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==Applications==
In the geometry of [[chemical compound]]s, it is common to visualize an [[atom cluster]] surrounding a central atom as a polyhedron—the [[convex hull]] of the surrounding atoms' locations. The [[tricapped trigonal prismatic molecular geometry]] describes clusters for which this polyhedron is a triaugmented triangular prism, although not necessarily one with equilateral triangle faces.{{r|kepert}} For example, the [[lanthanide]]s from [[lanthanum]] to [[dysprosium]] dissolve in water to form [[cation]]s surrounded by nine water molecules arranged as a triaugmented triangular prism.<ref name=Persson2022>{{
In the [[Thomson problem]], concerning the minimum-energy configuration of <math>n</math> charged particles on a sphere, and for the [[Tammes problem]] of constructing a [[spherical code]] maximizing the smallest distance among the points, the minimum solution known for <math>n=9</math> places the points at the vertices of a triaugmented triangular prism with non-equilateral faces, [[Circumscribed sphere|inscribed in a sphere]]. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is not known.{{r|whyte}}
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