A complete view of the tesseract in its various space presentations is given in the following figures or catalogue cubes, figs. 103-106. The first cube in each figure represents the view of a tesseract coloured as described as it begins to pass transverse to our space. The intermediate figure represents a sectional view when it is partly through, and the final figure represents the far end as it is just passing out. These figures will be explained in detail in the next chapter.
Fig. 105.
The tesseract, with red, white, blue axes in space. Opposite faces are coloured identically.
Fig. 106.
The tesseract, with blue, white, yellow axes in space. The blue axis runs downward from the base of the ochre cube as it stands originally. Opposite faces are coloured identically.
We have thus obtained a nomenclature for each of the regions of a tesseract; we can speak of any one of the eight bounding cubes, the twenty square faces, the thirty-two lines, the sixteen points.
CHAPTER XIII
REMARKS ON THE FIGURES
An inspection of above figures will give an answer to many questions about the tesseract. If we have a tesseract one inch each way, then it can be represented as a cube—a cube having white, yellow, red axes, and from this cube as a beginning, a volume extending into the fourth dimension. Now suppose the tesseract to pass transverse to our space, the cube of the red, yellow, white axes disappears at once, it is indefinitely thin in the fourth dimension. Its place is occupied by those parts of the tesseract which lie further away from our space in the fourth dimension. Each one of these sections will last only for one moment, but the whole of them will take up some appreciable time in passing. If we take the rate of one inch a minute the sections will take the whole of the minute in their passage across our space, they will take the whole of the minute except the moment which the beginning cube and the end cube occupy in their crossing our space. In each one of the cubes, the section cubes, we can draw lines in all directions except in the direction occupied by the blue line, the fourth dimension; lines in that direction are represented by the transition from one section cube to another. Thus to give ourselves an adequate representation of the tesseract we ought to have a limitless number of section cubes intermediate between the first bounding cube, the ochre cube, and the last bounding cube, the other ochre cube. Practically three intermediate sectional cubes will be found sufficient for most purposes. We will take then a series of five figures—two terminal cubes, and three intermediate sections—and show how the different regions appear in our space when we take each set of three out of the four axes of the tesseract as lying in our space.
In [fig. 107] initial letters are used for the colours. A reference to [fig. 103] will show the complete nomenclature, which is merely indicated here.
