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.
Modern browsers and operating systems impose security barriers to running Java code. You will probably be asked to agree to security waiver in order to run the applets provided here. If you are not able to run the applets at all, please refer to these suggestions.
General instructions: if you successfully run an applet to display a graphics image, you can rotate the image by clicking and moving your mouse over it. A "secondary click" will bring up a menu (for you, it might be right click or left click or control-click; on my Mac, it is option-click). One of the menu items is "New Display". Choosing this will bring up a copy of the graphics in a new window. The new window can be resized to show a large version of the picture. Associated to the new window, you will also have access to several menus that allow you to control various aspects of the display.
Symmetry axes of the cube:
Two fold axis (surface picture)
Three fold axis (surface picture)
Four fold axis (surface picture)
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:
Symmetry axes of the tetrahedron:
- Two fold axis (surface picture)
- Three fold axis (surface picture)
- 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.