5/16/2023 0 Comments Hypercube surface area![]() Square for a while, and if the hypercube comes through our space If a cube comes through the horizontal plane square first, we see a But just as the slices of an ordinary cube depend on the orientation of the cube with respect to the slicing plane, so too do the slices of the hypercube. This is analogous to the appearance of the slices of an ordinary ball passing through Flatland. No matter which way the hyperball is rotated, we will get the same sequence. The slicing sequence of the hyperball has already been compared to the inflation and deflation of a spherical balloon. ![]() How will the appearance of the slices change as the water level rises to cover the objects? We can imagine four-dimensional versions of Froebel's geometrical gifts, a hyperball and a hypercube suspended "above" our three-dimensional world, thought of as the analogue of the surface of the water. We can exhibit the slices even though we cannot construct the objects themselves that are being sliced! ![]() What would happen if the hypercube came through our universe from another direction, for example square first or edge first or corner first? For such questions, the modern graphics computer is ideally suited. We have already seen how a hypercube would appear if it penetrated our universe cube first: at the first instant, we would see an ordinary cube with 8 bright vertices, then a dull cube for a while, and finally another cube with 8 bright vertices. Slicing the Hypercube Slicing the Hypercube ![]()
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