  During the last decade of high-pressure research a whole new series of iron
oxides was discovered, like Fe$_4$O$_5$, Fe$_5$O$_6$, Fe$_7$O$_9$ etc.,
featuring closely related structures with arrays of one-dimensional (1D) chains
of trigonal prisms embedded between slabs of octahedra. Here, we develop a
unified approach to the series based on a specific crystallographic generation
mechanism which predicts the structures of these oxides and naturally
classifies them in terms of the slab cycle. When including magnetic
interactions, we show that the 1D chains have a symmetry protection against
magnetic perturbations from the iron ions in the slabs, and that the slab size
determines the type of magnetic order, which is either ferromagnetic or
antiferromagnetic. Dynamical mean-field theory calculations reveal the
orbitally selective Mott state of the Fe ions and tendency of conductivity to
low-dimensional behavior with particular enhancement along the 1D chains.
Across the series, the decoupling of the chains increases, and so with the
inherent charge ordering of the slabs, these structures have the potential to
allow experimental realization of the model system of coupled 1D wires. We
point out the possibility to stabilize these compounds in the thin-film form
that, together with a wide range of possible ionic substitutions and fact that
these compounds are recoverable at ambient pressure, makes them a very
promising platform to engineer physical systems with interesting
magnetotransport phenomena, as corroborated by the recent discovery of quantum
Hall effect in ZrTe$_5$.
