Thus, in the normal position, red goes up, white to the right, yellow away. Make white go up, yellow to the right, and red away. Learn the cube in this position by putting up the set of blocks of the normal cube, over and over again till it becomes as familiar to you as in the normal position. Then when this is learned, and the corresponding changes in the arrangements of the tesseract groups are made, another change should be made: let, in the normal cube, yellow go up, red to the right, and white away.
Learn the normal block of cubes in this new position by arranging them and re-arranging them till you know without thought where each one goes. Then carry out all the tesseract arrangements and turnings.
If you want to understand the subject, but do not see your way clearly, if it does not seem natural and easy to you, practise these turnings. Practise, first of all, the turning of a block of cubes round, so that you know it in every position as well as in the normal one. Practise by gradually putting up the set of cubes in their new arrangements. Then put up the tesseract blocks in their arrangements. This will give you a working conception of higher space, you will gain the feeling of it, whether you take up the mathematical treatment of it or not.
APPENDIX II
A LANGUAGE OF SPACE
The mere naming the parts of the figures we consider involves a certain amount of time and attention. This time and attention leads to no result, for with each new figure the nomenclature applied is completely changed, every letter or symbol is used in a different significance.
Surely it must be possible in some way to utilise the labour thus at present wasted!
Why should we not make a language for space itself, so that every position we want to refer to would have its own name? Then every time we named a figure in order to demonstrate its properties we should be exercising ourselves in the vocabulary of place.
If we use a definite system of names, and always refer to the same space position by the same name, we create as it were a multitude of little hands, each prepared to grasp a special point, position, or element, and hold it for us in its proper relations.
We make, to use another analogy, a kind of mental paper, which has somewhat of the properties of a sensitive plate, in that it will register, without effort, complex, visual, or tactual impressions.
But of far more importance than the applications of a space language to the plane and to solid space is the facilitation it brings with it to the study of four-dimensional shapes.