Sep 29
homework
To show that two 1/4 turns are similar, use notation
"x = rotation clockwise around x axis"
then y = z x z'
(the z rotation puts the y axis on top of where x was)
To show that two 1/3 corner turns are similar, we need
to move one corner to where another was; again this
can be done with a 1/4 face turn (depends on which corner
we want to get it to line up with). If
If 111 is the twist that goes through the x=y=z=1 axis,
and 101 is the one that goes through x=z=1, y=-1, then
101 = z' 111 z
This takes some visualization and keeping track of the
clockwise vs counterclockwise, and x,y,z right handedness rules,
but I think that's correct.
z
|_ y
/
x
parity of solid cube positions if 1/4 turn is one move
identity 0 quarter turns, i.e. even (1)
1/4 face 1 quarter twist, i.e. odd (6)
1/2 face 2 quarter turns, i.e. even (3)
1/3 corner 2 quarter turns, i.e. even (8)
1/2 edge 3 quarter turns, i.e. odd (6)
totals: 12 even, 12 odd
Aside: is this the same notion of parity as I defined before ?
Answer: Yes; cycle of 4 in S4 (1,2,3,4) => (2,3,4,1)
is 3 pair exchanges and therefore odd.
2 x 2 x 2 :
generators :
1/4 turns of face centers (I'll use all 12; same as 3x3x3)
Typical notation is R,L, U,D, F,B for clockwise on
right, left, up, down, front, back sides,
and R', L', U', D', F', B' for counter-clockwise.
number of group elements :
If any permutation, any twist, we'd have 8! * 3^8 .
Turns out you cannot have any twist (more on that later),
so actual number of positions is 8! * 3^7 = 88179840
*if* you're counting rotations of the whole puzzle
as different positions. If you don't think that way,
then you need to divide by 24 (solid cube rotations)
to get 3674160 : a mere three million and change.
Use CubeTwister to explore answers to the other questions; see below.
conjagacy
Correction from last time: if
X = A Y A'
then X and Y are said to be "conjugate" to each other. (I think I said the term was "similar", which is sometimes also used. But I think these days conjugate is more common.)
2x2x2 webpages and tools
solutions online
Discuss the notation usually used for the puzzle; compare with our group theory ideas.
R L U D F B clockwise right, left, up, down, front back sides
R' L' U' D' F' B' counterclockwise
R2 L2 U2 D2 F2 B2 half way
Look at the CubeTwister app and the script notation - note group theory concepts (commutator, conjugate, ...). Can use (), ()3, ()' notations as well. (The number means "repeat this many times"; the quote means inverse or undo.)
Examine a few of the "interesting move sequences" from the solutions. Compare with what we've seen by counting "do this" and "undo this" operations. How do you think people find these sequences?
Discuss contraints on twists of corners.
