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Updated: May 16, 2025
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.
He has now perfected a new method of representing the several dimensions by colours instead of by arbitrary written symbols. He states that this will very much simplify the study, as the reader will be able to distinguish instantly by sight any part or feature of the tesseract.
He found the little stud which, pressed by the pad's owner, had erased the previous drawings. He pressed it and the lines disappeared. And Tommy drew, crudely enough, that complicated diagram which is supposed to represent a cube which is a cube in four dimensions: a tesseract. Upon one surface of the cube he indicated the curving towers of the Golden City.
I am not certain that the power to do this is within the reach of everyone, as he thinks, for it appears to me to require considerable mathematical ability; but I can at any rate bear witness that the tesseract or fourth-dimensional cube which he describes is a reality, for it is quite a familiar figure upon the astral plane.
The large outside cube represents the original generating cube at the beginning of its motion into the fourth dimension, and the small inside cube represents it at the end of that motion. These two projections of the tesseract upon plane space are not the only ones possible, but they are typical.
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.
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