The Fourth Dimension

by Charles Howard Hinton

We have to keep in space the white and the yellow axes, and let the red go out, the blue come in.

Now, you will find on cube 10 a light yellow face; this should coincide with the base of 1, and the white and yellow lines on the two cubes should coincide. Then the blue axis running down you have the catalogue cube correctly placed, and it forms a guide for putting up the first representative block.

Catalogue cube 11 will represent what lies in the fourth dimension—now the red line runs in the fourth dimension. Thus the change from 10 to 11 should be towards red, corresponding to a null point is a red point, to a white line is a pink line, to a yellow line an orange line, and so on.

Catalogue cube 12 is like 10. Hence we see that to build up our blocks of tesseract faces we must take the bottom layer of the first block, hold that up in the air, underneath it place the bottom layer of the second block, and finally underneath this last the bottom layer of the last of our normal blocks.

Similarly we make the second representative group by taking the middle courses of our three blocks. The last is made by taking the three topmost layers. The three axes in our space before the transverse motion begins are blue, white, yellow, so we have light green tesseract faces, and after the motion begins sections transverse to the red light.

These three blocks represent the appearances as the tesseract group in its new position passes across our space. The cubes of contact in this case are those determinal by the three axes in our space, namely, the white, the yellow, the blue. Hence they are light green.

It follows from this that light green is the interior cube of the first block of representative cubic faces.

Practice in the manipulations described, with a realization in each case of the face or section which is in our space, is one of the best means of a thorough comprehension of the subject.

We have to learn how to get any part of these four-dimensional figures into space, so that we can look at them. We must first learn to swing a tesseract, and a group of tesseracts about in any way.

When these operations have been repeated and the method of arrangement of the set of blocks has become familiar, it is a good plan to rotate the axes of the normal cube 1 about a diagonal, and then repeat the whole series of turnings.