Rotating Cubes about Axes of Symmetry; 3D Rotation Is Non-Abelian

A cube has rotational axes of symmetry through the centers of a pair of opposite edges (two-fold symmetry, six axes), two opposite corners (three-fold symmetry, four axes), or the centers of two opposite faces (four-fold symmetry, three axes). The middle cube shows two examples of each axis; it is slightly transparent. Markers 0, 1, 2, and 3 are always visible, marking the vertices , , , and of each cube. Move the slider. The left cube rotates about the axis and the right rotates about the axis; the axes appear as stubs on the cubes. When is an integer, the edges are once more parallel to the central cube, demonstrating the symmetry, but the original coloring is not recovered until equals the axis number. Try different pairs of axes and change . You can click the picture and drag to alter the viewpoint.
Now click "AB, BA" in "maximum " and activate the slider. (Slow the animation down!) The rotation pauses after one complete turn (snapshot 3), giving orientations and . Then the axes of rotation are swapped. The new axes appear and the second rotations are performed. The image pauses again (snapshot 4), with the left cube in orientation and the right in orientation . These are different; in 3D rotations is not necessarily . In real and complex numbers (they are Abelian or commutative); 3D geometry (like many aspects of physics) is non-Abelian.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


An uncolored cube has rotational symmetry; this is Abelian and is not demonstrated here.
In elementary mathematics the order of operations does not matter, that is, gives the same result as . This simplifying assumption leads to many "common-sense" notions that have to be unlearned on proceeding to "higher" mathematics.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+