Group of rigid rotations of cube

In summary, the conversation discusses the difficulty in visualizing rotations within a group of 24 elements. The group can be thought of as having 6 faces that can be rotated in 4 different positions each. The book describes the rotations as 3 subgroups of order 4 and 4 subgroups of order 3, with the latter involving rotating a pair of diagonally opposite vertices through three possible positions. The speaker is struggling with understanding this concept and seeks any general advice on visualizing symmetry groups beyond the dihedral group of order 4.
  • #1
PhDorBust
143
0
I'm having trouble visualizing some of the rotations that compose this group.

Clearly the group has 24 elements by argument any of 6 faces can be up, and then cube can assume 4 different positions for each upwards face.

My book describes the rotations as follows:
3 subgroups of order 4 created by rotation about line passing through center of two faces.
4 subgroups of order 3 created by "taking hold of a pair of diagonally opposite vertices and rotating through the three possible positions, corresponding to the three edges emanating from each vertex."

My trouble lies with the second description, that is, I haven't the slightest idea of what it is saying. Any help? Also, any general comments on visualizing symmetry groups would be appreciated, I trouble going beyond dihedral group of order 4.
 
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  • #2
Hi PhDorBust! :wink:
PhDorBust said:
4 subgroups of order 3 created by "taking hold of a pair of diagonally opposite vertices and rotating through the three possible positions, corresponding to the three edges emanating from each vertex."

Any vertex has 3 edges …

rotate the cube about that vertex so as to interchange those 3 edges. :smile:

(there are 8 vertices altogether, but only 4 subgroups, because rotations about opposite vertices give the same effect)
 

1. What is a group of rigid rotations of a cube?

A group of rigid rotations of a cube is a mathematical concept that describes all possible ways to rotate a cube without changing its shape or position in space. It is a set of transformations that preserve the cube's properties, such as its edges, angles, and faces.

2. How many rotations are in a group of rigid rotations of a cube?

There are 24 rotations in a group of rigid rotations of a cube. These include 6 rotations around the vertical axis, 4 rotations around the horizontal axis, and 4 rotations around the diagonal axis. Each of these rotations can also be performed in the opposite direction, resulting in a total of 24 rotations.

3. What is the order of a group of rigid rotations of a cube?

The order of a group of rigid rotations of a cube is the number of elements or rotations in the group. As mentioned earlier, there are 24 rotations in a group of rigid rotations of a cube, so its order is 24.

4. Can a group of rigid rotations of a cube have subgroups?

Yes, a group of rigid rotations of a cube can have subgroups. For example, the rotations around the vertical axis form a subgroup of the entire group. Subgroups are smaller groups that have the same properties as the larger group and are closed under the same operation.

5. How is a group of rigid rotations of a cube useful in science?

A group of rigid rotations of a cube is useful in science because it allows for the study and understanding of symmetry, patterns, and transformations in three-dimensional space. It is also used in crystallography, chemistry, and physics to describe the symmetries and properties of molecules and crystals.

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