CHAPTER XIIIREMARKS 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, theochre 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.

Infig. 107initial letters are used for the colours. A reference tofig. 103will show the complete nomenclature, which is merely indicated here.

Fig. 107.

In this figure the tesseract is shown in five stages distant from our space: first, zero; second, 1/4 in.; third, 2/4 in.; fourth, 3/4 in.; fifth, 1 in.; which are calledb0,b1,b2,b3,b4, because they are sections taken at distances 0, 1, 2, 3, 4 quarter inches along the blue line. All the regions can be named from the first cube, theb0 cube, as before, simply by remembering that transference along the b axis gives the addition of blue to the colour of the region in the ochre, theb0 cube. In the final cubeb4, the colouring of the originalb0 cube is repeated. Thus the red line moved along the blue axis gives a red and blue or purple square. This purple square appears as the three purple lines in the sectionsb1,b2,b3, taken at 1/4, 2/4, 3/4 of an inch in the fourth dimension. If the tesseract moves transverse to our space we have then in this particular region, first of all a red line which lasts for a moment, secondly a purple line which takes itsplace. This purple line lasts for a minute—that is, all of a minute, except the moment taken by the crossing our space of the initial and final red line. The purple line having lasted for this period is succeeded by a red line, which lasts for a moment; then this goes and the tesseract has passed across our space. The final red line we call red bl., because it is separated from the initial red line by a distance along the axis for which we use the colour blue. Thus a line that lasts represents an area duration; is in this mode of presentation equivalent to a dimension of space. In the same way the white line, during the crossing our space by the tesseract, is succeeded by a light blue line which lasts for the inside of a minute, and as the tesseract leaves our space, having crossed it, the white bl. line appears as the final termination.

Take now the pink face. Moved in the blue direction it traces out a light purple cube. This light purple cube is shown in sections inb1,b2,b3, and the farther face of this cube in the blue direction is shown inb4—a pink face, called pinkbbecause it is distant from the pink face we began with in the blue direction. Thus the cube which we colour light purple appears as a lasting square. The square face itself, the pink face, vanishes instantly the tesseract begins to move, but the light purple cube appears as a lasting square. Here also duration is the equivalent of a dimension of space—a lasting square is a cube. It is useful to connect these diagrams with the views given in the coloured plate.

Take again the orange face, that determined by the red and yellow axes; from it goes a brown cube in the blue direction, for red and yellow and blue are supposed to make brown. This brown cube is shown in three sections in the facesb1,b2,b3. Inb4is the opposite orange face of the brown cube, the face called orangeb,for it is distant in the blue direction from the orange face. As the tesseract passes transverse to our space, we have then in this region an instantly vanishing orange square, followed by a lasting brown square, and finally an orange face which vanishes instantly.

Now, as any three axes will be in our space, let us send the white axis out into the unknown, the fourth dimension, and take the blue axis into our known space dimension. Since the white and blue axes are perpendicular to each other, if the white axis goes out into the fourth dimension in the positive sense, the blue axis will come into the direction the white axis occupied, in the negative sense.

Fig. 108.

Hence, not to complicate matters by having to think of two senses in the unknown direction, let us send the white line into the positive sense of the fourth dimension, and take the blue one as running in the negative sense of that direction which the white line has left; let the blue line, that is, run to the left. We have now the row of figures infig. 108. The dotted cube shows where we had a cube when the white line ran in our space—now it has turned out of our space, and another solid boundary, another cubic face of the tesseract comes into our space. This cube has red and yellow axes as before; but now, instead of a white axis running to the right, there is a blue axis running to the left. Here we can distinguish the regions by colours in a perfectly systematic way. The red line traces out a purplesquare in the transference along the blue axis by which this cube is generated from the orange face. This purple square made by the motion of the red line is the same purple face that we saw before as a series of lines in the sectionsb1,b2,b3. Here, since both red and blue axes are in our space, we have no need of duration to represent the area they determine. In the motion of the tesseract across space this purple face would instantly disappear.

From the orange face, which is common to the initial cubes infig. 107andfig. 108, there goes in the blue direction a cube coloured brown. This brown cube is now all in our space, because each of its three axes run in space directions, up, away, to the left. It is the same brown cube which appeared as the successive faces on the sectionsb1,b2,b3. Having all its three axes in our space, it is given in extension; no part of it needs to be represented as a succession. The tesseract is now in a new position with regard to our space, and when it moves across our space the brown cube instantly disappears.

