CHAPTER XIITHE SIMPLEST FOUR-DIMENSIONAL SOLID

A plane being, in learning to apprehend solid existence, must first of all realise that there is a sense of direction altogether wanting to him. That which we call right and left does not exist in his perception. He must assume a movement in a direction, and a distinction of positive and negative in that direction, which has no reality corresponding to it in the movements he can make. This direction, this new dimension, he can only make sensible to himself by bringing in time, and supposing that changes, which take place in time, are due to objects of a definite configuration in three dimensions passing transverse to his plane, and the different sections of it being apprehended as changes of one and the same plane figure.

He must also acquire a distinct notion about his plane world, he must no longer believe that it is the all of space, but that space extends on both sides of it. In order, then, to prevent his moving off in this unknown direction, he must assume a sheet, an extended solid sheet, in two dimensions, against which, in contact with which, all his movements take place.

When we come to think of a four-dimensional solid, what are the corresponding assumptions which we must make?

We must suppose a sense which we have not, a senseof direction wanting in us, something which a being in a four-dimensional world has, and which we have not. It is a sense corresponding to a new space direction, a direction which extends positively and negatively from every point of our space, and which goes right away from any space direction we know of. The perpendicular to a plane is perpendicular, not only to two lines in it, but to every line, and so we must conceive this fourth dimension as running perpendicularly to each and every line we can draw in our space.

And as the plane being had to suppose something which prevented his moving off in the third, the unknown dimension to him, so we have to suppose something which prevents us moving off in the direction unknown to us. This something, since we must be in contact with it in every one of our movements, must not be a plane surface, but a solid; it must be a solid, which in every one of our movements we are against, not in. It must be supposed as stretching out in every space dimension that we know; but we are not in it, we are against it, we are next to it, in the fourth dimension.

That is, as the plane being conceives himself as having a very small thickness in the third dimension, of which he is not aware in his sense experience, so we must suppose ourselves as having a very small thickness in the fourth dimension, and, being thus four-dimensional beings, to be prevented from realising that we are such beings by a constraint which keeps us always in contact with a vast solid sheet, which stretches on in every direction. We are against that sheet, so that, if we had the power of four-dimensional movement, we should either go away from it or through it; all our space movements as we know them being such that, performing them, we keep in contact with this solid sheet.

Now consider the exposition a plane being would makefor himself as to the question of the enclosure of a square, and of a cube.

He would say the squareA, in Fig. 96, is completely enclosed by the four squares,Afar,Anear,Aabove,Abelow, or as they are writtenAn,Af,Aa,Ab.

Fig. 96.

If now he conceives the squareAto move in the, to him, unknown dimension it will trace out a cube, and the bounding squares will form cubes. Will these completely surround the cube generated byA? No; there will be two faces of the cube made byAleft uncovered; the first, that face which coincides with the squareAin its first position; the next, that which coincides with the squareAin its final position. Against these two faces cubes must be placed in order to completely enclose the cubeA. These may be called the cubes left and right orAlandAr. Thus each of the enclosing squares of the squareAbecomes a cube and two more cubes are wanted to enclose the cube formed by the movement ofAin the third dimension.

Fig. 97.

The plane being could not see the squareAwith the squaresAn,Af, etc., placed about it, because they completely hide it from view; and so we, in the analogous case in our three-dimensional world, cannot see a cubeAsurrounded by six other cubes. These cubes we will callAnearAn,AfarAf,AaboveAa,AbelowAb,AleftAl,ArightAr, shown infig. 97. If now the cubeAmoves in the fourth dimension right out of space, it traces out a higher cube—a tesseract, as it may be called.Each of the six surrounding cubes carried on in the same motion will make a tesseract also, and these will be grouped around the tesseract formed byA. But will they enclose it completely?

All the cubesAn,Af, etc., lie in our space. But there is nothing between the cubeAand that solid sheet in contact with which every particle of matter is. When the cubeAmoves in the fourth direction it starts from its position, sayAk, and ends in a final positionAn(using the words “ana” and “kata” for up and down in the fourth dimension). Now the movement in this fourth dimension is not bounded by any of the cubesAn,Af, nor by what they form when thus moved. The tesseract whichAbecomes is bounded in the positive and negative ways in this new direction by the first position ofAand the last position ofA. Or, if we ask how many tesseracts lie around the tesseract whichAforms, there are eight, of which one meets it by the cubeA, and another meets it by a cube likeAat the end of its motion.

