assignments
due Tue Sep 9
getting started
- Play around with the online TopSpin puzzle. What sorts of things are you trying? How are you thinking about it?
- Play around with a Rubik's Cube. Again, pay attention to what sorts of things you're trying.
- Browse the links I've set up at the left of this page.
- Start looking at this "group" notion, as described at wikipedia:group (mathematics).
- Using the "submit work" link, tell me:
- What is your math background?
- How much experience do you have with Rubik-ish puzzles?
- How did the rest of this assigment go?
due Tue Sep 16
dihedral group and commutators
- Finish the group operation table for the dihedral group that we started in class. I've set up this dihedral page to write down the notation we decided on in class.
- The "commutator" of two group elements A and B is (A * B * A' * B') where A' is the inverse of A, with (A * A' = E) where E is the identify. If the commutator of two elements is the identity, those two are said to "commute".
- Do any of the topspin generators commute? Which ones? Do some *not* commute? Explain, and be specific. What is the commutator of two that don't commute?
- Answer the same questions for Rubik's cube.
- Not sure what the words mean? Check out the definitions page, or re-read some of the wikipedia articles on group theory, or ask someone.
- Last class we talked about several groups. Browse the following articles about them. (They get technical in places - I just want you to get the gist, not all the details.)
- Likewise, browse this article : wikipedia:permutation group. We'll be talking about those groups in our next class.
due Tue Sep 23
topspin
In class I described one method of solving the topspin puzzle. For this assignment, I'd like to walk you through that and have you explain what's going on.
To write a state (or part of a state) of the puzzle, write some sequence of numbers and put curly braces around those within the flipper. For example,
1 { 2 3 4 5 } 6 7
means that 1 is just outside the flipper to the left and 6 is just outside to the right.
Let's use these names for the three generators:
R = "right" : move all numbers clockwise
L = "left" : move all numbers counter-clockwise
F = "flip" : turn the small "flipper" dial half way
And call this the "solved" state :
E = "solved" : 1 { 2 3 4 5 } 6 7 ... 18 19 20
- What does the puzzle look like after the moves C = ( R F L F ) ?
- Explain why C is a commutator, namely (X Y X' Y'), where X' means "undo X".
- What does C C = C2 look like? C3? C4?
- In class I claimed that C2 was particularly useful, because of what happens to the "1". Explain.
- What does the puzzle look like after (C C L L L L ) = C2 L4 ? How many moves of the generators (i.e. R, L, F) is this altogether?
- Let's call S = (C2 L4). What does the puzzle look like after S5? How many moves of the generators is that altogether? Why is that a particularly interesting position?
- Any transformation done repeatedly leads back to the solved position. For example R20 = E. We call 20 the "cycle length" of R. What is the cycle length of F? Of C? Of S? (That last one is a bit tricky.)
- Finally, see if you can use all these ideas to solve a messed-up topspin puzzle, particularly the last few that are out of place, which is usually the hard part. Describe your experience, and feel free to come find me (or other folks in the class) for helpful hints.
due Tue Sep 30
small groups
This is week's assignment is aimed at understanding one question: how many different groups are there with up to eight elements? Please try to think about this without just looking up the answer (online or elsewhere); instead, I'd like you to see what you can figure out. Talking with other people in class is fine; just don't go google the answer and write it down.
- There are exactly two different groups with 4 elements: C4 (cyclic group with 4 elements; also called Z4), and D2 (also called Dih2, symmetries of a regular polygon with 2 sides).
- Write down the group operation table for these two groups.
- What is the order (cycle length) for each element of each of these two groups?
- For each, is it an Abelian group? ("Abelian" means that all the commutators are boring, in other words A * B * A-1 * B-1 = identity for any A and B.)
- How do you know that these two are not the "same"? (You'll have to think about what it might mean for two groups to be the same.)
- There are exactly two different groups with 6 elements.
- What are they?
- What is the order of each element of each?
- For each, is it Abelian?
- How many groups can you find with 5 elements? Discuss.
- Can you think of any ways to visualize one of these groups as a single "thing"?
due Tue Oct 7
edges only on rubik's cube
This week we're going to look at how the edges on the Rubik's Cube move under some the basic commutators, and how you can use that to put them in place ... while ignoring the corners.