In order to exhibit the other regions of the tesseract we must remember that now the white line runs in the unknown dimension. Where shall we put the sections at distances along the line? Any arbitrary position in our space will do: there is no way by which we can represent their real position.

However, as the brown cube comes off from the orange face to the left, let us put these successive sections to the left. We can call themwh0,wh1,wh2,wh3,wh4, because they are sections along the white axis, which now runs in the unknown dimension.

Running from the purple square in the white direction we find the light purple cube. This is represented in thesectionswh1,wh2,wh3,wh4,fig. 108. It is the same cube that is represented in the sectionsb1,b2,b3: infig. 107the red and white axes are in our space, the blue out of it; in the other case, the red and blue are in our space, the white out of it. It is evident that the face pinky, opposite the pink face infig. 107, makes a cube shown in squares inb1,b2,b3,b4, on the opposite side to thelpurple squares. Also the light yellow face at the base of the cubeb0, makes a light green cube, shown as a series of base squares.

The same light green cube can be found infig. 107. The base square inwh0is a green square, for it is enclosed by blue and yellow axes. From it goes a cube in the white direction, this is then a light green cube and the same as the one just mentioned as existing in the sectionsb0,b1,b2,b3,b4.

The case is, however, a little different with the brown cube. This cube we have altogether in space in the sectionwh0,fig. 108, while it exists as a series of squares, the left-hand ones, in the sectionsb0,b1,b2,b3,b4. The brown cube exists as a solid in our space, as shown infig. 108. In the mode of representation of the tesseract exhibited infig. 107, the same brown cube appears as a succession of squares. That is, as the tesseract moves across space, the brown cube would actually be to us a square—it would be merely the lasting boundary of another solid. It would have no thickness at all, only extension in two dimensions, and its duration would show its solidity in three dimensions.

It is obvious that, if there is a four-dimensional space, matter in three dimensions only is a mere abstraction; all material objects must then have a slight four-dimensional thickness. In this case the above statement will undergo modification. The material cube which is used as the model of the boundary of a tesseract will have a slight thickness in the fourth dimension, and when the cube ispresented to us in another aspect, it would not be a mere surface. But it is most convenient to regard the cubes we use as having no extension at all in the fourth dimension. This consideration serves to bring out a point alluded to before, that, if there is a fourth dimension, our conception of a solid is the conception of a mere abstraction, and our talking about real three-dimensional objects would seem to a four-dimensional being as incorrect as a two-dimensional being’s telling about real squares, real triangles, etc., would seem to us.

The consideration of the two views of the brown cube shows that any section of a cube can be looked at by a presentation of the cube in a different position in four-dimensional space. The brown faces inb1,b2,b3, are the very same brown sections that would be obtained by cutting the brown cube,wh0, across at the right distances along the blue line, as shown infig. 108. But as these sections are placed in the brown cube,wh0, they come behind one another in the blue direction. Now, in the sectionswh1,wh2,wh3, we are looking at these sections from the white direction—the blue direction does not exist in these figures. So we see them in a direction at right angles to that in which they occur behind one another inwh0. There are intermediate views, which would come in the rotation of a tesseract. These brown squares can be looked at from directions intermediate between the white and blue axes. It must be remembered that the fourth dimension is perpendicular equally to all three space axes. Hence we must take the combinations of the blue axis, with each two of our three axes, white, red, yellow, in turn.

Infig. 109we take red, white, and blue axes in space, sending yellow into the fourth dimension. If it goes into the positive sense of the fourth dimension the blue line will come in the opposite direction to that in which theyellow line ran before. Hence, the cube determined by the white, red, blue axes, will start from the pink plane and run towards us. The dotted cube shows where the ochre cube was. When it is turned out of space, the cube coming towards from its front face is the one which comes into our space in this turning. Since the yellow line now runs in the unknown dimension we call the sectionsy0,y1,y2,y3,y4, as they are made at distances 0, 1, 2, 3, 4, quarter inches along the yellow line. We suppose these cubes arranged in a line coming towards us—not that that is any more natural than any other arbitrary series of positions, but it agrees with the plan previously adopted.

Fig. 109.

The interior of the first cube,y0, is that derived from pink by adding blue, or, as we call it, light purple. The faces of the cube are light blue, purple, pink. As drawn, we can only see the face nearest to us, which is not the one from which the cube starts—but the face on the opposite side has the same colour name as the face towards us.

