Algebra: Abstract and Concrete

Frederick M. Goodman

This page leads to some three dimensional graphics related to regular figures and their symmetries, in Java. To view these figures, you must have Java enabled in your web browser. The applets use JavaView, a 3d geometry viewer and a geometric software. Make sure you have a recent version of Java installed on your computer, in order to use this software; you can get Java from

http://www.java.com/en/download/index.jsp

Symmetry axes of the cube:

Surface pictures

Two fold axis (surface picture)
Three fold axis (surface picture)
Four fold axis (surface picture)

Frame Pictures

Two fold axis (frame picture)
Three fold axis (frame picture)
Four fold axis (frame picture)

The action of the rotation group of the cube on the diagonals of the cube induces an isomorphism of the rotation group with S_4:

Diagonals of cube

Symmetry axes of the tetrahedron:

Surface pictures

Two fold axis (surface picture)
Three fold axis (surface picture)

Frame Pictures

Two fold axis (frame picture)
Three fold axis (frame picture)

Symmetry axes of the dodecahedron:

Tetrahedra imbedded in a cube:

Cube imbedded in a dodecahedron.

There are five cubes imbedded in the dodcecahedron, and the action of the group of the dodecahedron on this set of five cubes give an isomorphism of the symmetry group with the alternating group A_5.