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Updated: May 17, 2025
It is proposed to develop a representation of the tesseract or hyper-cube on the paper of this page, that is, on a space of two dimensions. Let us start as far back as we can: with a point. This line next moves in a direction at right angles to w, namely, x, a distance equal to its length, forming the square a b c d.
As y is known to be at right angles both to w and to x, z is at right angles to all three, and we have thus established the four mutually perpendicular axes necessary to complete the figure. This is the tesseract or hyper-cube in plane projection. It has the 16 points, 32 lines, and 8 cubes known to compose the figure. These cubes occur in pairs, and may be readily identified.
Now for a leap into the abyss, which won't be so terrifying, since it involves no change of method. The cube must move into the fourth dimension, developing there a hyper-cube. This is impossible, for the reason the cube would have to move out of our space altogether three-dimensional space will not contain a hyper-cube.
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