Schwarz triangle

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In mathematics, a Schwarz triangle is a spherical triangle that can be used to tile a sphere. Each Schwarz triangle defines a finite group — its triangle group.

A Schwarz triangle is represented by three rational numbers (a b c) each representing the angle at a vertex. A n/d value means the vertex angle is d/n of the halfcircle. If it has a 2 means a right triangle.

For whole numbers there are only 4 groups, also called Mobius triangles:

  1. (2 2 p) - Dihedral
  2. (2 3 3) - Tetrahedral
  3. (2 3 4) - Octahedral
  4. (2 3 5) - Icosahedral

Contents

[edit] A complete list of Schwarz triangles grouped by symmetry

There are four families of Schwarz triangles based on their symmetry.

  1. Dihedral symmetry - {}x{n}
    • Right: (2 2 n)
  2. Tetrahedral symmetry - {3,3}
    • Right: (2 3 3), (2 3/2 3), (2 3/2 3/2)
    • Others: (3/2 3 3), (3/2 3/2 3/2)
  3. Octahedral symmetry - {3,4}
    • Right: (2 3 4), (2 3/2 4), (2 3 4/3), (2 3/2 4/3)
    • Others: (3/2 4 4), (3 4/3 4), (3/2 4/3 4/3)
  4. Icosahedral symmetry - {3,5}
    • Right: (2 3 5), (2 3/2 5), (2 3 5/4), (2 3/2 5/4)
      • (2 5/2 5), (2 5/3 5), (2 5/2 5/4), (2 5/3 5/4)
      • (2 5/2 3), (2 5/3 3), (2 5/2 3/2), (2 5/3 3/2)
    • Others: (5/2 3 3), (5/3 3/2 3), (5/2 3/2 3/2)
      • (3/2 5 5), (3 5/4 5), (3/2 5/4 5/4)
      • (5/2 5/2 5/2), (5/3 5/3 5/2)
      • (3/2 3 5), (3 3 5/4), (3/2 3/2 5/4)
      • (5/4 5 5), (5/4 5/4 5/4)
      • (5/3 5/2 3), (5/2 5/2 3/2), (5/3 5/3 3/2)

[edit] See also

[edit] References

[edit] External links

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