Abstract
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.
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Notes
Here ordered means that the order of edges in the n-gon matters: e.g., a triangle with edges ordered from shortest to longest is distinct from the same triangle with edges ordered from longest to shortest.
Strictly speaking, scalene triangles are only chiral when viewed as living in a two-dimensional universe. In three dimensions a scalene triangle and its mirror image are related by a rotation.
Here the 1 should be thought of as the semiperimeter of the triangle. This is why we fixed the perimeter to be 2.
With a view towards generalizations to n-gons in \(\mathbb {R}^3\) and questions like “What is the probability that a random n-gon is knotted”, which is of interest for modeling ring polymers.
Note that a triangle can also be doubly-degenerate, with two sides of length 1 and one of length 0. This means that two vertices coincide.
Rassat and Fowler [14] showed that any non-isosceles triangle is the most chiral triangle according to some chirality measure in a particular family, but their chirality measures are slightly unnatural to define on the sphere.
Meaning furthest from the subset of unknotted polygons.
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Acknowledgements
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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Bowden, L., Haynes, A., Shonkwiler, C. et al. Spherical geometry and the least symmetric triangle. Geom Dedicata 198, 19–34 (2019). https://doi.org/10.1007/s10711-018-0327-4
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DOI: https://doi.org/10.1007/s10711-018-0327-4