The line segment has two "hyperfaces", which are its.The distance which was "swept out" by the point. It seems clear that every vertex willįorm an edge when it's swept through space. Through space perpendicular to the plane in which it lies.Įach edge will form two "image edges", each face will form two "image Where it started sweeping, one where it finished sweeping - we wouldĮxpect any hypercube to form two images of itself General, just as the single point formed two images General, we will see that an n-dimensional hypercube will have 2 n When it swept along: one (old) vertex at its startingĪnd one (new) vertex where it ended. Has, and by paying attention to the details as we build a cube we can N-cube has each of those is described by a "recurrence relation",īased on the number of vertices, edges, and hyperfaces an (n-1)-cube We want to know how many vertices, edges, and hyperfaces an For clarity, we'll usually call 2-dimensional faces " 2-faces",Īnd n-1 dimensional faces, " hyperfaces". We have a hard timeĭrawing anything higher than 3 dimensional figures so we stopped there.īy paying attention to what happens as we do this, we can learn what we Length, and sweeps out a square the square then moves perpendicularly Out a line segment the line segment then moves perpendicularly to its We start with a point, at the origin, which It through space perpendicular to the hyperplane in which it lies.ĭimensions. Some examples of "low-dimension" n-cubes:Īnd is occasionally just called a "hypercube". The rest of this page just expands on, explains, and In a direction perpendicular to itself, through a distance of r The magnitude of the volume of an n-cube with edge length r Like acres and square meters) is 2-volume. Ordinary "volume" (measured in things like quarts and liters) In general, we call the volume enclosed by a hypercube an n-volume. On each axis the hyperfaces are n-1 dimensional
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