  We study for the first time a three-dimensional octahedron constellation for
a space-based gravitational wave detector, which we call the Octahedral
Gravitational Observatory (OGO). With six spacecraft the constellation is able
to remove laser frequency noise and acceleration disturbances from the
gravitational wave signal without needing LISA-like drag-free control, thereby
simplifying the payloads and placing less stringent demands on the thrusters.
We generalize LISA's time-delay interferometry to displacement-noise free
interferometry (DFI) by deriving a set of generators for those combinations of
the data streams that cancel laser and acceleration noise. However, the
three-dimensional configuration makes orbit selection complicated. So far, only
a halo orbit near the Lagrangian point L1 has been found to be stable enough,
and this allows only short arms up to 1400 km. We derive the sensitivity curve
of OGO with this arm length, resulting in a peak sensitivity of about
$2\times10^{-23}\,\mathrm{Hz}^{-1/2}$ near 100 Hz. We compare this version of
OGO to the present generation of ground-based detectors and to some future
detectors. We also investigate the scientific potentials of such a detector,
which include observing gravitational waves from compact binary coalescences,
the stochastic background and pulsars as well as the possibility to test
alternative theories of gravity. We find a mediocre performance level for this
short-arm-length detector, between those of initial and advanced ground-based
detectors. Thus, actually building a space-based detector of this specific
configuration does not seem very efficient. However, when alternative orbits
that allow for longer detector arms can be found, a detector with much improved
science output could be constructed using the octahedron configuration and DFI
solutions demonstrated in this paper. (abridged)