The successive sections of the series,y0,y1,y2, etc., can be considered as derived from sections of theb0cube made at distances along the yellow axis. What is distant a quarter inch from the pink face in the yellow direction? This question is answered by taking a section from a point a quarter inch along the yellow axis in the cubeb0,fig. 107. It is an ochre section with lines orange and light yellow. This section will therefore take the place of the pink faceiny1when we go on in the yellow direction. Thus, the first section,y1, will begin from an ochre face with light yellow and orange lines. The colour of the axis which lies in space towards us is blue, hence the regions of this section-cube are determined in nomenclature, they will be found in full infig. 105.

There remains only one figure to be drawn, and that is the one in which the red axis is replaced by the blue. Here, as before, if the red axis goes out into the positive sense of the fourth dimension, the blue line must come into our space in the negative sense of the direction which the red line has left. Accordingly, the first cube will come in beneath the position of our ochre cube, the one we have been in the habit of starting with.

Fig. 110.

To show these figures we must suppose the ochre cube to be on a movable stand. When the red line swings out into the unknown dimension, and the blue line comes in downwards, a cube appears below the place occupied by the ochre cube. The dotted cube shows where the ochre cube was. That cube has gone and a different cube runs downwards from its base. This cube has white, yellow, and blue axes. Its top is a light yellow square, and hence its interior is light yellow + blue or light green. Its front face is formed by the white line moving along the blue axis, and is therefore light blue, the left-hand side is formed by the yellow line moving along the blue axis, and therefore green.

As the red line now runs in the fourth dimension, the successive sections can he calledr0,r1,r2,r3,r4, these letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch along the red axis we take all of the tesseract that can be found in a three-dimensional space, this three-dimensional space extending not at all in the fourth dimension, but up and down, right and left, far and near.

We can see what should replace the light yellow face ofr0, when the sectionr1comes in, by looking at the cubeb0,fig. 107. What is distant in it one-quarter of an inch from the light yellow face in the red direction? It is an ochre section with orange and pink lines and red points; see alsofig. 103.

This square then forms the top square ofr1. Now we can determine the nomenclature of all the regions ofr1by considering what would be formed by the motion of this square along a blue axis.

But we can adopt another plan. Let us take a horizontal section ofr0, and finding that section in the figures, offig. 107orfig. 103, from them determine what will replace it, going on in the red direction.

A section of ther0cube has green, light blue, green, light blue sides and blue points.

Now this square occurs on the base of each of the section figures,b1,b2, etc. In them we see that 1/4 inch in the red direction from it lies a section with brown and light purple lines and purple corners, the interior being of light brown. Hence this is the nomenclature of the section which inr1replaces the section ofr0made from a point along the blue axis.

Hence the colouring as given can be derived.

We have thus obtained a perfectly named group of tesseracts. We can take a group of eighty-one of them 3 × 3 × 3 × 3, in four dimensions, and each tesseract will have its name null, red, white, yellow, blue, etc., andwhatever cubic view we take of them we can say exactly what sides of the tesseracts we are handling, and how they touch each other.[5]

[5]At this point the reader will find it advantageous, if he has the models, to go through the manipulations described in the appendix.

Thus, for instance, if we have the sixteen tesseracts shown below, we can ask how does null touch blue.

Fig. 111.

In the arrangement given infig. 111we have the axes white, red, yellow, in space, blue running in the fourth dimension. Hence we have the ochre cubes as bases. Imagine now the tesseractic group to pass transverse to our space—we have first of all null ochre cube, white ochre cube, etc.; these instantly vanish, and we get the section shown in the middle cube infig. 103, and finally, just when the tesseract block has moved one inch transverse to our space, we have null ochre cube, and then immediately afterwards the ochre cube of blue comes in. Hence the tesseract null touches the tesseract blue by its ochre cube, which is in contact, each and every point of it, with the ochre cube of blue.

How does null touch white, we may ask? Looking at the beginning A,fig. 111, where we have the ochrecubes, we see that null ochre touches white ochre by an orange face. Now let us generate the null and white tesseracts by a motion in the blue direction of each of these cubes. Each of them generates the corresponding tesseract, and the plane of contact of the cubes generates the cube by which the tesseracts are in contact. Now an orange plane carried along a blue axis generates a brown cube. Hence null touches white by a brown cube.

Fig. 112.

