9867

The Group of Rotations of the Cube

Consider a particular representation, , of on that preserves a cube centered at the origin, with faces orthogonal to the axes. By examining the action of elements of the group on the cube, both singly and in composition with other elements, you can see that is isomorphic to the group of rotations of the cube. The brightest cube is fixed and the next two cubes show the actions of and . The axes of rotation are shown in gray. The thickest axis represents .

SNAPSHOTS

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

DETAILS

The two sliders can be used to select elements and from . These elements are shown in cycle form, as is their composition, . Below that are the matrices of , , and . The graphic shows how these matrices act on a cube centered at the origin, with faces orthogonal to the axes. The brightest cube is fixed, and displayed only as a point of reference. The middle cube (slightly darker) shows the action of on the fixed cube, where the gray line shows the axis of rotation. The action of is first, since the operation throughout is composition. The third and darkest cube shows both the action of on the cube already acted on by and the action of on the fixed cube. The thinner gray line shows the axis of rotation of and the thicker gray line shows the the axis of rotation of the composition . By experimenting with different elements and compositions of elements, you can verify that is isomorphic to the group of rotations of the cube.
    • 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.









 
RELATED RESOURCES
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 © 2014 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+