Spherical geometry and the least symmetric triangle
We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar n-gons and points in the Grassmannian of 2-planes in real n-space introduced by Hausmann and Knutson, this corresponds to finding the point in the fundamental domain of the hyperoctahedral group action on the Grassmannian which is furthest from the boundary, which we compute exactly. We also determine the least symmetric obtuse and acute triangles. These calculations provide prototypes for computations on polygon and shape spaces.
KeywordsGrassmannians Triangles Optimality Chirality Asymmetry
Mathematics Subject Classification53A04 52A10 52B15
We would like to thank Vance Blankers, Jason Cantarella, Renzo Cavalieri, Andy Fry, Tom Needham, Eric Rawdon, and Gavin Stewart for stimulating conversations about the geometry of polygon space. We are especially grateful to Noah Otterstetter for his participation in our early conversations about this project and to the anonymous referee for their very helpful suggestions for improving this paper. This work was supported by a grant from the Simons Foundation (#354225, CS).
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