Bring's curve

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

$v+w+x+y+z=v^{2}+w^{2}+x^{2}+y^{2}+z^{2}=v^{3}+w^{3}+x^{3}+y^{3}+z^{3}=0.$

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 0518240
• 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