# 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]
Local curvatureHyperbolic

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:

• ${\displaystyle \sum _{i=1}^{5}z_{i}=0}$
• ${\displaystyle \sum _{i=1}^{5}z_{i}^{2}=0}$
• ${\displaystyle \sum _{i=1}^{5}z_{i}^{3}=0}$

where ${\displaystyle z_{i}}$ uses homogeneous coordinates.

### Fundamental polygon

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