CHAPTER XINOMENCLATURE AND ANALOGIES PRELIMINARY TO THE STUDY OF FOUR-DIMENSIONAL FIGURES

In the following pages a method of designating different regions of space by a systematic colour scheme has been adopted. The explanations have been given in such a manner as to involve no reference to models, the diagrams will be found sufficient. But to facilitate the study a description of a set of models is given in an appendix which the reader can either make for himself or obtain. If models are used the diagrams in Chapters XI. and XII. will form a guide sufficient to indicate their use. Cubes of the colours designated by the diagrams should be picked out and used to reinforce the diagrams. The reader, in the following description, should suppose that a board or wall stretches away from him, against which the figures are placed.

Fig. 77.

Take a square, one of those shown in Fig. 77 and give it a neutral colour, let this colour be called “null,” and be such that it makes no appreciable differenceto any colour with which it mixed. If there is no such real colour let us imagine such a colour, and assign to it the properties of the number zero, which makes no difference in any number to which it is added.

Above this square place a red square. Thus we symbolise the going up by adding red to null.

Away from this null square place a yellow square, and represent going away by adding yellow to null.

Fig. 78.

To complete the figure we need a fourth square. Colour this orange, which is a mixture of red and yellow, and so appropriately represents a going in a direction compounded of up and away. We have thus a colour scheme which will serve to name the set of squares drawn. We have two axes of colours—red and yellow—and they may occupy as in the figure the direction up and away, or they may be turned about; in any case they enable us to name the four squares drawn in their relation to one another.

Now take, in Fig. 78, nine squares, and suppose that at the end of the going in any direction the colour started with repeats itself.

We obtain a square named as shown.

Let us now, infig. 79, suppose the number of squares to be increased, keeping still to the principle of colouring already used.

Here the nulls remain four in number. There are three reds between the first null and the null above it, three yellows between the first null and thenull beyond it, while the oranges increase in a double way.

Fig. 79.

Suppose this process of enlarging the number of the squares to be indefinitely pursued and the total figure obtained to be reduced in size, we should obtain a square of which the interior was all orange, while the lines round it were red and yellow, and merely the points null colour, as infig. 80. Thus all the points, lines, and the area would have a colour.

Fig. 80.

We can consider this scheme to originate thus:—Let a null point move in a yellow direction and trace out a yellow line and end in a null point. Then let the whole line thus traced move in a red direction. The null points at the ends of the line will produce red lines, and end innull points. The yellow line will trace out a yellow and red, or orange square.

Now, turning back tofig. 78, we see that these two ways of naming, the one we started with and the one we arrived at, can be combined.

By its position in the group of four squares, infig. 77, the null square has a relation to the yellow and to the red directions. We can speak therefore of the red line of the null square without confusion, meaning thereby the lineAB,fig. 81, which runs up from the initial null pointAin the figure as drawn. The yellow line of the null square is its lower horizontal lineACas it is situated in the figure.

Fig. 81.

If we wish to denote the upper yellow lineBD,fig. 81, we can speak of it as the yellow γ line, meaning the yellow line which is separated from the primary yellow line by the red movement.

In a similar way each of the other squares has null points, red and yellow lines. Although the yellow square is all yellow, its lineCD, for instance, can be referred to as its red line.

This nomenclature can be extended.

If the eight cubes drawn, infig. 82, are put close together, as on the right hand of the diagram, they form a cube, and in them, as thus arranged, a going up is represented by adding red to the zero, or null colour, a going away by adding yellow, a going to the right by adding white. White is used as a colour, as a pigment, which produces a colour change in the pigments with which it is mixed. From whatever cube of the lower set we start, a motion up brings us to a cube showing a change to red, thus light yellow becomes light yellow red, or light orange, which is called ochre. And going to theright from the null on the left we have a change involving the introduction of white, while the yellow change runs from front to back. There are three colour axes—the red, the white, the yellow—and these run in the position the cubes occupy in the drawing—up, to the right, away—but they could be turned about to occupy any positions in space.

Fig. 82.

Fig. 83.