We come here to a very curious thing. The whole solid cubeAis to be looked on merely as a boundary of the tesseract.

Yet this is exactly analogous to what the plane being would come to in his study of the solid world. The squareA(fig. 96), which the plane being looks on as a solid existence in his plane world, is merely the boundary of the cube which he supposes generated by its motion.

The fact is that we have to recognise that, if there is another dimension of space, our present idea of a solid body, as one which has three dimensions only, does not correspond to anything real, but is the abstract idea of a three-dimensional boundary limiting a four-dimensional solid, which a four-dimensional being would form. The plane being’s thought of a square is not the thought of what we should call a possibly existing real square,but the thought of an abstract boundary, the face of a cube.

Let us now take our eight coloured cubes, which form a cube in space, and ask what additions we must make to them to represent the simplest collection of four-dimensional bodies—namely, a group of them of the same extent in every direction. In plane space we have four squares. In solid space we have eight cubes. So we should expect in four-dimensional space to have sixteen four-dimensional bodies-bodies which in four-dimensional space correspond to cubes in three-dimensional space, and these bodies we call tesseracts.

Fig. 98.

Given then the null, white, red, yellow cubes, and those which make up the block, we notice that we represent perfectly well the extension in three directions (fig. 98). From the null point of the null cube, travelling one inch, we come to the white cube; travelling one inch away we come to the yellow cube; travelling one inch up we come to the red cube. Now, if there is a fourth dimension, then travelling from the same null point for one inch in that direction, we must come to the body lying beyond the null region.

I say null region, not cube; for with the introduction of the fourth dimension each of our cubes must become something different from cubes. If they are to have existence in the fourth dimension, they must be “filled up from” in this fourth dimension.

Now we will assume that as we get a transference from null to white going in one way, from null to yellow going in another, so going from null in the fourth direction we have a transference from null to blue, using thus thecolours white, yellow, red, blue, to denote transferences in each of the four directions—right, away, up, unknown or fourth dimension.

Fig. 99.A plane being’s representation of a block of eight cubes by two sets of four squares.

Hence, as the plane being must represent the solid regions, he would come to by going right, as four squares lying in some position in his plane, arbitrarily chosen, side by side with his original four squares, so we must represent those eight four-dimensional regions, which we should come to by going in the fourth dimension from each of our eight cubes, by eight cubes placed in some arbitrary position relative to our first eight cubes.

Fig. 100.

Our representation of a block of sixteen tesseracts by two blocks of eight cubes.[3]

[3]The eight cubes used here in 2 can be found in the second of the model blocks. They can be taken out and used.

Hence, of the two sets of eight cubes, each one will serveus as a representation of one of the sixteen tesseracts which form one single block in four-dimensional space. Each cube, as we have it, is a tray, as it were, against which the real four-dimensional figure rests—just as each of the squares which the plane being has is a tray, so to speak, against which the cube it represents could rest.

If we suppose the cubes to be one inch each way, then the original eight cubes will give eight tesseracts of the same colours, or the cubes, extending each one inch in the fourth dimension.

But after these there come, going on in the fourth dimension, eight other bodies, eight other tesseracts. These must be there, if we suppose the four-dimensional body we make up to have two divisions, one inch each in each of four directions.

The colour we choose to designate the transference to this second region in the fourth dimension is blue. Thus, starting from the null cube and going in the fourth dimension, we first go through one inch of the null tesseract, then we come to a blue cube, which is the beginning of a blue tesseract. This blue tesseract stretches one inch farther on in the fourth dimension.

Thus, beyond each of the eight tesseracts, which are of the same colour as the cubes which are their bases, lie eight tesseracts whose colours are derived from the colours of the first eight by adding blue. Thus—

The addition of blue to yellow gives green—this is anatural supposition to make. It is also natural to suppose that blue added to red makes purple. Orange and blue can be made to give a brown, by using certain shades and proportions. And ochre and blue can be made to give a light brown.

But the scheme of colours is merely used for getting a definite and realisable set of names and distinctions visible to the eye. Their naturalness is apparent to any one in the habit of using colours, and may be assumed to be justifiable, as the sole purpose is to devise a set of names which are easy to remember, and which will give us a set of colours by which diagrams may be made easy of comprehension. No scientific classification of colours has been attempted.

