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Why there are Finitely Many Plane and Sphere Patterns

We've seen that symmetry groups of surfaces correspond to orbifolds. Amazingly, most objects that look like orbifolds correspond to real symmetry groups. If we can figure out all of the orbifolds that correspond to plane patterns, we'll know how many different types of plane patterns there are!

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)/2n

Orbifolds 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.

Buying all the 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

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