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A new method to characterize the strength of magnetic frustration is proposedby calculating the minimum dimensionality of the absolute ground states of theclassical nearest-neighbor antiferromagnetic $n$-vector model with arbitrary$n$. Platonic solids in three and four dimensions and Archimedean solids havelowest-energy configurations in a number of spin dimensions equal to theirreal-space dimensionality. Fullerene molecules and geodesic icosahedra canproduce ground states in as many as five spin dimensions. Frustration is alsocharacterized by the maximum value of the ground-state energy when the exchangeinteractions are allowed to vary.

Competition between frustration and spin dimensionality in the classical antiferromagnetic $n$-vector model with arbitrary $n$