Starting, then, with these sixteen colour names, we have a catalogue of the sixteen tesseracts, which form a four-dimensional block analogous to the cubic block. But the cube which we can put in space and look at is not one of the constituent tesseracts; it is merely the beginning, the solid face, the side, the aspect, of a tesseract.

We will now proceed to derive a name for each region, point, edge, plane face, solid and a face of the tesseract.

The system will be clear, if we look at a representation in the plane of a tesseract with three, and one with four divisions in its side.

The tesseract made up of three tesseracts each way corresponds to the cube made up of three cubes each way, and will give us a complete nomenclature.

In this diagram,fig. 101, 1 represents a cube of 27 cubes, each of which is the beginning of a tesseract. These cubes are represented simply by their lowest squares, the solid content must be understood. 2 represents the 27 cubes which are the beginnings of the 27 tesseracts one inch on in the fourth dimension. These tesseracts are represented as a block of cubes put side by side withthe first block, but in their proper positions they could not be in space with the first set. 3 represents 27 cubes (forming a larger cube) which are the beginnings of the tesseracts, which begin two inches in the fourth direction from our space and continue another inch.

Fig. 101.123Each cube is the beginning of the first tesseract going in the fourth dimension.Each cube is the beginning of the second tesseract.Each cube is the beginning of the third tesseract.

Fig. 102.[4]1234A cube of 64 cubes each 1. in × 1 in., the beginning of a tesseract.A cube of 64 cubes, each 1 in. × 1 in. × 1 in. the beginning of tesseracts 1 in. from our space in the 4th dimension.A cube of 64 cubes, each 1 in. × 1 in. × 1 in. the beginning of tesseracts 2 in. from our space in the 4th dimension.A cube of 64 cubes, each 1 in. × 1 in. × 1 in. the beginning of tesseracts 3 in. from our space in the 4th dimension.

Fig. 102.[4]

[4]The coloured plate, figs. 1, 2, 3, shows these relations more conspicuously.

Infig. 102, we have the representation of a block of 4 × 4 × 4 × 4 or 256 tesseracts. They are given in four consecutive sections, each supposed to be taken one inch apart in the fourth dimension, and so giving fourblocks of cubes, 64 in each block. Here we see, comparing it with the figure of 81 tesseracts, that the number of the different regions show a different tendency of increase. By taking five blocks of five divisions each way this would become even more clear.

We see,fig. 102, that starting from the point at any corner, the white coloured regions only extend out in a line. The same is true for the yellow, red, and blue. With regard to the latter it should be noticed that the line of blues does not consist in regions next to each other in the drawing, but in portions which come in in different cubes. The portions which lie next to one another in the fourth dimension must always be represented so, when we have a three-dimensional representation. Again, those regions such as the pink one, go on increasing in two dimensions. About the pink region this is seen without going out of the cube itself, the pink regions increase in length and height, but in no other dimension. In examining these regions it is sufficient to take one as a sample.

The purple increases in the same manner, for it comes in in a succession from below to above in block 2, and in a succession from block to block in 2 and 3. Now, a succession from below to above represents a continuous extension upwards, and a succession from block to block represents a continuous extension in the fourth dimension. Thus the purple regions increase in two dimensions, the upward and the fourth, so when we take a very great many divisions, and let each become very small, the purple region forms a two-dimensional extension.

In the same way, looking at the regions marked l. b. or light blue, which starts nearest a corner, we see that the tesseracts occupying it increase in length from left to right, forming a line, and that there are as many lines of light blue tesseracts as there are sections between thefirst and last section. Hence the light blue tesseracts increase in number in two ways—in the right and left, and in the fourth dimension. They ultimately form what we may call a plane surface.

Now all those regions which contain a mixture of two simple colours, white, yellow, red, blue, increase in two ways. On the other hand, those which contain a mixture of three colours increase in three ways. Take, for instance, the ochre region; this has three colours, white, yellow, red; and in the cube itself it increases in three ways.

Now regard the orange region; if we add blue to this we get a brown. The region of the brown tesseracts extends in two ways on the left of the second block, No. 2 in the figure. It extends also from left to right in succession from one section to another, from section 2 to section 3 in our figure.

Hence the brown tesseracts increase in number in three dimensions upwards, to and fro, fourth dimension. Hence they form a cubic, a three-dimensional region; this region extends up and down, near and far, and in the fourth direction, but is thin in the direction from left to right. It is a cube which, when the complete tesseract is represented in our space, appears as a series of faces on the successive cubic sections of the tesseract. Compare fig. 103 in which the middle block, 2, stands as representing a great number of sections intermediate between 1 and 3.

