To each character of the orbifold notation we assign a certain *cost*,
as listed below. (We'll explain how these costs were computed in a later section.) The items in the lower
table are half price; they correspond to characters in the symbol
after the first * or x.

Name || Handle | Gyration Points --------------------------------------------------- Symbol || o | 2 3 4 5 6 ... n --------------------------------------------------- Cost || 2 | 1/2 2/3 3/4 4/5 5/6 ... (n-1)/n Name || Cross Cap / Mirror | Kaleidoscopic/Corner Points ----------------------------------------------------------------- Symbol || x or * | 2 3 4 5 6 ... n ----------------------------------------------------------------- Cost || 1 | 1/4 2/6 3/8 4/10 5/12 ... (n-1)/2nOrbifolds whose symbols cost less than two dollars correspond to symmetry groups of the sphere. Those whose cost equals two dollars correspond to symmetry groups of the plane. Ones that cost more than $2 correspond to symmetry groups of the hyperbolic plane.

Using these facts, we can easily enumerate the seventeen crystallographic groups in the plane, the seven infinite families of finite symmetry groups of the sphere, and seven other spherical symmetry groups.

Now we are ready to classify all the finite spherical symmetry groups.
This example should be very helpful to you in your homework -- you'll
be classifying all the **plane** symmetry groups!

The finite symmetry groups of the sphere are those that cost strictly less than two dollars and are listed in the left column of the table below. (There are two exceptions to this statement: the groups *mn and mn can exist only if m=n. The reason is that if a spherical polygon has just 2 angles, they are necessarily equal.) We see that there are seven isolated groups and seven infinite families of groups of symmetries of the surface of the sphere. The seventeen crystallographic groups in the plane are those whose symbol costs exactly $2. They are listed in the right column.

Sphere groups | Plane groups ------------------------------------------------------ achiral icosahedral *532 | hexascopic *632 chiral icosahedral 532 | hexatropic 632 achiral octahedral *432 | tetrascopic *442 chiral octahedral 432 | tetragyro 4*2 achiral tetrahedral *332 | tetratropic 442 pyritohedral 3*2 | triscopic *333 chiral tetrahedral 332 | trigyro 3*3 | tritropic 333 polydiscopic *22n | discopic *2222 polydigyros 2*n | dirhombic 2*22 polyditropic 22n | digyros 22* polyscopic *nn | didromic 22x polygyros n* | ditropic 2222 polydromic nx | monoscopic ** polytropic nn | monorhombic *x | monodromic xx (n > 0) | monotropic o

Author: Heidi Burgiel

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Created: Dec 7 1995 ---
Last modified: Jun 11 1996

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