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Consider the cube with vertices at (+-1, +-1, +-1)
how many rotations of this cube have the property that is you compose a rotation with itself, you get the inverse of the rotation?
Any hints or help with this, I have no idea where to begin with this?
Hi darren86!Quote:
Originally Posted by darren86
Which rotations can you think up?
Perhaps for starters rotations around an axis through the center of a face?
How many of those are there, or more relevant, how many of those, composed with themselves, are their inverse?
Which other types of rotations are there that rotate the cube onto itself?
if R2 = R-1,
then R(R2) = R(R-1), that is:
R3 = I.
a little reflection (hah! i kill myself!) shows R must be a rotation of 120 degrees in some plane.
which 3 vertices of a cube are equidistant from a given vertex? might they form the plane we are looking for? how many rotations can we form this way? (hint: this number must divide 24).