We can conveniently represent a block of cubes by three sets of squares, representing each the base of a cube.

Thus the block,fig. 83, can be represented by thelayers on the right. Here, as in the case of the plane, the initial colours repeat themselves at the end of the series.

Fig. 84.

Proceeding now to increase the number of the cubes we obtainfig. 84, in which the initial letters of the colours are given instead of their full names.

Here we see that there are four null cubes as before, but the series which spring from the initial corner will tend to become lines of cubes, as also the sets of cubes parallel to them, starting from other corners. Thus, from the initial null springs a line of red cubes, a line of white cubes, and a line of yellow cubes.

If the number of the cubes is largely increased, and the size of the whole cube is diminished, we get a cube with null points, and the edges coloured with these three colours.

The light yellow cubes increase in two ways, forming ultimately a sheet of cubes, and the same is true of the orange and pink sets. Hence, ultimately the cubethus formed would have red, white, and yellow lines surrounding pink, orange, and light yellow faces. The ochre cubes increase in three ways, and hence ultimately the whole interior of the cube would be coloured ochre.

We have thus a nomenclature for the points, lines, faces, and solid content of a cube, and it can be named as exhibited infig. 85.

Fig. 85.

We can consider the cube to be produced in the following way. A null point moves in a direction to which we attach the colour indication yellow; it generates a yellow line and ends in a null point. The yellow line thus generated moves in a direction to which we give the colour indication red. This lies up in the figure. The yellow line traces out a yellow, red, or orange square, and each of its null points trace out a red line, and ends in a null point.

This orange square moves in a direction to which we attribute the colour indication white, in this case the direction is the right. The square traces out a cube coloured orange, red, or ochre, the red lines trace out red to white or pink squares, and the yellow lines trace out light yellow squares, each line ending in a line of its own colour. While the points each trace out a null + white, or white line to end in a null point.

Now returning to the first block of eight cubes we can name each point, line, and square in them by reference to the colour scheme, which they determine by their relation to each other.

Thus, infig. 86, the null cube touches the red cube bya light yellow square; it touches the yellow cube by a pink square, and touches the white cube by an orange square.

Fig. 86.

There are three axes to which the colours red, yellow, and white are assigned, the faces of each cube are designated by taking these colours in pairs. Taking all the colours together we get a colour name for the solidity of a cube.

Let us now ask ourselves how the cube could be presented to the plane being. Without going into the question of how he could have a real experience of it, let us see how, if we could turn it about and show it to him, he, under his limitations, could get information about it. If the cube were placed with its red and yellow axes against a plane, that is resting against it by its orange face, the plane being would observe a square surrounded by red and yellow lines, and having null points. See the dotted square,fig. 87.

Fig. 87.

We could turn the cube about the red line so that a different face comes into juxtaposition with the plane.

Suppose the cube turned about the red line. As itis turning from its first position all of it except the red line leaves the plane—goes absolutely out of the range of the plane being’s apprehension. But when the yellow line points straight out from the plane then the pink face comes into contact with it. Thus the same red line remaining as he saw it at first, now towards him comes a face surrounded by white and red lines.

Fig. 88.

If we call the direction to the right the unknown direction, then the line he saw before, the yellow line, goes out into this unknown direction, and the line which before went into the unknown direction, comes in. It comes in in the opposite direction to that in which the yellow line ran before; the interior of the face now against the plane is pink. It is a property of two lines at right angles that, if one turns out of a given direction and stands at right angles to it, then the other of the two lines comes in, but runs the opposite way in that given direction, as infig. 88.

Now these two presentations of the cube would seem, to the plane creature like perfectly different material bodies, with only that line in common in which they both meet.

Again our cube can be turned about the yellow line. In this case the yellow square would disappear as before, but a new square would come into the plane after the cube had rotated by an angle of 90° about this line. The bottom square of the cube would come in thus in figure 89. The cube supposed in contact with the plane is rotated about the lower yellow line and then the bottom face is in contact with the plane.

Here, as before, the red line going out into the unknown dimension, the white line which before ran in theunknown dimension would come in downwards in the opposite sense to that in which the red line ran before.