- First study the commutator T = F R' F' R .
- How many edges are disturbed, and where do they go?
- To do this on a physical cube, hold it in your hands and turn the "left" and "right" sides. I think of this as "down, down, up, up", where "down" and "up" refer to whether the edge in the center is moving down or up.
- How many times to you need to do T to return the edges to their starting positions? How many times to return the edges and corners?
- The operator T moves pieces near a single corner.
- What happens if you turn the whole cube in your hands around that corner, and then apply a similar sequence to two different sides?
- In other words if Q = U F' U' F, then where are the sides after T Q ?
- Now study the commutator S = F' R' F R .
- (I think of this as "up down down up", looking at leftside/rightside twists.)
- How many edges are disturbed, and where do they go?
- See how far you can get in trying to solve just the edges using these two sequences on different pairs of sides, cycling just a few edges at a time. Work on nearby edges, so that the ones remaining out of place are near each other. Typically as you get nearly done two things can go wrong:
- case 1 : everything is in place, but two of them are flipped. The question up above about using the operator T on two different sides near the same corner can fix that.
- case 2 : Two edges are out of place. In that case, the parity is wrong. (We've mentioned this concept once before, and will discuss again next class. Basically, an even number of moves will leave the edges in an even parity permutation, while an odd number of moves leaves in odd parity.) So, make one quarter turn move, to put one of the wrong edges in its proper place. Now there are three edges out of place. Then continue with the S and T commutators described above.
- Play around with as much of this as you can with real and/or virtual Rubik's cube, tell me how it came out, and we'll continue to discuss in class next week.
due Tue Oct 14
open Rubik's Cube exploration
- Use any of the ideas we've discussed in class to try to figure out a way to solve Rubik's cube.
- Or pick another method from one of the many online. (There's a list at the bottom of the rubiks cube page.)
- Describe what you've managed to do, and what you're still having trouble with.
due Tue Oct 28
Rubik's paper
- -- mid-term project grade --
- Write up a short (3-5 pages or equivalent) description of what the Rubik's Cube is all about, and what it has to do with group theory.
- Include a description of your exploration of Jim's or other online solution techniques.
- Include how many positions it has, and where that number comes from.
- Say the sorts of things you say to an academically minded friend who asked "So, what's the story with that puzzle, and why is that course you're taking say it's about 'group theory'?"
- Do include references to online or other sources you use, as you would in any academic work.
- Do make it clear what parts of all this make sense to you.
due Tue Nov 11
final project proposal
- Choose a puzzle to investigate using the methods we've discussed this semester.
- A good place to look is Jaap's puzzle pages listed on the resources page.
- I'd prefer it to be a group theory puzzle, unless you can make an argument to convince me otherwise.
- Describe a bit about what the puzzle is, and how it's similar or different to the TopSpin and Rubik's Cube that we've already done.
due Tue Nov 18
Wallpaper homework
- Encrypt and decrypt some words, sentences and famous phrases using matrices and their inverses. (It is best to use matrices that represent a rotation or refection).
- Take your favorite wallpaper pattern (if it can be found online please give a link to it) and describe the symmetries of the wallpaper, in particular its point group and the lattice used.
due Tue Nov 25
Wallpaper homework II
- Create your own wallpaper pattern and state the wallpaper group used.
- From any of the 17 wallpaper groups discussed, pick one that has two elements that do not commute, and calculate their commutator.
due Tue Dec 9
Project presentations
- Come to class ready to discuss your puzzle, what you know about it, and what is still problematic.
due Fri Dec 12
Final Project
- As described above, submit an analysis and discussion of your chosen puzzle.
- It should be roughly equivalent to the work of a five page paper, though depending on the format (diagrams, move sequences, prose) the actual number of words may vary.
- Be clear about all sources used, what is your own work, and what are ideas from other places.
- Do include a discussion of the group theory of this puzzle, including things like
- how many positions can be reached, and why
- what positions aren't possible, and why
- what moves can be considered generators,
- what is permuted by the generators,
- examples of which of the generators commute, and which don't
- And of course you can talk about solving it. But don't have that be the only thing, eh?
Course grade
- ... will be filled in here by Jim.
