![]() Moving to the other woudl sweep out the entire hypercube. These pairs are “opposite”Ĭubes in the hypercube, as they have no face in common. The four movies below show the same set of rotations, but with aĭifferent pair of cubes highlighted in each. Similarly, the other four cubes form the “back” of the hypercube, and they cover the shape a second time. The four cubes that share the front corner, when taken together, would fill up this three-dimensional shape, forming the “front” of the hypercube (just as the front three faces of a cube cover the entire hexagon of the corner view of a cube). ![]() This is the most symmetric view of the hypercube. This rotation opens up the last pair of cubes (the green and yellow ones) and now all eight cubes are projected as congruent shapes in three-space. Here, both the closest and farthest vertex are seen at the center of the diagram, just as was the case witht he corner view of a 3D cube. The final rotation in four-space brings our viewpoint to one that looks directly at a vertex of the hypercube. ![]() The remaining six cubes are formed by attaching corresponding faces of the two flattened cubes, and these six all are the same shape in the projection. The front corner is of each is attached to the front edge, while the back corner is attached to the back edge, but both appear at the same location in the projection. Two of the cubes are still flat at the top and bottom, and we see them as the corner view of a 3D cube. This rotation opens up two more of the cubes (the cyan and purple ones). Both the closest and the farthest edges appear at the same location in the projection, just as they do in a 3D cube viewed edge first. Our next rotation moves us to where we look directly at one of the hypercube’s edges, again, the one at the center. This opens up two of the cubes that were flattened in the previous view (the orange and brown ones), and produces the analog of the “revolving door” illusion. We then rotate our viewpoint to look directly at one of the square faces of the hypercube (the one at the center of the figure). The “front” and “back” cubes of the hypercube both appear to be the same size, and the remaining six cubes are projected “flat” as the faces of the cube. Users should refer to the original published version of the material for the full abstract.The initial (four-dimensional) viewpoint is looking directly at one of the cubical faces of the hypercube, so the projection appears to be a cube. No warranty is given about the accuracy of the copy. However, users may print, download, or email articles for individual use. Hypercube n constructs an -dimensional hypercube.
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