Fig. 89.

Now if we usei,j,k, for the three space directions,ileft to right,jfrom near away,kfrom below up; then, using the colour names for the axes, we have that first of all white runsi, yellow runsj, red runsk; then after the first turning round thekaxis, white runs negativej, yellow runsi, red runsk; thus we have the table:—

Here white with a negative sign after it in the column underjmeans that white runs in the negative sense of thejdirection.

We may express the fact in the following way:— In the plane there is room for two axes while the body has three. Therefore in the plane we can represent any two. If we want to keep the axis that goes in the unknown dimension always running in the positive sense, then the axis which originally ran in the unknowndimension (the white axis) must come in in the negative sense of that axis which goes out of the plane into the unknown dimension.

It is obvious that the unknown direction, the direction in which the white line runs at first, is quite distinct from any direction which the plane creature knows. The white line may come in towards him, or running down. If he is looking at a square, which is the face of a cube (looking at it by a line), then any one of the bounding lines remaining unmoved, another face of the cube may come in, any one of the faces, namely, which have the white line in them. And the white line comes sometimes in one of the space directions he knows, sometimes in another.

Now this turning which leaves a line unchanged is something quite unlike any turning he knows in the plane. In the plane a figure turns round a point. The square can turn round the null point in his plane, and the red and yellow lines change places, only of course, as with every rotation of lines at right angles, if red goes where yellow went, yellow comes in negative of red’s old direction.

This turning, as the plane creature conceives it, we should call turning about an axis perpendicular to the plane. What he calls turning about the null point we call turning about the white line as it stands out from his plane. There is no such thing as turning about a point, there is always an axis, and really much more turns than the plane being is aware of.

Taking now a different point of view, let us suppose the cubes to be presented to the plane being by being passed transverse to his plane. Let us suppose the sheet of matter over which the plane being and all objects in his world slide, to be of such a nature that objects can pass through it without breaking it. Let us suppose it to be of the same nature as the film of a soap bubble, so thatit closes around objects pushed through it, and, however the object alters its shape as it passes through it, let us suppose this film to run up to the contour of the object in every part, maintaining its plane surface unbroken.

Then we can push a cube or any object through the film and the plane being who slips about in the film will know the contour of the cube just and exactly where the film meets it.

Fig. 90.

Fig. 90 represents a cube passing through a plane film. The plane being now comes into contact with a very thin slice of the cube somewhere between the left and right hand faces. This very thin slice he thinks of as having no thickness, and consequently his idea of it is what we call a section. It is bounded by him by pink lines front and back, coming from the part of the pink face he is in contact with, and above and below, by light yellow lines. Its corners are not null-coloured points, but white points, and its interior is ochre, the colour of the interior of the cube.

If now we suppose the cube to be an inch in each dimension, and to pass across, from right to left, through the plane, then we should explain the appearances presented to the plane being by saying: First of all you have the face of a cube, this lasts only a moment; then you have a figure of the same shape but differently coloured. This, which appears not to move to you in any direction which you know of, is really moving transverse to your plane world. Its appearance is unaltered, but each moment it is something different—a section further on, in the white, the unknown dimension. Finally, at theend of the minute, a face comes in exactly like the face you first saw. This finishes up the cube—it is the further face in the unknown dimension.

The white line, which extends in length just like the red or the yellow, you do not see as extensive; you apprehend it simply as an enduring white point. The null point, under the condition of movement of the cube, vanishes in a moment, the lasting white point is really your apprehension of a white line, running in the unknown dimension. In the same way the red line of the face by which the cube is first in contact with the plane lasts only a moment, it is succeeded by the pink line, and this pink line lasts for the inside of a minute. This lasting pink line in your apprehension of a surface, which extends in two dimensions just like the orange surface extends, as you know it, when the cube is at rest.

But the plane creature might answer, “This orange object is substance, solid substance, bounded completely and on every side.”

Here, of course, the difficulty comes in. His solid is our surface—his notion of a solid is our notion of an abstract surface with no thickness at all.

