Bring's curve
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An early picture of Bring's curve as a floor mosaic by Paolo Uccello, 1430
In mathematics, Bring's curve (also called Bring's surface) is the curve given by the equations
It was named by Klein (2003, p.157) after Erland Samuel Bring who studied a similar construction in 1786 in a Promotionschrift submitted to the University of Lund.
The automorphism group of the curve is the symmetric group S5 of order 120, given by permutations of the 5 coordinates. This is the largest possible automorphism group of a genus 4 complex curve.
The curve can be realized as a triple cover of the sphere branched in 12 points, and is the Riemann surface associated to the small stellated dodecahedron. It has genus 4.
References
- Bring, Erland Samuel; Sommelius, Sven Gustaf (1786), Meletemata quædam mathematica circa transformationem æquationem algebraicarum, Promotionschrift, University of Lund
- Edge, W. L. (1978), "Bring's curve", Journal of the London Mathematical Society, 18 (3): 539–545, doi:10.1112/jlms/s2-18.3.539, ISSN 0024-6107, MR 518240
- Klein, Felix (2003) [1884], Lectures on the icosahedron and the solution of equations of the fifth degree, Dover Phoenix Editions, New York: Dover Publications, ISBN 978-0-486-49528-6, MR 0080930
- Weber, Matthias (2005), "Kepler's small stellated dodecahedron as a Riemann surface", Pacific J. Math., 220: 167–182
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