  Among crystal structures of N-component metal alloys, far fewer examples are
known with N>=4 than with N=2 or 3, in apparent contradiction to the
exponentially growing number of possible combinations of elements. Two effects
contribute to this shortfall. Since the N-component composition space resides
within a d-dimensional simplex with d=N-1, the vanishing volume in high
dimensions reduces the distinct N-component compositions. Additionally, the
increasing surface area makes it more probable that stable structures reside on
the surface of the simplex (containing fewer than N components) as opposed to
its interior. Despite their rarity, we propose that the actual number of N=4-
and 5-component alloys greatly exceeds the number that are currently known.
Specific estimates are developed through application of the empirical Miedema
enthalpy model.
