# Bring's surface

Bring's surface | |
---|---|

A fundamental polygon of Bring's surface. | |

Dimensions | 2 |

Connected | Yes |

Compact | Yes |

Euler characteristic | -6 |

Orientable | Yes |

Genus | 4 |

Symmetry | S_{5}, order 120^{[note 1]} |

Local curvature | Hyperbolic |

**Bring's surface** is a genus-4 Riemann surface. It has the highest order symmetry group of any genus-4 Riemann surface.

## Construction[edit | edit source]

### Equations[edit | edit source]

An immersion of Bring's surface in can be defined as the solutions to the equations:

where uses homogeneous coordinates.

### Fundamental polygon[edit | edit source]

Bring's surface can also be constructed by associating specific sides of a hyperbolic icosagon. If the edges of the icosagon are numbered clockwise starting from 0 then the following associations are made:

- 0, 7
- 1, 10
- 2, 13
- 3, 16
- 4, 11
- 5, 14
- 6, 17
- 8, 15
- 9, 18
- 12, 19

Edges are associated without a half twist as Bring's surface is orientable.

## Tessellations of Bring's surface[edit | edit source]

Schläfli type | Image | Order of symmetry group | Euclidean realizations |
---|---|---|---|

{5,5} | 120 | Great dodecahedron Small stellated dodecahedron | |

{5,4} | 240 | Dodecadodecahedron | |

{4,5} | 240 | Medial rhombic triacontahedron |

Other related non-regular polyhedra are also topologically equivalent to tessellations of Bring's surface. For example, the truncated great dodecahedron, a uniform polyhedron, is a truncation of the great dodecahedron and thus topologically equivalent to a tessellation of Bring's surface.

## See also[edit | edit source]

## External links[edit | edit source]

- Wikipedia contributors. "Bring's curve".

## Notes[edit | edit source]

- ↑ Orientation preserving conformal mappings. Does not include reflections.

## Bibliography[edit | edit source]

- Weber, Matthias (2005). "Kepler's Small Stellated Dodecahedron as a Riemann Surface".
*Pacific Journal of Mathematics*.**220**.