Spheres Sliced in 2D and 3D
Flatlanders can understand a sphere as a sequence of circles changing over time. The flatlanders see time as a third dimension, but we see the third dimension as a physical one. Similarly, we can understand a hypersphere from the fourth dimension as a sequence of spheres changing over time. We use time as a means of representing a fourth physical dimension.
These pages walk you through the analogs of the cube in lower and higher dimensions, developing the sequence: point, line, square, cube, hypercube. It begins the investigation of the hypercube by counting some of its parts, and by locating the cubes that form the faces of the hypercube.
Rotating Cubes and Hypercubes
Rather than look at a single two-dimensional shadow of a cube, we can look at a sequence of shadows as the cube rotates. This gives our two-dimensional Flatlanders a better understanding of the cube as they watch which pieces shrink (when they are far from the light) and which grow (when they are near the light). We can readily interpret such two-dimensional pictures as a three-dimensional cube rotating. Similarly, we can look at a sequence of views of a hypercube as it rotates in four-space.
Folding Cubes and Hypercubes
Here we look at how unfolding the square faces of a cube can help us to explain a cube to Flatlanders. The shadows can be seen by the people in Flatland, and they can try to use these shadows to interpret the folding that we are doing in three-space. Similarly, we can unfold a hypercube into three-space, and watch its shadow as it folds together in four-space.
Orthographic and Perspective Projections
Our shadows of cubes and hypercubes have used stereographic projections, in which things that are farther away appear smaller than things that are nearby; but we can also use orthographic projection, which corresponds to a viewpoint (or lightsource) that is "infinitely" far away. In this view, far away things are not reduced in size. This makes our views more symmetric, and reduces the picture to one that seems to be a lower-dimensional one when our viewing direction is parallel to faces of the cube or hypercube.
The Faces of the Hypercube in Orthographic Projection
On this page, we show the sequence of orthographic views of the hypercube that we first introduced in the movies above, but this time, we highlight various pairs of cubes, and track the changes that occur to them as we move from viewpoint to viewpoint, first looking at a cubical face of the hypercube, then a square face, then an edge, and finally a corner.
A Cube Falls Through Flatland
One way to show a three-dimensional cube to a Flatlander is to let the cube pass through the plane of Flatland and see the sequence of shapes that it produces. If the cube hits flatland face first, then the Flatlanders will see a square appear, then remain for a while, and then disappear. But this is not the only way a cube and pass through flatland. Here we see three symmetric slicing sequences for the cube.
A Hypercube Falls Through Spaceland
These movies show the slicing sequences for the hypercube when it is sliced cube first, square first, edge first, or corner first. These are the analogous views to the slices of the cube passing through Flatland given above. In each case, we view the hypercube from "above"; that is, the square-first view is shown within the orthographic view of the hypercube looking at it square first. You should be able to recognize the similarities with three-dimensional cube slices, and may even see the various cube-slicing sequences as the individual faces of the hypercube are cut by the slicing hyperplane. The edge-first sequence is the first interesting one, and you should be able to see the corner-first slicing of the flattened cubes on top and bottom.
Projections of Sliced Cubes
Our views in the movies of cubes falling through Flatland showed the slices from a viewpoint directly overhead. That is, we saw the square-first slices looking at the cube square first, the edge-first slices looking edge first, and the corner-first slices corner first. We could look at any of the slicing sequences from any of the viewpoints, however. In these movies, we show all nine combinations of viewpoints with slicing sequences. From most views, the sequence will look different depending on which face (or edge or corner) is hit first, and so there are several sequences in each movie. Both stereographic and orthographic views are given.
Hypercube Slices Viewed Corner First
The previous set of movies show the four symmetric slicing sequences of the hypercube, each viewed from a direction looking perpendicular to the slicing hyperplane. Just as will the cube, however, we can view the slices from different directions. The cube had three slicing directions and three viewing directions, giving nine combinations in all; for the hypercube, there are four of each, for sixteen all together. It seemed excessive to produce this many, so we chose to show all four sequences from one viewpoint, the corner view. In this view, all eight cubes are visible, so we can see the entire slicing sequence clearly in each case.
Hypercube Slices Colored by Cube
Here we show the slices of the hypercube viewed in orthographic projection corner first, but this time we color the faces of the slices according to which of the eight cubes of the hypercube they are slicing through. The goal here is to see how the various faces of the slice sweep out the cubical faces of the hypercube. You should be able to see the three slicing sequences of the three-dimensional cube appear in different combinations in these movies.
Stereographic Projections of Hypercube Slices
Our previous movies of the hypercube have almost all been in orthographic projection. These movies show the slicing sequences of the hypercube in stereographic projection, instead. Since in these stereographic views, none of the cubes are flattened out, you can see how the faces of the slices sweep out the cubes of the hypercube more easily. The edge-first stereographic view of the hypercube is one you have not seen before.
Stereographic Projection of Tori in the Three-Sphere
The surface of the hypercube can be viewed as a polyhedral model for the three-sphere, so our understanding of the eight cubical faces of the hypercube can help us understand the three-sphere in four-space. In particular, we can see that the hypercube can be broken down into two solid (polyhedral) tori each made from four of the cubes. These movies investigate the smooth analogue of this decomposition.
Interactive Models of 4-Dimensional Objects
It is not always easy to understand a 3-dimensional model from just one view; it often helps to be able to walk around the object to see it from a variety of viewpoint. All the more so when the 3-dimensional object is a projection of an object from four dimensions. Many of the objects illustrated in the links above are made available heres as 3D models that can be rotated using a Java applet called JavaView.