What are the symmetries of a cube?

Answer 1

The symmetries of a cube include rotations, reflections, and combinations of these transformations that leave the cube unchanged. There are 24 rotational symmetries of a cube, including rotations by 90, 180, and 270 degrees around different axes. Additionally, there are reflections across various planes of symmetry that preserve the cube's shape and orientation. These symmetries form the cube's symmetry group, known as the octahedral group O(24).

Answer 2

Well honey, a cube has rotational symmetries of order 4, which means you can rotate that bad boy by 90, 180, or 270 degrees and it still looks the same. It also has reflectional symmetries across its faces and diagonals, giving it a total of 48 symmetries. So basically, a cube is a symmetrical diva that knows how to work it from every angle.

Answer 3

From the perspective of a symmetry group, a cube has 48 symmetries total. They include:

• 24 rotational symmetries:
  • the identity
  • 6 90° rotations about axes through the centers of opposite faces
  • 3 180° rotations about the same axes
  • 8 120° rotations about the space diagonals connecting opposite vertices
  • 6 180° rotations about axes through the centers of opposite edges
• 24 reflection symmetries that involve one of the above rotations, followed (or, equivalently) preceded by the same reflection