3
$\begingroup$

Possible Duplicate:
Proof that cube has 24 rotational symmetries

What is the set of physical symmetries of the cube? Ie, what are the ways you can rotate, reflect, and/or flip the cube without cutting and pasting it back together? Can you also please explain how you get your answer.

Thank You!

  • 0
    Also, please tell us w$h$ic$h$ kind and level of course you're doing the problem for. It can be asked at various different levels of sophistication that require different styles of answer.2011-10-17

2 Answers 2

4

Label the vertices of the cube 1, 2, ..., 8, in such a way that 1 is adjacent to 2, 3, and 4. After you've done a physical symmetry, the vertex labeled 1 will wind up where one of the vertices was, originally. So, how many places are there for the vertex labeled 1 to go? Once you've decided where 1 goes, how many places are there where 2 can go? Once you've decided where 1 and 2 go, does that determine where all the otehr vertices go, or is there still some choice to be made? So, all told, how many physical symmetries are there? And what are they?

0

You can rotate $90^\circ$ about an axis that runs through the centers of two opposite faces; or $180^\circ$ about an axis that runs through the centers of two opposite edges, or $120^\circ$ about an axis that runs through two opposite vertices. Then close under composition.

  • 0
    There are also reflections2011-10-17