We should have to explain to him that, from every point of what he called a solid, a new dimension runs away. From every point a line can be drawn in a direction unknown to him, and there is a solidity of a kind greater than that which he knows. This solidity can only be realised by him by his supposing an unknown direction, by motion in which what he conceives to be solid matter instantly disappears. The higher solid, however, which extends in this dimension as well as in those which he knows, lasts when a motion of that kind takes place, different sections of it come consecutively in the plane of his apprehension, and take the place of the solid which he at first conceives to be all. Thus, the higher solid—oursolid in contradistinction to his area solid, his two-dimensional solid, must be conceived by him as something which has duration in it, under circumstances in which his matter disappears out of his world.

We may put the matter thus, using the conception of motion.

A null point moving in a direction away generates a yellow line, and the yellow line ends in a null point. We suppose, that is, a point to move and mark out the products of this motion in such a manner. Now suppose this whole line as thus produced to move in an upward direction; it traces out the two-dimensional solid, and the plane being gets an orange square. The null point moves in a red line and ends in a null point, the yellow line moves and generates an orange square and ends in a yellow line, the farther null point generates a red line and ends in a null point. Thus, by movement in two successive directions known to him, he can imagine his two-dimensional solid produced with all its boundaries.

Now we tell him: “This whole two-dimensional solid can move in a third or unknown dimension to you. The null point moving in this dimension out of your world generates a white line and ends in a null point. The yellow line moving generates a light yellow two-dimensional solid and ends in a yellow line, and this two-dimensional solid, lying end on to your plane world, is bounded on the far side by the other yellow line. In the same way each of the lines surrounding your square traces out an area, just like the orange area you know. But there is something new produced, something which you had no idea of before; it is that which is produced by the movement of the orange square. That, than which you can imagine nothing more solid, itself moves in a direction open to it and produces a three-dimensionalsolid. Using the addition of white to symbolise the products of this motion this new kind of solid will be light orange or ochre, and it will be bounded on the far side by the final position of the orange square which traced it out, and this final position we suppose to be coloured like the square in its first position, orange with yellow and red boundaries and null corners.”

This product of movement, which it is so easy for us to describe, would be difficult for him to conceive. But this difficulty is connected rather with its totality than with any particular part of it.

Any line, or plane of this, to him higher, solid we could show to him, and put in his sensible world.

We have already seen how the pink square could be put in his world by a turning of the cube about the red line. And any section which we can conceive made of the cube could be exhibited to him. You have simply to turn the cube and push it through, so that the plane of his existence is the plane which cuts out the given section of the cube, then the section would appear to him as a solid. In his world he would see the contour, get to any part of it by digging down into it.

The Process by which a Plane Being would gain a Notion of a Solid.

If we suppose the plane being to have a general idea of the existence of a higher solid—our solid—we must next trace out in detail the method, the discipline, by which he would acquire a working familiarity with our space existence. The process begins with an adequate realisation of a simple solid figure. For this purpose we will suppose eight cubes forming a larger cube, and first we will suppose each cube to be coloured throughout uniformly.Let the cubes infig. 91be the eight making a larger cube.

Fig. 91.

Now, although each cube is supposed to be coloured entirely through with the colour, the name of which is written on it, still we can speak of the faces, edges, and corners of each cube as if the colour scheme we have investigated held for it. Thus, on the null cube we can speak of a null point, a red line, a white line, a pink face, and so on. These colour designations are shown on No. 1 of the views of the tesseract in the plate. Here these colour names are used simply in their geometrical significance. They denote what the particular line, etc., referred to would have as its colour, if in reference to the particular cube the colour scheme described previously were carried out.

If such a block of cubes were put against the plane and then passed through it from right to left, at the rate of an inch a minute, each cube being an inch each way, the plane being would have the following appearances:—

First of all, four squares null, yellow, red, orange, lasting each a minute; and secondly, taking the exact places of these four squares, four others, coloured white, light yellow, pink, ochre. Thus, to make a catalogue of the solid body, he would have to put side by side in his world two sets of four squares each, as infig. 92. The firstare supposed to last a minute, and then the others to come in in place of them, and also last a minute.