In a similar way from the pink region by addition of blue we have the light purple region, which can be seen to increase in three ways as the number of divisions becomes greater. The three ways in which this region of tesseracts extends is up and down, right and left, fourth dimension. Finally, therefore, it forms a cubic mass of very small tesseracts, and when the tesseract is given in space sections it appears on the faces containing the upward and the right and left dimensions.

We get then altogether, as three-dimensional regions, ochre, brown, light purple, light green.

Finally, there is the region which corresponds to a mixture of all the colours; there is only one region such as this. It is the one that springs from ochre by the addition of blue—this colour we call light brown.

Looking at the light brown region we see that it increases in four ways. Hence, the tesseracts of which it is composed increase in number in each of four dimensions, and the shape they form does not remain thin in any of the four dimensions. Consequently this region becomes the solid content of the block of tesseracts, itself; it is the real four-dimensional solid. All the other regions are then boundaries of this light brown region. If we suppose the process of increasing the number of tesseracts and diminishing their size carried on indefinitely, then the light brown coloured tesseracts become the whole interior mass, the three-coloured tesseracts become three-dimensional boundaries, thin in one dimension, and form the ochre, the brown, the light purple, the light green. The two-coloured tesseracts become two-dimensional boundaries, thin in two dimensions,e.g., the pink, the green, the purple, the orange, the light blue, the light yellow. The one-coloured tesseracts become bounding lines, thin in three dimensions, and the null points become bounding corners, thin in four dimensions. From these thin real boundaries we can pass in thought to the abstractions—points, lines, faces, solids—bounding the four-dimensional solid, which in this case is light brown coloured, and under this supposition the light brown coloured region is the only real one, is the only one which is not an abstraction.

It should be observed that, in taking a square as the representation of a cube on a plane, we only represent one face, or the section between two faces. The squares,as drawn by a plane being, are not the cubes themselves, but represent the faces or the sections of a cube. Thus in the plane being’s diagram a cube of twenty-seven cubes “null” represents a cube, but is really, in the normal position, the orange square of a null cube, and may be called null, orange square.

A plane being would save himself confusion if he named his representative squares, not by using the names of the cubes simply, but by adding to the names of the cubes a word to show what part of a cube his representative square was.

Thus a cube null standing against his plane touches it by null orange face, passing through his plane it has in the plane a square as trace, which is null white section, if we use the phrase white section to mean a section drawn perpendicular to the white line. In the same way the cubes which we take as representative of the tesseract are not the tesseract itself, but definite faces or sections of it. In the preceding figures we should say then, not null, but “null tesseract ochre cube,” because the cube we actually have is the one determined by the three axes, white, red, yellow.

There is another way in which we can regard the colour nomenclature of the boundaries of a tesseract.

Consider a null point to move tracing out a white line one inch in length, and terminating in a null point, seefig. 103or in the coloured plate.

Then consider this white line with its terminal points itself to move in a second dimension, each of the points traces out a line, the line itself traces out an area, and gives two lines as well, its initial and its final position.

Thus, if we call “a region” any element of the figure, such as a point, or a line, etc., every “region” in moving traces out a new kind of region, “a higher region,” and gives two regions of its own kind, an initial and a finalposition. The “higher region” means a region with another dimension in it.

Now the square can move and generate a cube. The square light yellow moves and traces out the mass of the cube. Letting the addition of red denote the region made by the motion in the upward direction we get an ochre solid. The light yellow face in its initial and terminal positions give the two square boundaries of the cube above and below. Then each of the four lines of the light yellow square—white, yellow, and the white, yellow opposite them—trace out a bounding square. So there are in all six bounding squares, four of these squares being designated in colour by adding red to the colour of the generating lines. Finally, each point moving in the up direction gives rise to a line coloured null + red, or red, and then there are the initial and terminal positions of the points giving eight points. The number of the lines is evidently twelve, for the four lines of this light yellow square give four lines in their initial, four lines in their final position, while the four points trace out four lines, that is altogether twelve lines.

Now the squares are each of them separate boundaries of the cube, while the lines belong, each of them, to two squares, thus the red line is that which is common to the orange and pink squares.

