# Bring's surface

Bring's surface
A fundamental polygon of Bring's surface.
Dimensions2
ConnectedYes
CompactYes
Euler characteristic-6
OrientableYes
Genus4
SymmetryS5, order 120[note 1]

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

## Construction

### Equations

An immersion of Bring's surface in ${\displaystyle \mathbb{P}_4(\mathbb{C})}$ can be defined as the solutions to the equations:

• ${\displaystyle \sum_{i=1}^5z_i=0}$
• ${\displaystyle \sum_{i=1}^5z_i^2=0}$
• ${\displaystyle \sum_{i=1}^5z_i^3=0}$

where ${\displaystyle z_i}$ uses homogeneous coordinates.

### Fundamental polygon

A fundamental polygon for Bring's surface. Identified edges are connected with a line and labeled with the same number.

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

Regular tessellations of Bring's surface
Schläfli type Image Order of symmetry group Euclidean realizations
{5,5} 120 Great dodecahedron
Small stellated dodecahedron