Hypercube Slices

The image below is a still screenshot of an interactive hypercube-slice applet found at the site Projections of Hypercube Slices by Davide P. Cervone. It is one of the links from his phenomenal page Some Notes on the Fourth Dimension. This site has many other interactive links.When you go to this site, play with the controls you see here under the applet for some cool effects!

As in the wonderful work of fiction Flatland by Edwin A. Abbot, objects from higher dimensions are often studied by imaging slices of those objects as they pass through a lower dimension. Another site that contains applets showing animated 3D slices of a hypercube is Alice begegnet der vierten Dimension . I like the continuous motion of the animations. The longer you watch the better feel you can get for what is going on.


Below are still shots of a hypercube going through 3-dimensional space. These are taken from the site HyperSpace, User Manual by Paul Bourke. Though it is possible to comprehend each solid shown here one at a time, in order to grasp the full 4-dimensional object, you need to keep all of them in mind at one time and "stack them up," just as someone in a 2-dimensional world would have to "stack up" a series of smaller to larger to smaller circles to grasp the idea of a sphere.

To help you get an idea of what is going on here I created a series of slices of a sphere going through "Flatland." (Imagine a ball being submerged in water and the part of it that is at water level as it passes through the surface of the water.) In order to understand what a sphere is, a thing that cannot exist inside of Flatland, the Flatlanders would have to keep all of these slices in mind and stack them up.