Now suppose that there is a direction, the fourth dimension, which is perpendicular alike to every one of the space dimensions already used—a dimension perpendicular, for instance, to up and to right hand, so that the pink square moving in this direction traces out a cube.

A dimension, moreover, perpendicular to the up and away directions, so that the orange square moving in this direction also traces out a cube, and the light yellow square, too, moving in this direction traces out a cube.Under this supposition, the whole cube moving in the unknown dimension, traces out something new—a new kind of volume, a higher volume. This higher volume is a four-dimensional volume, and we designate it in colour by adding blue to the colour of that which by moving generates it.

It is generated by the motion of the ochre solid, and hence it is of the colour we call light brown (white, yellow, red, blue, mixed together). It is represented by a number of sections like 2 infig. 103.

Now this light brown higher solid has for boundaries: first, the ochre cube in its initial position, second, the same cube in its final position, 1 and 3,fig. 103. Each of the squares which bound the cube, moreover, by movement in this new direction traces out a cube, so we have from the front pink faces of the cube, third, a pink blue or light purple cube, shown as a light purple face on cube 2 infig. 103, this cube standing for any number of intermediate sections; fourth, a similar cube from the opposite pink face; fifth, a cube traced out by the orange face—this is coloured brown and is represented by the brown face of the section cube infig. 103; sixth, a corresponding brown cube on the right hand; seventh, a cube starting from the light yellow square below; the unknown dimension is at right angles to this also. This cube is coloured light yellow and blue or light green; and, finally, eighth, a corresponding cube from the upper light yellow face, shown as the light green square at the top of the section cube.

The tesseract has thus eight cubic boundaries. These completely enclose it, so that it would be invisible to a four-dimensional being. Now, as to the other boundaries, just as the cube has squares, lines, points, as boundaries, so the tesseract has cubes, squares, lines, points, as boundaries.

The number of squares is found thus—round the cube are six squares, these will give six squares in their initial and six in their final positions. Then each of the twelve lines of the cube trace out a square in the motion in the fourth dimension. Hence there will be altogether 12 + 12 = 24 squares.

If we look at any one of these squares we see that it is the meeting surface of two of the cubic sides. Thus, the red line by its movement in the fourth dimension, traces out a purple square—this is common to two cubes, one of which is traced out by the pink square moving in the fourth dimension, and the other is traced out by the orange square moving in the same way. To take another square, the light yellow one, this is common to the ochre cube and the light green cube. The ochre cube comes from the light yellow square by moving it in the up direction, the light green cube is made from the light yellow square by moving it in the fourth dimension. The number of lines is thirty-two, for the twelve lines of the cube give twelve lines of the tesseract in their initial position, and twelve in their final position, making twenty-four, while each of the eight points traces out a line, thus forming thirty-two lines altogether.

The lines are each of them common to three cubes, or to three square faces; take, for instance, the red line. This is common to the orange face, the pink face, and that face which is formed by moving the red line in the sixth dimension, namely, the purple face. It is also common to the ochre cube, the pale purple cube, and the brown cube.

The points are common to six square faces and to four cubes; thus, the null point from which we start is common to the three square faces—pink, light yellow, orange, and to the three square faces made by moving the three lineswhite, yellow, red, in the fourth dimension, namely, the light blue, the light green, the purple faces—that is, to six faces in all. The four cubes which meet in it are the ochre cube, the light purple cube, the brown cube, and the light green cube.

Fig. 103.

The tesseract, red, white, yellow axes in space. In the lower line the three rear faces are shown, the interior being removed.]

Fig. 104.The tesseract, red, yellow, blue axes in space, the blue axis running to the left, opposite faces are coloured identically.

A complete view of the tesseract in its various space presentations is given in the following figures or catalogue cubes, figs. 103-106. The first cube in each figurerepresents the view of a tesseract coloured as described as it begins to pass transverse to our space. The intermediate figure represents a sectional view when it is partly through, and the final figure represents the far end as it is just passing out. These figures will be explained in detail in the next chapter.

Fig. 105.The tesseract, with red, white, blue axes in space. Opposite faces are coloured identically.

Fig. 106.The tesseract, with blue, white, yellow axes in space. The blue axis runs downward from the base of the ochre cube as it stands originally. Opposite faces are coloured identically.

We have thus obtained a nomenclature for each of the regions of a tesseract; we can speak of any one of the eight bounding cubes, the twenty square faces, the thirty-two lines, the sixteen points.

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