Understanding energy landscapes is a major challenge in chemistry and biology. Although a wide variety of methods have been invented and applied tothis problem, very little is understood about the actual mathematical structures underlying such landscapes. Perhaps the most general assumption is the idea that energy landscapes are low-dimensional manifolds embedded in high-dimensional Euclidean space. While this is a very mild assumption, we have discovered an example of an energy landscape which is non-manifold, demonstrating previously unknown mathematical complexity. The example occurs in the energy landscape of cyclooctane, which was found to have the structure of a reducible algebraicvariety, composed of the union of a sphere and a Klein bottle, intersecting in two rings.
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