If we ask again how red touches light blue tesseract, let us rearrange our group,fig. 112, or rather turn it about so that we have a different space view of it; let the red axis and the white axis run up and right, and let the blue axis come in space towards us, then the yellow axis runs in the fourth dimension. We have then two blocks in which the bounding cubes of the tesseracts are given, differently arranged with regard to us—the arrangement is really the same, but it appears different to us. Starting from the plane of the red and white axes we have the four squares of the null, white, red, pink tesseracts as shown in A, on the red, white plane, unaltered, only from them now comes out towards us the blue axis.Hence we have null, white, red, pink tesseracts in contact with our space by their cubes which have the red, white, blue axis in them, that is by the light purple cubes. Following on these four tesseracts we have that which comes next to them in the blue direction, that is the four blue, light blue, purple, light purple. These are likewise in contact with our space by their light purple cubes, so we see a block as named in the figure, of which each cube is the one determined by the red, white, blue, axes.

The yellow line now runs out of space; accordingly one inch on in the fourth dimension we come to the tesseracts which follow on the eight named in C,fig. 112, in the yellow direction.

These are shown in C.y1,fig. 112. Between figure C and C.y1is that four-dimensional mass which is formed by moving each of the cubes in C one inch in the fourth dimension—that is, along a yellow axis; for the yellow axis now runs in the fourth dimension.

In the block C we observe that red (light purple cube) touches light blue (light purple cube) by a point. Now these two cubes moving together remain in contact during the period in which they trace out the tesseracts red and light blue. This motion is along the yellow axis, consequently red and light blue touch by a yellow line.

We have seen that the pink face moved in a yellow direction traces out a cube; moved in the blue direction it also traces out a cube. Let us ask what the pink face will trace out if it is moved in a direction within the tesseract lying equally between the yellow and blue directions. What section of the tesseract will it make?

We will first consider the red line alone. Let us take a cube with the red line in it and the yellow and blue axes.

Fig. 113.

The cube with the yellow, red, blue axes is shown infig. 113. If the red line is moved equally in the yellow and in the blue direction by four equal motions of ¼ inch each, it takes the positions 11, 22, 33, and ends as a red line.

Now, the whole of this red, yellow, blue, or brown cube appears as a series of faces on the successive sections of the tesseract starting from the ochre cube and letting the blue axis run in the fourth dimension. Hence the plane traced out by the red line appears as a series of lines in the successive sections, in our ordinary way of representing the tesseract; these lines are in different places in each successive section.

Fig. 114.

Thus drawing our initial cube and the successive sections, calling themb0,b1,b2,b3,b4,fig. 115, we have the red line subject to this movement appearing in the positions indicated.

We will now investigate what positions in the tesseract another line in the pink face assumes when it is moved in a similar manner.

Take a section of the original cube containing a vertical line, 4, in the pink plane,fig. 115. We have, in the section, the yellow direction, but not the blue.

From this section a cube goes off in the fourth dimension, which is formed by moving each point of the section in the blue direction.

Fig. 115.

Fig. 116.

Drawing this cube we havefig. 116.

Now this cube occurs as a series of sections in our original representation of the tesseract. Taking four steps as before this cube appears as the sections drawn inb0,b1,b2,b3,b4,fig. 117, and if the line 4 is subjected to a movement equal in the blue and yellow directions, it will occupy the positions designated by 4, 41, 42, 43, 44.

Fig. 117.

Hence, reasoning in a similar manner about every line, it is evident that, moved equally in the blue and yellow directions, the pink plane will trace out a space which is shown by the series of section planes represented in the diagram.

Thus the space traced out by the pink face, if it is moved equally in the yellow and blue directions, is represented by the set of planes delineated in Fig. 118, pinkface or 0, then 1, 2, 3, and finally pink face or 4. This solid is a diagonal solid of the tesseract, running from a pink face to a pink face. Its length is the length of the diagonal of a square, its side is a square.

Let us now consider the unlimited space which springs from the pink face extended.

This space, if it goes off in the yellow direction, gives us in it the ochre cube of the tesseract. Thus, if we have the pink face given and a point in the ochre cube, we have determined this particular space.

Similarly going off from the pink face in the blue direction is another space, which gives us the light purple cube of the tesseract in it. And any point being taken in the light purple cube, this space going off from the pink face is fixed.

Fig. 118.

The space we are speaking of can be conceived as swinging round the pink face, and in each of its positions it cuts out a solid figure from the tesseract, one of which we have seen represented infig. 118.

