Dwell in Possibility

The following is taken from the site "What is Four-Dimensional Space Like?"

If you were to live in a tesseract, you might choose to live in its three dimensional surface, much as a two dimensional person might choose live in the 6 square rooms that form the two dimensional surface of a cube. So your house would be the eight cubes that form the surface of the tesseract. Imagine that there are doors where ever two of these cubes meet. If you are in one of these rooms, how many doors would you see? What would the next room look like if you passed through one of the doors? How many doors must you pass through to get to the farthest room? How many paths lead to that farthest room? Could you have any windows to outside the tesseract? What about windows to inside the tesseract?

Some of these questions are not easy. To answer them, go back to the easy case of a three dimensional cube with faces consisting of squares. Ask the analogous questions there and just extrapolate the answers to the tesseract.

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.

Rotating Hypercube

The object you see is the four-dimensional analog of a cube, just like a cube is a three-dimensional analog of a square. As you watch this video, realize there are no cubes on the "inside." The cubes you see are the outside of the shape just as squares are the outside of a cube. Look at the perspective drawings of a cube below. In the one on the right the smaller square is NOT "inside" the cube. It's just a matter of perspective, as if you are looking into a box, and the back is farther away than the front. Same thing with this video. The hypercube is ROTATING. It is NOT turning itself inside out.