  We develop the geometrical, analytical, and computational framework for
reactive island theory for three degrees-of-freedom time-independent
Hamiltonian systems. In this setting, the dynamics occurs in a 5-dimensional
energy surface in phase space and is governed by four-dimensional stable and
unstable manifolds of a three-dimensional normally hyperbolic invariant sphere.
The stable and unstable manifolds have the geometrical structure of spherinders
and we provide the means to investigate the ways in which these spherinders and
their intersections determine the dynamical evolution of trajectories. This
geometrical picture is realized through the computational technique of
Lagrangian descriptors. In a set of trajectories, Lagrangian descriptors allow
us to identify the ones closest to a stable or unstable manifold. Using an
approximation of the manifold on a surface of section we are able to calculate
the flux between two regions of the energy surface.