Each of these solid figures is given by one position of the swinging space, and by one only. Hence in each of them, if one point is taken, the particular one of the slanting spaces is fixed. Thus we see that given a plane and a point out of it a space is determined.

Now, two points determine a line.

Again, think of a line and a point outside it. Imagine a plane rotating round the line. At some time in its rotation it passes through the point. Thus a line and apoint, or three points, determine a plane. And finally four points determine a space. We have seen that a plane and a point determine a space, and that three points determine a plane; so four points will determine a space.

These four points may be any points, and we can take, for instance, the four points at the extremities of the red, white, yellow, blue axes, in the tesseract. These will determine a space slanting with regard to the section spaces we have been previously considering. This space will cut the tesseract in a certain figure.

One of the simplest sections of a cube by a plane is that in which the plane passes through the extremities of the three edges which meet in a point. We see at once that this plane would cut the cube in a triangle, but we will go through the process by which a plane being would most conveniently treat the problem of the determination of this shape, in order that we may apply the method to the determination of the figure in which a space cuts a tesseract when it passes through the 4 points at unit distance from a corner.

We know that two points determine a line, three points determine a plane, and given any two points in a plane the line between them lies wholly in the plane.

Fig. 119.

Let now the plane being study the section made by a plane passing through the nullr, nullwh, and nullypoints,fig. 119. Looking at the orange square, which, as usual, we suppose to be initially in his plane, he sees that the line from nullrto nully, which is a line in the section plane, the plane, namely, through the three extremities of the edges meeting in null, cuts the orangeface in an orange line with null points. This then is one of the boundaries of the section figure.

Let now the cube be so turned that the pink face comes in his plane. The points nullrand nullwhare now visible. The line between them is pink with null points, and since this line is common to the surface of the cube and the cutting plane, it is a boundary of the figure in which the plane cuts the cube.

Again, suppose the cube turned so that the light yellow face is in contact with the plane being’s plane. He sees two points, the nullwhand the nully. The line between these lies in the cutting plane. Hence, since the three cutting lines meet and enclose a portion of the cube between them, he has determined the figure he sought. It is a triangle with orange, pink, and light yellow sides, all equal, and enclosing an ochre area.

Let us now determine in what figure the space, determined by the four points, nullr, nully, nullwh, nullb, cuts the tesseract. We can see three of these points in the primary position of the tesseract resting against our solid sheet by the ochre cube. These three points determine a plane which lies in the space we are considering, and this plane cuts the ochre cube in a triangle, the interior of which is ochre (fig. 119will serve for this view), with pink, light yellow and orange sides, and null points. Going in the fourth direction, in one sense, from this plane we pass into the tesseract, in the other sense we pass away from it. The whole area inside the triangle is common to the cutting plane we see, and a boundary of the tesseract. Hence we conclude that the triangle drawn is common to the tesseract and the cutting space.

Fig. 120.

Now let the ochre cube turn out and the brown cube come in. The dotted lines show the position the ochre cube has left (fig. 120).

Here we see three out of the four points through which the cutting plane passes, nullr, nully, and nullb. The plane they determine lies in the cutting space, and this plane cuts out of the brown cube a triangle with orange, purple and green sides, and null points. The orange line of this figure is the same as the orange line in the last figure.

Now let the light purple cube swing into our space, towards us,fig. 121.

Fig. 121.

The cutting space which passes through the four points, nullr,y,wh,b, passes through the nullr,wh,b, and therefore the plane these determine lies in the cutting space.

This triangle lies before us. It has a light purple interior and pink, light blue, and purple edges with null points.

This, since it is all of the plane that is common to it, and this bounding of the tesseract, gives us one of the bounding faces of our sectional figure. The pink line in it is the same as the pink line we found in the first figure—that of the ochre cube.

Finally, let the tesseract swing about the light yellow plane, so that the light green cube comes into our space. It will point downwards.

Fig. 122.

The three points,n.y,n.wh,n.b, are in the cuttingspace, and the triangle they determine is common to the tesseract and the cutting space. Hence this boundary is a triangle having a light yellow line, which is the same as the light yellow line of the first figure, a light blue line and a green line.

We have now traced the cutting space between every set of three that can be made out of the four points in which it cuts the tesseract, and have got four faces which all join on to each other by lines.

Fig. 123.

