dynamics. We
therefore introduce a `2+epsilon' dimensional model of pipe flow which is a
minimal 3-dimensionalisation of the axisymmetric case: only sinusoidal
variation in azimuth plus azimuthal shifts are retained, yet the same dynamics
familiar from experiments are found. In particular the model retains the
subcritical dynamics of fully resolved pipe flow, capturing realistic localised
`puff'-like structures which can decay abruptly after long times, as well as
global `slug' turbulence. Relaminarisation statistics of puffs reproduce the
memoryless feature of pipe flow and indicate the existence of a Reynolds number
about which lifetimes diverge rapidly, provided that the pipe is sufficiently
long. Exponential divergence of the lifetime is prevalent in shorter periodic
domains. In a short pipe, exact travelling-wave solutions are found nearby to
flow trajectories on the boundary between laminar and turbulent flow. In a long
pipe, the attracting state on the laminar-turbulent boundary is a localised
structure which resembles a smoothened puff. This `edge' state remains
localised even for Reynolds numbers where the turbulent state is global.
