assignments
due Tue Sep 8
getting started
- Play around with either (or both)
- the online TopSpin puzzle, or
- Rubik's Cube.
- What sorts of things are you trying as you fool around with the puzzles? How are you thinking about them?
- 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 15
commutators and the dihedral group
- In class we defined (R, L, F) as the (right, left, flip) operations on the TopSpin. We examined the sequence (R F L F), found how many times that can be repeated before it loops around back to the original, and discussed why that is the "commutator" of R and F, written [R, F], and what that means.
- First, review what we did from your notes (or a friends), and look at the "commutator" definition on the definitions page (see the links in the left margin).
- Second, do a similar analysis for R2 and F. In other words,
- Write down the [RR, F] commutator in terms of R, F, and L. (In words its "rotate right twice, flip, undo the rotations, undo the flips").
- Do that on the TopSpin puzzle, and write down where the numbers go.
- Figure out how many repetitions of that it takes to return the puzzle to its original state. Does anything interesting happen along the way?
- Also in class we started to work out the group operation table (sometimes called the "Cayley table") of D4, rotating and flipping the numbers (1,2,3,4) in a square grid.
- Do a similar analysis for D3, the symmetries of solid equilateral triangle. There are 6 group elements: the identity, two rotations, and three ways to flip it upside down. Name these six elements, and work out the 6 x 6 group operation table describing what happens when you do two operations in a row. Try this on your own first. If you need help, you can look online at wikipedia:Cayley Table and wikipedia:dihedral group of order 6.
- Start thinking about the rotations of a solid cube: how many there are, and how they combine.
due Tue Sep 22
permutations and cycle notation
- S4 (the symmetric or permutation group of 4 symbols) has 4! = 24 elements. The rotational symmetries of a solid cube also has 24 elements, as discussed in class. Are these the same groups? Why or why not? (Think about the cycles of the different group elements and which ones are similar to each other.)
- We've now discussed in class several times one way to solve TopSpin, namely to do (R F L F) twice, which shifts once piece relative to the others. (I'm using left-to-right notation here.) Then by lining that piece up where it was and doing this many times you can shift one piece all the way around, eventually shifting it by just one place. Use this trick (and any others you like) to solve the TopSpin puzzle (if you haven't already), and discuss how that goes.
- Consider a 2x2x2 Rubik's cube, which is mostly like just looking at the corners of a 3x3x3 and ignoring everything else. If I unfold it and put different number for each it would look like this :
+----------+
| |
| 1 2 |
| |
| 4 3 |
| |
+----------+----------+----------+----------+
| | | | |
| 6 5 | 15 16 | 17 20 | 21 24 |
| | | | |
| 7 8 | 14 13 | 18 19 | 22 23 |
| | | | |
+----------+----------+----------+----------+
| |
| 9 12 |
| |
| 10 11 |
| |
+----------+
Now if I rotate the top face that has the numbers one
through four, the numbers below also cycle. In the
permutation notation from class, this move is then
rotate_top = (1,2,3,4)(15,6,21,17)(16,5,24,20)
What are some of the other moves on the 2x2x2 cube
in this notation?
due Tue Sep 29
odd, even, 2x2x2
- Describe a method of solving the TopSpin puzzle, either what I did in class or another method, in your own words.
- Consider S6, the permutations of 6 things.
- If the configuration of the identity if [1,2,3,4,5,6], is [2,3,4,5,6,1] an even or odd permutation? Why?
- Do that last operation again - what do you get? Is that odd or even? Why?
- If you didn't look at the cycle notation on the 2x2x2 cube from the last assignment, revisit it now that I've talked about it in class.
- Start working through a group theory examination of the 2x2x2 rubik's cube (i.e. a regular cube ignoring everything but the corners). In particular, we would like to know things like
- What are the "generators"?
- How many group elements (i.e. positions) are there?
- What is the commutator of two of the 1/4 turn generators?
- What is the order (i.e. length of the cycle) of that commutator?
- What is commutator of a 1/4 turn move with a 1/2 turn on an adjacent side.
- What is the order of that?
- Is it possible for a commutator to move only two of the corners? Why or why not? (Hint: think odd and even ...)
due Tue Oct 6
conjugacy & continue 2x2x2 Rubik
- Last class I defined what it meant for two group operations to be conjugate : A and B are conjugate if there is X such that A = X B X-1 .
- For the 24 element solid cube group, find X in the relation above to show that two different 1/4 turn face-center axis rotations (F and R, say) are conjugate.
- Do the same for two corner-to-corner 1/3 turn rotations.
- If a 1/4 rotation is one move, which of the 24 positions are an odd or even number of moves from the solved position?
- Continue exploring the 2x2x2 cube (i.e. a regular cube ignoring everything but the corners), using the resources I posted in class and some group theory.
- Pick a commutator of some simple moves. (We did one of these a bit in class.)
- What is its order (i.e. length of the cycle) of that commutator?
- Does anything interesting happen along the way when you repeat it?
- Is it possible for a commutator to move only two of the corners? Why or why not?
- Come to class ready to discuss what you've learned about ways to manipulate the 2 x 2 x 2.
due Tue Oct 13
solving the cube - class discussion
- As agreed last class, we're going to have a class discussion of what you folks found this week while looking at recipes for solving (at leat) the 2x2x2 , and we'll discuss some approaches to the 3x3x3 as well.
- No need to turn in anything this week.
due Tue Oct 20
hendricks days
due Tue Oct 27
solving the cube - your method
- Practice solving the 3 x 3 x 3 , but any of the methods
- Describe your approach and your experience, and come to class ready to discuss.
due Tue Nov 3
change ringing
- Browse through the links on my Oct 27 lecture notes. Pay particular attention to the plain hunt on 6 bells, and make sure you know how that works.
- Come to class on Thursday ready to try the plain hunt.
- Come up with your own version of the exercise that we did in class, namely, invent and/or explain how to change ring all 24 permutations of 4 bells. Only neighbor swaps are allowed. Explain what each person 1 through 4 does by giving the pattern that they ring (i.e. twenty four numbers 1 though 4, for where they ring their bell) and how you think they might remember that pattern. (You might want to start with the plain hunt, and find something to add.)
due Tue Nov 10
final project proposal
- Propose a puzzle for your final project. Discuss briefly what you have found out so far, and what you're going to do next.
- Look through articles listed at the top of the Nov 10 lecture notes - we'll discuss 'em next week.
due Tue Nov 17
wallpapers
- Check out these three images : ( tiles , card , wall ). (You'll need to be logged in to access them.)
- Using the resources from my Nov 10 notes, or other online sources, decide which of the 17 wallpaper groups each pattern belongs to, and explain why.
- Next week I'd like to discuss matrix multiplication and how that fits into groups and group theory. If you haven't done that in a previous math topic, please read about how that works. Googling "matrices" or "matrix multiplication" brings up sites like
- Here's one to try : \( \left[ \begin{smallmatrix} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{smallmatrix} \right] \left[ \begin{smallmatrix} 10 \\ 20 \\ 30 \end{smallmatrix} \right] \)
due Tue Nov 24
four dimensions
due Tue Dec 1
physics
due Tue Dec 8
project presentations
- Come to class ready to explain what you've learned about the puzzle that you chose.
due Fri Dec 11
final project submission
- Submit the written version of your final puzzle project: a short paper explaining things like how many positions, how to solve it, what pretty positions it has, etc. Please do include a bibilography and discussion of how you went about exploring and learning about the puzzle.
course grade
- a place for end of semester comments