The short edges disappear as our slice passes three more of the vertices of the cube, leaving us with a triangular slice. Moving on from here, the sequence progresses in reverse, with three of the edges of the hexagon getting longer and three shorter (but the opposite ones from before). At this point, all six faces of the cube are sliced in exactly the same way. As we progress further, the triangle becomes more truncated, and at half way through the cube we arrive at a slice that is a perfectly regular hexagon. As we slice further, the corners of this triangle become truncated (as our slicing plane begins to cut into the remaining three faces of the cube). Eventually, the slice reaches the three corners of the cube that are adjacent to the original corner. As we slice further, the triangle grows in size. That is, we start at one corner, and cut off more and more of the cube until we reach the opposite corner.Īs we first start to cut off the corner, our slice is an equilateral triangle (one vertex along each edge emanating from the corner, and one edge for each of the three faces meeting at the corner). Of course, a cube can be sliced in many different ways (see the movies of slices of a cube), but the most interesting views come when we slice it by planes perpendicular to its long diagonal. To do this, we first look very carefully at how the process lets us understand a simple three-dimensional object such as a cube.
#TH10 HYPERCUBE SERIES#
We can slice it by a series of parallel three-dimensional hyperplanes, view the three-dimensional images so produced, and try to reconstruct in hour heads the four-dimensional object from the three-dimensional slices. We can employ a similar strategy in trying to understand a four-dimensional object. For example, in a CAT scan, a doctor obtains a series of two-dimensional images of slices of the human body, and can use these to reconstruct a three-dimensional image of the person being scanned. Our goal is to try to use the slices of the standard cube together with how these correspond to the cube itself (which we understand) to try to use the slicing sequence for the hypercube to generate in our minds an analogous understanding of the hypercube in four dimensions.Īn important means of understanding a three-dimensional object is to cut it by a series of parallel planes, and look at the two-dimensional slices so produced. The light blue figures represent slices of the cube at five different heights. The dark blue hexagon made from six equilateral triangles is actually an orthographic view of a cube when viewed from a direction parallel to its long diagonal (see the movies of orthographic projections).
The symmetry of the view is reflected in the regularity of the slices, which are in the form of regular or semi-regular solids: the tetrahedron, truncated tetrahedron, and octahedron.īelow each hypercube is a plane containing an analogous view of a three-dimensional cube being sliced. The slices at five different heights are shown in light blue. The dark blue ribbing is an orthographic view of the hypercube itself from a viewpoint that is parallel to the long diagonal, so the closest and farthest points on the hypercube are both projected to the same point in three-space, namely the center of the figure. In this image, we see a sequence of slices of a hypercube that is being cut by a hyperplane perpendicular to the hypercube's long diagonal. The reference is a short description in an article where the illustrations came from Polaroid pictures taken directly off the computer screen! For modern interactive versions of the object, see for example the two cube sequences in the interactive art exhibit site "Para Além da Terceira Dimensão''. After a quarter of a century, this film, now available in video, is still in demand, especially in schools and colleges. Our main contribution was a scripted tour of the 4-dimensional cube with three movements, orthographic projections, central projections, and slicing by planes and hyperplanes. Even then the topic had been treated by several researchers, most notablyA. Instead of demanding a workstation, it is possible to realize scenes on a laptop computer. In these days, programming images of hypercubes is a beginning exercise in introductory courses in computer graphics. For a more thorough description of this film, see Banchoff. We then slice each figure by hyperplanes perpendicular to the vectors (1,0,0,0) then (1,1,0,0) then (1,1,1,0) and finally (1,1,1,1). In each case we rotate in the coordinate planes xy, yu, xw, yw, and zw, ending at the original position. This film treats the convex hull of the sixteen points (Â☑, Â☑, Â☑, Â☑) in 4-space, first by orthogonal projection then by central projection from 4-space to 3-space.