Bring's surface

From Polytope Wiki
Jump to navigation Jump to search
Bring's surface
Fundamental domain of Bring curve.svg
A fundamental polygon of Bring's surface.
Euler characteristic-6
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[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]

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[edit | edit source]

Regular tessellations of Bring's surface
Schläfli type Image Order of symmetry group Euclidean realizations
{5,5} Great dodecahedron fundamental domain.svg 120 Great dodecahedron
Small stellated dodecahedron
{5,4} Dodecadodecahedron fundamental domain.svg 240 Dodecadodecahedron
{4,5} Medial rhombic triacontahedron fundamental domain.svg 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]

  1. 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.