Talk:Bring's curve

Curve vs. surface

Complex-analytic manifolds of complex dimension 1, and hence of real dimension 2, are sometimes called curves and sometimes called surfaces. This page should point out that the name "Bring's surface" is also used for what this page calls "Bring's curve". And note that the Wikipedia page for "Riemann surface" uses the name "Bring's surface". LyleRamshaw (talk) 05:05, 11 January 2015 (UTC)

Side identifications for the fundamental domain

The side identifications for the fundamental polygon are wrong. They disagree with those listed in the paper by Riera and Rodriguez as well as those in a recent paper by Ramshaw.

With the current side identifications, the geodesics extending the sides of the central pentagon close up once they hit the boundary of the fundamental domain (in sides 5, 6, 7, 8, or 9), but with an angle of pi/2 at the intersection point. But Bring's surface is kaleidoscopic, so the reflection in any side of any triangle in the tiling extends to an isometry of the whole surface. In particular, there should be global isometries of the surface acting by a reflection across any of the curves I am describing above. But if an isometry fixes two segments that intersect at an angle different from 0 or pi it is necessarily the identity, which is a contradiction. (Another way to see the contradiction is that the small angular sector at the intersection point has to be mapped to two different places by the "reflection") 70.29.237.98 (talk) 03:29, 31 March 2023 (UTC)

This has since been fixed. 173.178.144.119 (talk) 13:55, 14 April 2023 (UTC)

Riemann surface

@R.e.b. and David Eppstein: The article states since 2016 that the associated Riemann surface is the small stellated dodecahedron. Would that really be the non-convex polyhedron of 12 pentagrams? Or might it just be the pentakis dodecahedron, a convex polyhedron of 60 triangles? Watchduck (quack) 21:12, 13 March 2024 (UTC)

It could only be the pentakis dodecahedron if it is topologically a sphere. Is it? I imagine what it really means is that the associated Riemann surface is a smooth Riemann surface topologically equivalent to a complex of pentagons glued in the pattern of a small stellated dodecahedron. Or maybe pentagons with a cross-cap in the middle corresponding to the fact that the faces are pentagrams instead of pentagons? It looks both vague and unsourced to me. The first step should be to find a source and then clarify it based on what that source says.

Unfortunately, R.e.b. has not been editing Wikipedia for years. —David Eppstein (talk) 00:31, 14 March 2024 (UTC)

Bring's curve with dodecadodecahedral graph in green and its dual in violet. It is a quotient of the order-4 pentagonal tiling and its dual square tiling.
20-gon edges marked with the same letter are equal.

The article small stellated dodecahedron does indeed have a source. (Weber 2005) I can not claim to understand it. But I get the impression, that the dodecadodecahedron (shown in the source) is topologically equivalent to the green graph shown on the right. (So the violet graph would be equivalent to the medial rhombic triacontahedron.) The proposed image description on the right is equivalent to the one in Klein quartic. --Watchduck (quack) 20:40, 14 March 2024 (UTC)

@LyleRamshaw's preprint arXiv:2205.08196 is pretty neat, and has some parts related to this question. –jacobolus (t) 18:55, 15 March 2024 (UTC)