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 then every even edge is associated with the edge 7 steps clockwise and every odd edge is associated with the edge 7 steps anticlockwise. Edges are associated without a half twist as Bring's surface is orientable.

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.