Systems of particles interacting via inverse-power law potentials have an invariance with respect to changes in length and temperature, implying a correspondence in the dynamics and thermodynamics between different 'isomorphic' sets of temperatures and densities. In a recent series of works, it has been argued that such correspondences hold to a surprisingly good approximation in a much more general class of potentials, an observation that summarizes many properties that have been observed in

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... e past. In this paper we show that such relations are exact in high-dimensional liquids and glasses, a limit in which the conditions for these mappings to hold become transparent. The special role played by the exponential potential is also confirmed.