§ John Conway: The symmetries of things
Original way to classify wallpaper groups: think of geometric transforms
that fix the pattern. Thurston's orbifold solution: think of quotients of
by groups --- this gives you an orbifold (orbit manifold).
Take a chair, surround it around by a sphere. The symmetries of a physical
object fixes the center of gravity. So we pick the center of the sphere to
be the center of gravity. The "celestial sphere" (the sphere around the chair)
is a nice manifold (We only have the surface of the sphere). The vertical
line that divides the chair also divides the sphere into two parts.
Conway came up with his notation for wallpaper groups/orbifolds. There are only
four types of features.
- The points of the orbifold are orbits of the group.
- So now the orbifold gives us a hemisphere in this case.
- The topology of the orbifold determines the group.
- This is astonishing, because the group is a metrical object: elements of the group preserve the inner product of the space.
- And yet, geometrical groups are determined by the topology of their orbifolds!
- Thurston's metrization conjecture: certain topological problems reduce to geometrical ones.
An orbifol can be made to carry some amount of metrical information. The cone
point only has 90 degrees, so it is in some sense, "a quarter of a point".
- The hemisphere orbifold is
*. (group of order 2).
* denotes the effect on the orbifold.
* really means: what is left out of a sphere when I cut out a hemispherical hole.
* is the name for a disk, because a hemisphere is a disk topologically. It has metrical information as well, but we're not going to speak about it, because all we need is the topological information.
- One-fourth of a sphere (symmetry group of rectangular table) is denoted by
* 2 2. The
* for the hemisphere, and
2, 2for the angles of
- If the table is a sphere, then we have diagonal symmetry as well. In this case, the orbifold has angle
pi/4. So the table is
* 4 4.
- If we take a cube, then we have an even more complicated orbifold. The "fundamental region" of the cube has 2, 3, and 4 mirrors going through them. So in the orbifold, we get triangles of angles
pi/2, pi/3, pi/4. This would be
* 4 3 2.
- Draw a swastika. This has no reflection symmetry. This has a gyration : a point about which the figure can be rotated, but the point is NOT on a line of reflection. We can tear the paper and make it into a cone. This gives us a cone point . The angle around the cone point is
2pi/4. This is the orbifold of the original square with a swastika on it.
Let us try to work out the euler characteristic of the rectangular table
orbifold by using . The orbifold as one face.
The wrong thing to say is that the orbifold has two edges and two vertices.
It is untrue because the edge of the orbifold is only half an edge --- let's
say that lines have thickness. In this case, we will have , ,
and . The euler characteristic works out to be a half. This is appropriate,
because the orbifold is a type of divided manifold.
- Draw a cube with swastikas marked on each face. This has no reflection symmetry. Once again, we have a gyration, and again, only the gyration/singularities matter. This group is again
4, 3, 2 , but in blue . In this notation, red is reflection, blue is "true motion" (?).
Now, see that we started with positive euler characteristic (2), and we divide
it by some
- If we work this out for a cube, we get . This is because the sphere gets divided into 48 pieces, and the sphere has an euler characteristic of 2!
- Alternatively, we can think that we started out with 2 dollars, and we are then buying the various features of our orbifold.
1$, a blue number after a star, for example:
1/2 a dollar.
2/3 of a dollar,
3/4 of a dollar. In general,
1 - 1/n. The red numbers are children, so they cost half an much:
1/2(1 - 1/n) = (n-1)/2n.
n (the order of the group). So we end up with a positive euler characteric.
By a sort of limiting argument, the euler characteristic of the wallpaper groups,
which are infinite, is zero. However, see that we must get to the zero by starting
with two dollars and buying things off the menu! If we try and figure out what
all the possible ways are to start with 2 dollars and buy things till we are
left with exactly 0 dollars, we get that there are 17 possible ways of buying
things on the menu! Thus, this the reason for there being 17 wallpaper groups.
Because we can completely enumerate 2-manifolds, we can completely enumerate
2-orbifolds, which are essentially the same thing as symmetry groups. The real
power is in the 3D case. We don't have a full classification of 3-manifolds. But
we maybe able to go the other way. This is the metrization theorem.
- To buy more than two dollars, you are buying symmetries from the hyperbolic plane!