Fig. 92.

In speaking of them he would have to denote what part of the respective cube each square represents. Thus, at the beginning he would have null cube orange face, and after the motion had begun he would have null cube ochre section. As he could get the same coloured section whichever way the cube passed through, it would be best for him to call this section white section, meaning that it is transverse to the white axis. These colour-names, of course, are merely used as names, and do not imply in this case that the object is really coloured. Finally, after a minute, as the first cube was passing beyond his plane he would have null cube orange face again.

The same names will hold for each of the other cubes, describing what face or section of them the plane being has before him; and the second wall of cubes will come on, continue, and go out in the same manner. In the area he thus has he can represent any movement which we carry out in the cubes, as long as it does not involve a motion in the direction of the white axis. The relation of parts that succeed one another in the direction of the white axis is realised by him as a consecution of states.

Now, his means of developing his space apprehension lies in this, that that which is represented as a time sequence in one position of the cubes, can become a real co-existence,if something that has a real co-existence becomes a time sequence.

We must suppose the cubes turned round each of the axes, the red line, and the yellow line, then something, which was given as time before, will now be given as the plane creature’s space; something, which was given as space before, will now be given as a time series as the cube is passed through the plane.

The three positions in which the cubes must be studied are the one given above and the two following ones. In each case the original null point which was nearest to us at first is marked by an asterisk. In figs. 93 and 94 the point marked with a star is the same in the cubes and in the plane view.

Fig. 93.The cube swung round the red line, so that the white line points towards us.

Infig. 93the cube is swung round the red line so as to point towards us, and consequently the pink face comes next to the plane. As it passes through there are two varieties of appearance designated by the figures 1 and 2 in the plane. These appearances are named in the figure, and are determined by the order in which the cubescome in the motion of the whole block through the plane.

With regard to these squares severally, however, different names must be used, determined by their relations in the block.

Thus, infig. 93, when the cube first rests against the plane the null cube is in contact by its pink face; as the block passes through we get an ochre section of the null cube, but this is better called a yellow section, as it is made by a plane perpendicular to the yellow line. When the null cube has passed through the plane, as it is leaving it, we get again a pink face.

Fig. 94.The cube swung round yellow line, with red line running from left to right, and white line running down.

The same series of changes take place with the cube appearances which follow on those of the null cube. In this motion the yellow cube follows on the null cube, and the square marked yellow in 2 in the plane will be first “yellow pink face,” then “yellow yellow section,” then “yellow pink face.”

Infig. 94, in which the cube is turned about the yellow line, we have a certain difficulty, for the plane being willfind that the position his squares are to be placed in will lie below that which they first occupied. They will come where the support was on which he stood his first set of squares. He will get over this difficulty by moving his support.

Then, since the cubes come upon his plane by the light yellow face, he will have, taking the null cube as before for an example, null, light yellow face; null, red section, because the section is perpendicular to the red line; and finally, as the null cube leaves the plane, null, light yellow face. Then, in this case red following on null, he will have the same series of views of the red as he had of the null cube.

Fig. 95.

There is another set of considerations which we will briefly allude to.

Suppose there is a hollow cube, and a string is stretched across it from null to null,r,y,wh, as we may call the far diagonal point, how will this string appear to the plane being as the cube moves transverse to his plane?

Let us represent the cube as a number of sections, say 5, corresponding to 4 equal divisions made along the white line perpendicular to it.

We number these sections 0, 1, 2, 3, 4, corresponding to the distances along the white line at which they aretaken, and imagine each section to come in successively, taking the place of the preceding one.

These sections appear to the plane being, counting from the first, to exactly coincide each with the preceding one. But the section of the string occupies a different place in each to that which it does in the preceding section. The section of the string appears in the position marked by the dots. Hence the slant of the string appears as a motion in the frame work marked out by the cube sides. If we suppose the motion of the cube not to be recognised, then the string appears to the plane being as a moving point. Hence extension on the unknown dimension appears as duration. Extension sloping in the unknown direction appears as continuous movement.

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