Sep 8
ground rules
first explorations
Discuss first assignment. How did you think about these things?
Let's look at topspin systematically.
- How many short things are there we can do?
- What do short (up to 4 moves, say) sequences of those moves do?
- How many of those are there? Are there any interesting ones?
- What happens if we repeat a sequence? (That's easy to do, and easy to remember.)
- How can we organize our thinking?
- Write it down.
- So we need a notation system
Let's define
R rotate to the right
L rotate to the left
F flip the center wheel
and then write a sequence like
RFLF right then flip then left then flip
and also sketch the numbers on the wheel, and follow what happens to 'em.
A key idea: any position we can get to (by a sequence of moves) is itself a transformation that we can apply by doing those moves. So we could give that a new name, and add to the (R, L, F) list.
Before doing this for top spin, let's look at a really small example.
D4
Start with something sort of like topspin but much smaller.
Consider a circle of four numbers.
1
4 2
3
We're going only two transformations: a rotation and a vertical flip.
What positions can we get to, and how do they relate to each other?
Let
R = rotate clockwise
4
3 1
2
F = flip top and bottom turning the original into
3
4 2
1
Then consider what happens when we several in a row:
we get some other transform.
Work this out in class and then write down the whole
"multiplication table" for this thing.
groups
The collection of things we just discussed and the "do two in a row" is, in fact, what a mathematical "group" is all about. Here are the wikipedia articles for the one we just discussed.
Here's the math technobabble definition
A "group"
a set of things G = {a, b, c, ...} and
a binary operation a * b
that have the following properties :
closure : For all x,y in G, x*y=z is also in G.
identity: There exists an element e of G
such that for all x in G, e*x = x*e = x .
associativity: For all x,y,z in G, (x*y)*z = x*(y*z)
inverse: For each x in G, there is x'
such that x*x' = x'*x = e.
- Discuss some other examples of groups, such as C3.
- Find some sets with an operation which are not groups.
some buzzwords
(I'm collecting all these on the "definitions" page; see links to left.)
- order of a group : number of elements.
- abelian : x * y = y * x for all x,y.
- generators: a set {a,b,c,...} such that any element z = a*b*... using only elements in that set.
our puzzles as groups
- Key point: there's a 1-to-1 match between a position of a puzzle and the transformation that takes the puzzle from the solved state to that position.
- Discuss that idea: it can be a subtle one.
- Example: for the dihedral clock, "do nothing" is the "identity" element that leaves it unchanged.
- The primitive "flips" and "turns" of each puzzle are then "generators" which when applied (with the binary group operation) reach any position (i.e. any transformation, which is any group element).
That was a lot. Discuss it in the context of D4: what does that look like when thought of as a puzzle? As a group?
group theory background
what's next
- next assignment posted soon; stay tuned.
