  Brackets are an essential component in aircraft manufacture and design,
joining parts together, supporting weight, holding wires, and strengthening
joints. Hundreds or thousands of unique brackets are used in every aircraft,
but manufacturing a large number of distinct brackets is inefficient and
expensive. Fortunately, many so-called "different" brackets are in fact very
similar or even identical to each other. In this manuscript, we present a
data-driven framework for constructing a comparatively small group of
representative brackets from a large catalog of current brackets, based on
hierarchical clustering of bracket data. We find that for a modern commercial
aircraft, the full set of brackets can be reduced by 30\% while still
describing half of the test set sufficiently accurately. This approach is based
on designing an inner product that quantifies a multi-objective similarity
between two brackets, which are the "bra" and the "ket" of the inner product.
Although we demonstrate this algorithm to reduce the number of brackets in
aerospace manufacturing, it may be generally applied to any large-scale
component standardization effort.