The triangles are shown infig. 123as they join on to the triangle in the ochre cube. But they join on each to the other in an exactly similar manner; their edges are all identical two and two. They form a closed figure, a tetrahedron, enclosing a light brown portion which is the portion of the cutting space which lies inside the tesseract.

We cannot expect to see this light brown portion, any more than a plane being could expect to see the inside of a cube if an angle of it were pushed through his plane. All he can do is to come upon the boundaries of it in a different way to that in which he would if it passed straight through his plane.

Thus in this solid section; the whole interior lies perfectly open in the fourth dimension. Go round it as we may we are simply looking at the boundaries of the tesseract which penetrates through our solid sheet. If the tesseract were not to pass across so far, the trianglewould be smaller; if it were to pass farther, we should have a different figure, the outlines of which can be determined in a similar manner.

The preceding method is open to the objection that it depends rather on our inferring what must be, than our seeing what is. Let us therefore consider our sectional space as consisting of a number of planes, each very close to the last, and observe what is to be found in each plane.

Fig. 124.

The corresponding method in the case of two dimensions is as follows:—The plane being can see that line of the sectional plane through nully, nullwh, nullr, which lies in the orange plane. Let him now suppose the cube and the section plane to pass half way through his plane. Replacing the red and yellow axes are lines parallel to them, sections of the pink and light yellow faces.

Where will the section plane cut these parallels to the red and yellow axes?

Let him suppose the cube, in the position of the drawing,fig. 124, turned so that the pink face lies against his plane. He can see the line from the nullrpoint to the nullwhpoint, and can see (comparefig. 119) that it cutsABa parallel to his red axis, drawn at a point half way along the white line, in a pointB, half way up. I shall speak of the axis as having the length of an edge of the cube. Similarly, by letting the cube turn so that the light yellow square swings against his plane, he can see (comparefig. 119) that a parallel to his yellow axis drawn from a point half-way along the white axis, is cut at half its length by the trace of the section plane in the light yellow face.

Hence when the cube had passed half-way through he would have—instead of the orange line with null points, which he had at first—an ochre line of half its length, with pink and light yellow points. Thus, as the cube passed slowly through his plane, he would have a succession of lines gradually diminishing in length and forming an equilateral triangle. The whole interior would be ochre, the line from which it started would be orange. The succession of points at the ends of the succeeding lines would form pink and light yellow lines and the final point would be null. Thus looking at the successive lines in the section plane as it and the cube passed across his plane he would determine the figure cut out bit by bit.

Coming now to the section of the tesseract, let us imagine that the tesseract and its cuttingspacepass slowly across our space; we can examine portions of it, and their relation to portions of the cutting space. Take the section space which passes through the four points, nullr,wh,y,b; we can see in the ochre cube (fig. 119) the plane belonging to this section space, which passes through the three extremities of the red, white, yellow axes.

Now let the tesseract pass half way through our space. Instead of our original axes we have parallels to them, purple, light blue, and green, each of the same length as the first axes, for the section of the tesseract is of exactly the same shape as its ochre cube.

But the sectional space seen at this stage of the transference would not cut the section of the tesseract in a plane disposed as at first.

To see where the sectional space would cut these parallels to the original axes let the tesseract swing so that, the orange face remaining stationary, the blue line comes in to the left.

Fig. 125.

Here (fig. 125) we have the nullr,y,bpoints, and of the sectional space all we see is the plane through these three points in it.

In this figure we can draw the parallels to the red and yellow axes and see that, if they started at a point half way along the blue axis, they would each be cut at a point so as to be half of their previous length.

Swinging the tesseract into our space about the pink face of the ochre cube we likewise find that the parallel to the white axis is cut at half its length by the sectional space.

Fig. 126.

Hence in a section made when the tesseract had passed half across our space the parallels to the red, white, yellow axes, which are now in our space, are cut by the section space, each of them half way along, and for this stage of the traversing motion we should havefig. 126. The section made of this cube by the plane in which the sectional space cuts it, is an equilateral triangle with purple, l. blue, green points, and l. purple, brown, l. green lines.

Thus the original ochre triangle, with null points and pink, orange, light yellow lines, would be succeeded by a triangle coloured in manner just described.

This triangle would initially be only a very little smaller than the original triangle, it would gradually diminish, until it ended in a point, a null point. Each of its edges would be of the same length. Thus the successivesections of the successive planes into which we analyse the cutting space would be a tetrahedron of the description shown (fig. 123), and the whole interior of the tetrahedron would be light brown.

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