Visualizing the Hypercube

Let's look at the familiar picture of a box split open, unfolded and laid flat in a plane:


The labels and arrows shown on the edges tell us how to reglue the box back together again. Alternatively we can think of these instructions as specifying abstract gluings, and we can imagine our 2-dimensional crab swimming in the resulting 2-dimensional space, which would be indistinguishable from its point of view to swimming within the boundary surface of a 3-dimensional cube. We can use the same trick to imagine the boundary of a 4-dimensional cube, often referred to as the hypercube or tesseract. The following Java animation shows what this would look like. Please be patient, it might take a minute or two for the animation to download from the net.

You need a JAVA enabled browser to view this animation such as Netscape Navigator v3.0 for the Macintosh (or v2.0 for Windows)












The gluing instructions for the hypercube are as follows: you are supposed to abstractly glue together squares having the same color. More precisely there are 12 pairs of "hinges" each arranged along an edge of the central cube hidden within the four arms of the figure. You are supposed to abstractly glue the two squares in each pair together by closing the hinge (without moving the cubes of which they are faces of). In addition there are 4 squares on the outer arms of the figure. These are supposed to be abstractly glued to the 4 lateral faces of the bottommost cube, by rotating them 180 degrees along their bottom edges and then moving them down towards the bottom cube, keeping the squares parallel to their original positions. (The 180 degree rotations are indicated by arrowheads in some of the frames of the above animation. Again the gluings are abstract, with only the squares being glued and the cubes of which they are faces being left undisturbed.) Finally the bottom square face of the bottommost cube is supposed to be glued to the top square face of the topmost square by moving it straight up along the central axis of the figure.

If you have trouble following some of these abstract gluing instructions, you might try them on one of the strips along the side of the unfolded hypercube such as:


The gluing instructions given above can be physically realized along these strips by folding up the strip so that pairs of squares with the same color get folded into a single square.

There is a science fiction story by Robert Heinlein, "-And He Built a Crooked House", which depicts life in such a space. In the story, an architect builds a house having the shape of an unfolded hypercube. During an earthquake, the house gets folded into the boundary of a hypercube. If you were an occupant of such a house, initially you would have no idea that anything unusual has happened: each individual room looks exactly the same as before. However you will notice something is amiss as soon as you try to leave the house. You will find that you are trapped inside the folded house, yet you never encounter any sealed barriers. If you open the front door in the bottom floor of the house, you reemerge upside down through one of the (formerly outward facing) windows in a room in one of the arms of the third floor. If you try to leave through one of the windows in the room on the second floor, you reemerge through a trapdoor in the floor of a room in one of the arms of the third floor. If you try to leave via a skylight in a room in the one of the arms on the third floor, you reemerge via a window into the topmost room. Finally if you try to leave via the skylight in topmost room, you reemerge via a trapdoor into (what used to be) the ground floor of the house.

Scientists seriously consider the possibility that the universe might have a shape like that of the hypercube. Before you dismiss this as an absurd idea, repeat the thought experiment of the crooked house with huge rooms billions and billions of light years across. You will then find it very difficult to recognize that the universe is finite in extent. If the speed of light were infinite, you would be able to see the back of your head, while looking forward. (Eg. think of lying on the ground floor of the crooked house with the trapdoor open and looking straight up through the stairwell in the central axis through the skylight in the top floor.) However since the speed of light is finite, what you would see instead of the back of your head is a cloud of dust billions of years ago from which our solar system formed.

There are infinitely many other candidates for the possible shapes of the universe which may be constructed by abstract gluings of polyhedra. Here is Dutch artist's Maurits Escher's conception of such a universe from his work "Another World":


Some of these candidate universes are 3-dimensional spaces which are unorientable. If the universe has such a shape, you might find after a long space flight around the universe that your right and left sides got inverted upon your return. While you might think that this would be only mildly annoying (eg. all books, newspapers, etc. on Earth would have their print look mirror-reversed to your eyes), you would find yourself in serious difficulties unless you brought a large supply of food along with you. For as is pointed out in another science fiction story "Technical Error" by Arthur C. Clarke, you would then find yourself unable to digest any food on earth, since all biological matter on earth is built out of right-handed amino acid molecules, whereas your inverted body would only be able to digest the left-handed forms of these molecules.

For an interesting discussion of some evidence pointing to the universe being closed and finite see the following article on the distribution of gamma ray bursters.


The unfolded hypercube is a subject of Salvador Dali's painting "Crucifixion" (aka "Corpus Hypercubus")





You might also take a look at this brief discussion of higher dimensional regular polyhedra, of which the hypercube is an example.


For an alternative take on some of these topics, check out this innovative multi-disciplinary project by a group of Yale undergraduates: