Space Names.

If the words written in the squares drawn in [fig. 1] are used as the names of the squares in the positions in which they are placed, it is evident that a combination of these names will denote a figure composed of the designated squares. It is found to be most convenient to take as the initial square that marked with an asterisk, so that the directions of progression are towards the observer and to his right. The directions of progression, however, are arbitrary, and can be chosen at will.

Fig. 1.

Thus et, at, it, an, al will denote a figure in the form of a cross composed of five squares.

Here, by means of the double sequence, e, a, i and n, t, l, it is possible to name a limited collection of space elements.

The system can obviously be extended by using letter sequences of more members.

But, without introducing such a complexity, the principles of a space language can be exhibited, and a nomenclature obtained adequate to all the considerations of the preceding pages.

1. Extension.

Fig. 2.

Call the large squares in [2] by the name written in them. It is evident that each can be divided as shown in [fig. 1]. Then the small square marked 1 will be “en” in “En,” or “Enen.” The square marked 2 will be “et” in “En” or “Enet,” while the square marked 4 will be “en” in “Et” or “Eten.” Thus the square 5 will be called “Ilil.”

This principle of extension can be applied in any number of dimensions.

2. Application to Three-Dimensional Space.

To name a three-dimensional collocation of cubes take the upward direction first, secondly the direction towards the observer, thirdly the direction to his right hand.

These form a word in which the first letter gives the place of the cube upwards, the second letter its place towards the observer, the third letter its place to the right.

We have thus the following scheme, which represents the set of cubes of column 1, [fig. 101], page 165.

We begin with the remote lowest cube at the left hand, where the asterisk is placed (this proves to be by far the most convenient origin to take for the normal system).

Thus “nen” is a “null” cube, “ten” a red cube on it, and “len” a “null” cube above “ten.”

By using a more extended sequence of consonants and vowels a larger set of cubes can be named.

To name a four-dimensional block of tesseracts it is simply necessary to prefix an “e,” an “a,” or an “i” to the cube names.

Thus the tesseract blocks schematically represented on page 165, [fig. 101] are named as follows:—

2. Derivation of Point, Line, Face, etc., Names.

The principle of derivation can be shown as follows: Taking the square of squares

the number of squares in it can be enlarged and the whole kept the same size.

Compare [fig. 79], p. 138, for instance, or the bottom layer of [fig. 84].

Now use an initial “s” to denote the result of carrying this process on to a great extent, and we obtain the limit names, that is the point, line, area names for a square. “Sat” is the whole interior. The corners are “sen,” “sel,” “sin,” “sil,” while the lines are “san,” “sal,” “set,” “sit.”

I find that by the use of the initial “s” these names come to be practically entirely disconnected with the systematic names for the square from which they are derived. They are easy to learn, and when learned can be used readily with the axes running in any direction.

To derive the limit names for a four-dimensional rectangular figure, like the tesseract, is a simple extension of this process. These point, line, etc., names include those which apply to a cube, as will be evident on inspection of the first cube of the diagrams which follow.

All that is necessary is to place an “s” before each of the names given for a tesseract block. We then obtain apellatives which, like the colour names on page 174, [fig. 103], apply to all the points, lines, faces, solids, and to the hyper-solid of the tesseract. These names have the advantage over the colour marks that each point, line, etc., has its own individual name.

In the diagrams I give the names corresponding to the positions shown in the coloured plate or described on p. 174. By comparing cubes 1, 2, 3 with the first row of cubes in the coloured plate, the systematic names of each of the points, lines, faces, etc., can be determined. The asterisk shows the origin from which the names run.

These point, line, face, etc., names should be used in connection with the corresponding colours. The names should call up coloured images of the parts named in their right connection.

It is found that a certain abbreviation adds vividness of distinction to these names. If the final “en” be dropped wherever it occurs the system is improved. Thus instead of “senen,” “seten,” “selen,” it is preferable to abbreviate to “sen,” “set,” “sel,” and also use “san,” “sin” for “sanen,” “sinen.”

We can now name any section. Take e.g. the line in the first cube from senin to senel, we should call the line running from senin to senel, senin senat senel, a line light yellow in colour with null points.

Here senat is the name for all of the line except its ends. Using “senat” in this way does not mean that the line is the whole of senat, but what there is of it is senat. It is a part of the senat region. Thus also the triangle, which has its three vertices in senin, senel, selen, is named thus:

The tetrahedron section of the tesseract can be thought of as a series of plane sections in the successive sections of the tesseract shown in [fig. 114], p. 191. In b0 the section is the one written above. In b1 the section is made by a plane which cuts the three edges from sanen intermediate of their lengths and thus will be:

The sections in b2, b3 will be like the section in b1 but smaller.

Finally in b4 the section plane simply passes through the corner named sin.

Hence, putting these sections together in their right relation, from the face setat, surrounded by the lines and points mentioned above, there run:

and these faces and lines run to the point sin. Thus the tetrahedron is completely named.

The octahedron section of the tesseract, which can be traced from [fig. 72], p. 129 by extending the lines there drawn, is named:

Front triangle selin, selat, selel, setal, senil, setit, selin with area setat.

The sections between the front and rear triangle, of which one is shown in 1b, another in 2b, are thus named, points and lines, salan, salat, salet, satet, satel, satal, sanal, sanat, sanit, satit, satin, satan, salan.

The rear triangle found in 3b by producing lines is sil, sitet, sinel, sinat, sinin, sitan, sil.

The assemblage of sections constitute the solid body of the octahedron satat with triangular faces. The one from the line selat to the point sil, for instance, is named selin, selat, selel, salet, salat, salan, sil. The whole interior is salat.

Shapes can easily be cut out of cardboard which, when folded together, form not only the tetrahedron and the octahedron, but also samples of all the sections of the tesseract taken as it passes cornerwise through our space. To name and visualise with appropriate colours a series of these sections is an admirable exercise for obtaining familiarity with the subject.