Bring's surface

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

Equations
An immersion of Bring's surface in $$\mathbb{P}_4(\mathbb{C})$$ can be defined as the solutions to the equations: where $$z_i$$ uses homogeneous coordinates.
 * $$\sum_{i=1}^5z_i=0$$
 * $$\sum_{i=1}^5z_i^2=0$$
 * $$\sum_{i=1}^5z_i^3=0$$

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: Edges are associated without a half twist as Bring's surface is orientable.
 * 0, 7
 * 1, 10
 * 2, 13
 * 3, 16
 * 4, 11
 * 5, 14
 * 6, 17
 * 8, 15
 * 9, 18
 * 12, 19

Tessellations of Bring's surface
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.