English    Deutsch
Reader Testimonials
“Your book virtually replaces entire libraries. Simply PERFECT!”
Peter Dimmroth

Animated Hypercube


This Java applet (courtesy of Mark Newbold) shows the projection of a four-dimensional cube (hypercube) into three-dimensional space in a stereoscopic view, so you can watch it in actual 3D using anaglyph glasses (there is also a viewing mode not requiring glasses). This shows more clearly than the figures in the book that a hypercube is composed of eight cubes that appear in different sizes and distortions depending on the angle of perspective, even though they are actually all equally sized and orthogonal in four-dimensional space.

If you see only an empty field above, there is no Java installed for your browser. The latest version is available free of charge from Oracle.

Instructions:

Click on the Stereo button to switch between three viewing modes:

  1. For red-blue glasses (also suitable for red-cyan glasses), red = left eye
  2. For red-green glasses, red = left eye
  3. Separate images for viewing without glasses

In the third mode, you may need some practice to bring the two images together by crossing your eyes. The following may help: Hold up a finger between your face and the monitor. Look at the finger, then shift your attention to the images on the monitor. When they are properly merged, you will see three images. The center image is stereoscopic. Ignore the outer images. If you have difficulty crossing your eyes, try moving back away from the monitor. This reduces the degree of eye-crossing required. Once you have the stereoscopic image in focus, try using your hands to frame the center image and block out the outer images. When you get your hands in the correct position, there is an illusion that the 3D image is floating in space between your hands. Since the double image requires more width than height, I recommend that you use the “Detach” button to put the applet into its own window so you can resize it.

Projection varies the viewpoint distance: low values = “telephoto”, high values = “wide-angle”. The higher the value, the stronger the perspective distortion. The Stop/Start button stops and restarts the rotation. With Speed you can adjust the rotation speed. The hypercube rotates in three-dimensional space (ordinary rotation) as well as in four-dimensional space (the “surface cubes” of the hypercube are shifted against each other).

Click on the hypercube and keep the mouse button pressed to rotate the object manually by moving the mouse. If you press the Shift button at the same time, the hypercube will rotate in four-dimensional space.

Clicking on Detach places the Java applet in a separate window, which you can resize to your needs.

If you really want to delve into higher dimensions, here are two challenges for you: