Expert: Sherry Wallin - 7/31/2010

Question
QUESTION: look at wikipedia, burnside's lemma. look at the blue bullet points. i can follow the argument but i cannot see why the powers of 3 are the powers written. why on earth are 3 cubed of the possible colourings of the cube left unchanged by the six 90 degree face rotations? for example.

ANSWER: Hi Tony~

I'm not quite sure what it is you want to know. Is it how there are 3 cubed ways to rotate 90 degrees? Choose any face and let's say you decide to rotate it 90 degrees, you have 4 choices as far as directions to turn. Let's say you follow the cube to the right, so that is one. Now you are again faced with 4 choices but you eliminate the one you just traversed so you have three choices to turn and go 90 degrees. Again you have 3 choices without traversing what you just traveled. If you were to open the cube and flatten it out it might be easier to see and count the number of different ways to turn 90 degrees.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: what they are saying on the burnside's lemma page, wikipedia is that there are 6 possible 90 degree face rotations. i.e. a turn to the left or right through 90 degrees (thats 2), around either of the 3 axes through the centre of opposite faces of the cube (thats 6 in total). I am happy with that, they go on to say that each of these rotations leaves       '3 cubed' of all '3 to the power 6' possible colourings of a cube, 'in three colours', unchanged. Why '3 cubed'?

Answer
Tony~
 
Ok I think I see where you need some clarification now...

You can still picture that cube opened up and keep in mind that there will be 5 squares laying flat and one 'hovering' over the flat space. Ask yourself how many choices there are for the first square and first color? Well you have 3 choices of a color to begin with in any of the squares. Let's use red, blue, yellow to simplify things a bit:
       _______
        |         |
        |   Y     |
 ____|         |_____
|          |
| B        G       B     |
|____          _____ |
      |          |
      |   *Y     |
      |______ |

For simplicity let's say you chose yellow for the first square, then you could put blue or green in like I will above as just one of the choices. Notice that yellow or blue would have to go next to the green, so I will put blue there. Now the only color that could go next is the yellow because when the cube is folded each of the sides will be touching and if I was to put blue there where the asterisk is then you would have 3 blue sides in a row.
So I will put yellow there. Now the only color that can be placed for the 'hovering square' is green. I have just given you one combination. How many faces are there? Yes 6, and how many possible 90 degree faces? Yes 6. So there are 3 colors and 6 faces for a total of 3^6 possible colorings of the cube.

Math Prof