The Hypercube

The hypercube is perhaps the most well-known of all the 4D objects. It is known by many names, among which are the tesseract, the 8-cell, the 4D measure polytope, and the tetracube. Many of these names describe its different special properties. It has been the subject of several stories, such as Robert A. Heinlein's And He Built a Crooked House. It has also been the subject of countless 4D wireframe rotation programs, screensavers, and Java applets.

Oblique Projection

There are several ways of constructing the hypercube. The simplest way is to extrude the 3D cube along the W-axis. The following oblique projection of the hypercube underlines this method of constructing the hypercube.

The red cube shows the starting 3D cube, and the blue cube shows the endpoint of the extrusion. The black lines trace the path of the 8 vertices of the cube as it is extruded. If one examines the above diagram carefully, one will see that the hypercube in fact consists of 8 cubes. The 6 cubes besides the two obvious ones are formed by the extrusion of each of the 6 square faces of the red cube into the W-axis. These 8 cubes form the outer boundary of the hypercube.

Perspective Projections

The difficulty with the above diagram is that there are too many intersecting lines, and it is difficult to discern the 8 constituent cubes. The following diagram tries to correct this defect by using a perspective projection instead:

In this diagram the blue “inner” cube is actually the same size as the red “outer” cube, but it appears to be smaller because it is farther away along the W-axis. The 6 frustums connecting these two cubes are actually identical cubes; but they appear distorted into frustums because they are being viewed at from an angle. Furthermore, all 8 cubes lie on the outer boundary of the hypercube. Even though it appears that the inner cube is on the “inside” whereas the outer cube is on the “outside”, they actually lie on the outside of the hypercube, on two opposite sides. The following animation shows what happens when we rotate the hypercube in the XW plane.

We use dotted lines for edges that project inside the envelope of the image so that it is easier to see.

Hidden Surface Removal

One thing that is often neglected to be mentioned when such wire diagrams of the hypercube are presented is the fact that they represent projections of the hypercube without the removal of hidden surfaces. This is like showing the rotation of the wireframe of a 3D cube, where you can see through its faces and see what is on the other side of the cube. While this is useful in seeing the entire structure of the hypercube, it sometimes gives too much detail and becomes confusing. The following diagrams try to complement the picture by showing projections of the hypercube where obscured 4D surfaces are not shown.

For example, when viewed from the angle that corresponds with the cube-within-a-cube diagram shown earlier, the hypercube in fact appears as a simple 3D cube:

When rotated 45 degrees in the XW plane, the hypercube appears as follows:

Only two cells are visible because the rest are obscured behind them in the 4th direction.

Vertex-first projection

Another fact that is often neglected when hypercube projection images and diagrams are shown is that projections such as the cube-within-a-cube actually view the hypercube from a “flat” angle, akin to looking at a 3D cube directly at one face, or perhaps at an edge, and seeing only two faces at a time. Just as we intuitively imagine the 3D cube as viewed from an angle such that we can see three of its faces at a time, so a more “intuitive” angle of looking at the hypercube is from an angle where we can see four of its cells at once. The following diagram shows one such view of the hypercube.

The 3D surface of this projection is called a rhombic dodecahedron. It is a 12-faceted polyhedron where each face is a rhombus. The four cells of the hypercube visible from this angle are shown below:

The other four cells of the hypercube are behind these four in the 4th direction, so they are not visible. The center of this rhombic dodecahedron, where the blue edges meet, is the corner of the hypercube closest to the viewer.