About Hypercubes
Hypercubes in Literature and Art. The idea of the hypercube as a somewhat mystical object has appeared in many places. One well-known science fiction story, —And He Built a Crooked House— published
by Robert Heinlein in 1941, uses the hypercube as an essential plot element. A radical architect builds a
house based on a 3D unfolding of the faces of a hypercube, giving a strange configuration of eight cubical
rooms; the 3D analogy is the unfolding of a paper cube’s six squares to form a flat cross-shaped object.
Nothing is amiss in the story until a strange earthquake occurs, collapsing the house into the fourth dimension, so all that can be seen from the real world is a single cube, and yet that cube connects correctly to
all the other (now outwardly-invisible) cubes as it would in four dimensions. Finally, after another earthquake, the house disappears from the 3D world altogether. The artist Salvador Dalı́ used a figure essentially
identical in shape to the description of Heinlein’s Crooked House architecture in his surreal painting of the
crucifixion (Corpus Hypercubus), showing Christ crucified on an arrangement of the eight cubic surfaces of
an unfolded hypercube, essentially a doubled solid cross.
Hypercubes in Electronic Media. Creating pictures as well as animations of hypercubes has occupied
artists, students, hobbyists, scientists, amateurs, and professionals since the earliest times of electronic
media. Thomas Banchoff and his collaborators at Brown University made some of the first documented
computer animations of 4D objects, and those that have followed are almost too numerous to count. A
search for “hypercube” on YouTube results in 1,250 hits! Our own YouTube contribution to this genre is
4Dice, http://www.youtube.com/watch?v=fx7ehl7YvMY; it is one of the first animated hypercube presentations of which we are aware to include back-face culling, simulating the view that would be
seen by an actual 4D viewer with a proper 4D rendering model.
Making a Hypercube. The hypercube is a member of a family of figures created by drawing lines between
the corresponding points of two copies of the same figure one dimension lower. In the Figure, we illustrate
the classic construction of a hypercube by sweeping a line into a 2D square, the square into a 3D cube, and
finally using 8 lines to connect the corners of two copies of the 3D cube to form a 4D hypercube. The first
version of this final view is an orthogonal projection, and the second is a perspective projection, in which
the more distant edges of the hypercube’s structure appear smaller, exactly analogous to ties on a railroad
track shrinking in the distance. The last image on the right is the back-face culled hypercube image that is
our ultimate result.
Construction of a hypercube or tesseract by building up from lower dimensions.
1
We observe that the 1D cube has two points that bound its interior, one at either end of the single x-axis;
the 2D square’s interior is bounded by two pairs of lines, one at each end of the x-axis, and one at each end
of the y-axis. The cube has three pairs of squares bounding a 3D volume centered around the origin. We thus
can see that the hypercube’s four pairs of cubes, with one cube at each end of the x-, y-, z-, and w-axes, must
be the boundary of a 4D solid volume if we could perceive such a thing directly. That is, at the center of this
4D volume, a 4D ant could walk in any of four directions, and would have to move a considerable distance
before it finally bumped into one of the eight boundary cubes that we usually associate with a hypercube.
2