  Networks of atom-centered coordination octahedra commonly occur in inorganic
and hybrid solid-state materials. Characterizing their spatial arrangements and
characteristics is crucial for relating structures to properties for many
materials families. The traditional method using case-by-case inspection
becomes prohibitive for discovering trends and similarities in large datasets.
Here, we operationalize chemical intuition to automate the geometric parsing,
quantification, and classification of coordination octahedral networks. We find
axis-resolved tilting trends in ABO$_{3}$ perovskite polymorphs, which assist
in detecting oxidation state changes. Moreover, we develop a scale-invariant
encoding scheme to represent these networks, which, combined with
human-assisted unsupervised machine learning, allows us to taxonomize the
inorganic framework polytypes in hybrid iodoplumbates (A$_x$Pb$_y$I$_z$).
Consequently, we uncover a violation of Pauling's third rule and the design
principles underpinning their topological diversity. Our results offer a
glimpse into the vast design space of atomic octahedral networks and inform
high-throughput, targeted screening of specific structure types.
