Are you into cubes? I made a reference sheet / infographic of the cube’s symmetry group, and every single subgroup. There are 98 subgroups, so… I hope you like lots of cubes.
This image is under CC BY-NC-SA 2.0. Click for the full size (hosted on Flickr).
Group theory
This diagram is derived from group theory, a mathematical subject that you would study in an abstract algebra course. You can think of a group as a set of transformations or symmetries, although technically it’s a bit broader than that. Group theory is useful in chemistry and physics–and of course, origami, although most origamists I know don’t think much about group theory at all.
The symmetry group of the cube is formally known as the octahedral group. It contains 24 rotations and 24 reflections. In total, that’s 48 symmetries, so we say that the group has order 48.
The octahedral group has 98 subgroups (including itself). Each subgroup contains a subset of the original 48 symmetries. Not just any subset will do; subgroups obey certain rules. For instance, if a rotation and reflection are in the group, then the group must also contain the symmetry consisting of the rotation followed by the reflection.
Each subgroup is represented by a cube that obeys all its symmetries. Orange and blue are just used to visually distinguish the front and the back of the cube. As far as the symmetries are concerned, it’s okay to rotate/reflect blue triangles into orange triangles and vice versa.
The subgroups have been organized into conjugacy classes. If two symmetry groups are in the same conjugacy class, then it is possible to rotate/reflect one group into another. This process is called conjugation. For example, suppose we have a symmetry group consisting of 90 degree rotations around the y axis. Then we have another symmetry group consisting of 90 degree rotations around the x axis. These two groups are conjugates, because you can rotate the y axis into the x axis.
Conjugacy classes depend on what sorts of conjugation are allowed. For instance, can you conjugate groups by rotating them 45 degrees, even though 45-degree rotations aren’t among the original 48 symmetries of the octahedral group? A “conjugacy class in Oh” only allows conjugation using rotations/reflections that were in the octahedral group. A “conjugacy class in O(3)” allows conjugations using any rotations/reflections. (Oh is the notation for the octahedral group, and O(3) is the notation for the group consisting of all possible rotations and reflections.)
I’ve labeled each of the conjugacy classes (in O(3)) with Schoenflies notation. I don’t think the notation system matters too much, but it’s helpful to have names that we can refer to.
The making of the diagram
I think a lot about symmetry groups, and have found myself on many occasions referring to this diagram on Wikipedia. It’s a beautiful diagram, but it’s not very intuitive, and I think most people would not realize at a glance that it represents the same thing as my diagram above. Even though I know how to read it, it still takes me a while each time to figure out what each of the colors means. So I’ve had this idea for a while to make a diagram that would be more intuitively readable. And though most people still won’t understand what it represents, I hope it’s at least a bit more evocative.
I didn’t realize until I started making it just how hard it would be. Especially when I decided to expand out all the conjugacy classes, that’s so many cubes. And there just isn’t a good way to make all the arrows seem anything other than chaotic. It’s so big that I probably made a mistake somewhere. If you spot a mistake, please let me know so I can fix it. I also have vector graphics if anybody is interested.
I wanted to give the same treatment to the icosahedral group, but I counted and there are 174 164 subgroups, so I’m not rushing into that one. It would also be neat to do the wallpaper groups, but there are some unique challenges related to those.
I would have loved to have a diagram like this when I took abstract algebra in undergrad. Abstract algebra is really tough to visualize, I came away not really understanding what it meant to have a normal subgroup. Normal subgroups really pop out in a diagram like this.
I’m more familiar with Lie groups. Discrete groups would be more a solid-state thing, I guess.
Yeah, there’s some discrete group theory in crystals, and geometry was one of the things that drew me to condensed matter physics in the first place. Sadly I never had much chance to learn it or use it in that context.
I did my PhD in finite group theory, but that was a long, long time ago. There’s a close relative of O_h which is the the binary octahedral group. (Aka double cover of O, or central extension of O by Z_2). I think it can be regarded as the rotations of a ‘fermionic cube’ where it takes 720 degrees to get back to the start.
@grahamjones,
So, you’re talking about a finite subgroup of U(2)? That’s really neat, I never thought about that. Are there any exotic regular “polyhedra” you can get from that, or is it only double-covers of familiar polyhedra?
Yes, it’s a finite subgroup of SU(2) and so of U(2). I don’t know the answer to your other question. I think in terms of abstract groups, not polyhedra. It’s also isomorphic to GL(2,3) – perhaps you can make something geometric out of that. It’s also the semidirect poduct of the quaternion group of order 8 with the permutation S_3 group on three letters (which you think of as i,j,k).
For groups like this it can be very helpful to consider the group’s action on various features of (in this case) the cube, and then consider what the corresponding stabilizers must look like. As an exercise look at the above diagrams and try to identify the stabilizers for: a face; a body diagonal; a single vertex (note that a symmetry that fixes a body diagonal may either fix or swap the vertices at the ends!); an edge; an inscribed tetrahedron; etc.; there are less immediately obvious stabilizers as well, such as of the cube’s “handedness”.
Essentially if you can “project” the cube down to some smaller set of things or “distill” a property like handedness from it, the group has an action on that set, the things in that set have an orbit under that action, and there are subgroups of the group that stabilize one or another of the things in the set. Subgroups like the group of rotations also can be further broken down in this way. The orbit-stabilizer theorem (the orbit’s size equals the index of the stabilizer) helps enormously here, of course.
So the inscribed tetrahedron case must have index two, since there are only two such tetrahedra and there are indibutably symmetries of the cube that exchange them (e.g., inversion through the center). That stabilizer must therefore be one of the three order-24 subgroups near the top (and in fact it’s the rightmost in your diagram, since it must include reflection symmetries of the tetrahedron, which the left one does not, and must not have the inversion as a subgroup, which the middle one does). A vertex stabilizer must have index 8 in both the full group and the rotation subgroup since the orbit of a vertex under either is all eight vertices; so the only cluster of order 3 subgroups must be the rotation subgroup’s vertex stabilizers (and indeed there are four, one for each body diagonal, whose elements will be 120 degree rotations about the relevant diagonal), and the full group’s vertex stabilizers must contain those as index-2 subgroups, so are a conjugacy class of order 6 subgroups containing reflections. The left conjugacy class of order 6 subgroups is only rotations, so it’s not that one, and members of the middle one contain the inversion subgroup, whose nontrivial element fixes no vertices, so it cannot be that one either and is therefore the one on the right. (Or, note that the vertex stabilizers must be subgroups of the tetrahedron stabilizer, which is true only for the rightmost cluster of order-6 subgroups. They have index 4 rather than 8 in that, because the orbit of a vertex under the action of the tetrahedron stabilizer is only it and the other three vertices that belong to the same inscribed tetrahedron.)
Icosahedral group next?
@B. Bound,
My method of representing each symmetry group was to start with a fundamental domain (a region which, when rotated/reflected, covers the whole cube exactly once)–a triangle. Then for each subgroup, I colored in the orbit (within the subgroup) of that triangle. The construction implies that the stabilizer of the set of colored triangles is the subgroup being represented. Therefore, it uniquely determines the subgroup.
And you’re right, the symmetry of a tetrahedron is T_d (rather than T_h)
@mathman85,
Yeah, eventually. I did an initial assessment, and while there are 174 subgroups of the icosahedral group, it actually has fewer conjugacy classes. There’s also no difference between the conjugacy classes in I_h and O(3). So in a way it’s simpler.
I have no idea how to do the wallpaper groups though, since those are all infinite order.