Up: Math 53 Selected Course Notes

# Orthographic Views of the Hypercube:

 The initial frame of this movie shows stereographic projection of the hypercube looking directly at one of its eight cubical faces, so we see a cube within a cube. One is colored red and the other blue, but the two faces are removed from each to make it easier to see the interior structure of the projection. As the movie progresses, the distance to the light source is increased toward infinity, and at the end, the light source is infinitely far away, so its rays are parallel. Thus the images of both the closest and farthest cubes are the same size. We only see one cube in the projection as both cubes now overlap. This movie shows our (orthographic) view of a hypercube as we move from looking directly at a cubical face to looking directly at a square face. At several points in the movie, we rotate the hypercubecube slightly so that we can see its other cubical faces better. This should help you understand the "sliding cubes" image that you are viewing. The front face is colored red and the back face blue. In the orthographic projection, the two cubes seem to slide past each other. Four of the other six cubes are flattened out and form the front, back, top and bottom of the rectangular box. These are exhibiting the "sliding square" behaviour that we saw in the orthographic views of the 3-dimensional cube; you can see the bottom and top cubes better when we tilt the hypercube at two points within the movie. The final two cubes form a complementary pair to the red and blue ones, and are shown in the next movie. This movie shows the same sequence of motions as the previous one, but this time two of the side cubes of the hypercube cube are colored. In the initial view, the sides are projected as squares (since they are parallel to the light source) so we don't see them as cubes. As the movie progresses, they open up and become more recognizable as cubes. This movie again shows the same sequence of motions as the previous two, but this time two different side cubes of the hypercube cube are colored. In the initial view, the sides are projected as th etop and bottom squares. As the movie progresses, they undergo the "sliding squares" behaviour that we saw in the projections of the three-dimensional cube. The remaining two cubes in the hypercube also have essentially this same set of projections (but thay would appear as the front and back faces of the projected rectangular box). This movie shows the hypercube rotating from a view that is directly at a square face to a view that is directly at an edge. We begin at by tilting the hypercube backward slightly toward one of its cubical faces (the "sliding cubes" illusion) and then start rotating the cube toward the edge. You should be able to see the two colored cubes easily this way. Partway through, this initial tilting is undone, and the rest of the rotation occurs in a more symmetric view. The final projection is a hexagonal prism, which is rotated in three-dimensions to give a better view. Finally, the colored cubes are removed so you see the interior structure more easily. In this view, six of the eight cubes are formed by congruent rhomboidal prisms (like the pair of colored ones), but the remaining two are still flattened out as the hexagons on the top and bottom. These correspond to views of these cubes down their long diagonals projected into a plane. This movie shows same sequence of motions as the previous one, but with a different set of cubes being colored. These are the two flattened out cubes that form the top and bottom faces of the projection. They open up into full cubes as we rotate the hypercube to a viewpoint that is directly at an edge. This movie shows the same basic rotation as the previous two, but from a different viewpoint in three-space. The initial is from directly at a square face of the hypercube, with two cubical faces of the hypercube colored. The the hypercube is rotated so that we are looking directly at an edge, which gives us the hexagonal prism view that we saw in the previous movies. This movie shows the same motions as the previous movie, but with two different faces of the hypercube colored. This time, the two faces are the ones that remain flat in the final view of the hypercube. They form the hexagons at the top and bottom of the hexagonal prism that is the projection of the hypercube. Our viewpoint in fourspace is along the long diagonals of these two cubes. Note that these two cubes undergow the same transformation as the projection of the three-dimensional cube when we rotated from a viewpoint along its edge to directly at its corner. The unusual vibrating color effects are due to the fact that the cube is flattened out, so every point is covered twice, and the computer is having a hard time deciding which part of the cube is on top. This movie shows the hypercube rotating from a view that is directly at an edge to a view that is directly at a corner. Two of the eight cubical faces are colored, so you can see the structure of the hypercube better. The final position is looking directly down the long diagonal of the hypercube. This is the most symmetric view of the hypercube, where each of the eight cubical faces is projected to congruent shapes in three-space. This shape is called the rhombic dodecahedron, which is formed by 12 rhombic faces meeting in sets of 3 at the wider angle and 4 at the smaller one. This shape is rotated in three-space at the end of the movie so that you can see it better. This movie shows the same basic sequence of motions as the previous one, but with a different pair of cubes colored. This time, the two flattened cubes that form the hexagonal top and bottom of the projection are shown. First, we rotate back slightly to a less symmetric view so that these cubes look more like flattened cubes. Then we rotate in three-space to a view that is parallel to the top and bottom hexagons, and perform the four-dimensional rotation toward the corner of the hypercube. partway through, we undo the initial rotation and go back to the more symmetric view before completing the rotation. The final viewpoint is along the long diagonal of the hypercube, as before. This movie shows the hypercube rotating from a viewpoint were we look directly at a cubical face to where we look direclty at a corner, down the long diagonal of the hypercubecube. This corresponds to our earlier view of the hypercube as two cubes moving apart. That original movie was a correct shadow (as the light source moves from directly above to a slanted direction), but it does not represent a view of the hypercube that four-dimensional viewers could ever actually see, since such a view would always be into a "plane" that is perpendicular to the direction of sight. This movie shows a more correct view of the hypercube. In this view, the corner of the blue cube that is at the center of the figure is the corner we are looking directly at, and the red one that is projected to the same position is the opposite corner of the hypercube, the point that is farthest away from us. The long diagonal has been projected down to a point since we are looking along it. This view corresponds to our view of the three-dimensional cube as a hexagon. Try to see how these distorted cubes correspond to the faces of the cube that we saw in the orthogonal views of the cube. This movie shows the same sequence as the previous one, but with two different cubes colored. These ones start out projected as just squares (the cubes are completely flattened out), but open up to form distored cubes in the final projection. This movie show the rotation from looking directly at a cubical face, to looking directly at a suare face, to looking direclty at an edge and then to looking directly at a corner of the hypercube. Now that you have seen the other ones, this one should make more sense.

Up: Math 53 Selected Course Notes