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Rotational symmetries of cube

Posted by ubpdqn on February 6, 2011

There are 48 symmetries of the cube: 24 rotational x 2 reflectional.

The twenty three non-identity rotational symmetries are decomposed in the following graphic. The animated images are examples of particular axis of rotation.

Non-identity rotational symmetries of cube


Post script:

See Yaroslav’s comment  below. The representation of these symmetries using CayleyGraph[SymmetricGroup[4]] is shown below.

Thank you Yaroslav.


3 Responses to “Rotational symmetries of cube”

  1. That’s a cool way of visualizing 2-cycle, 3-cycle and 4-cycle generators of the symmetric group. Only two generators are needed, so you can get any rotation of the cube by picking two axes from the 3 above, and combining several rotations of those two axes. Standard choice is 2-cycle,3-cycle, so if you do CayleyGraph[SymmetricGroup[4]], it’ll show you the map where red arrow represents 180 degree rotation (2-cycle), blue arrow gives 120 degree (3-cycle)

    • ubpdqn said

      Thank you Yaroslav for this insight. I thought Mathematica provided a nice way of visualizing invariance under trasnformation and coudl show the axes of the 2,3,and 4-cycles.

  2. […] was inspired by the comment of Yaroslav on the post regarding rotational symmetries of the […]

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