APPENDIX ITHE MODELS
In Chapter XI. a description has been given which will enable any one to make a set of models illustrative of the tesseract and its properties. The set here supposed to be employed consists of:—
1. Three sets of twenty-seven cubes each.2. Twenty-seven slabs.3. Twelve cubes with points, lines, faces, distinguished by colours, which will be called the catalogue cubes.
1. Three sets of twenty-seven cubes each.
2. Twenty-seven slabs.
3. Twelve cubes with points, lines, faces, distinguished by colours, which will be called the catalogue cubes.
The preparation of the twelve catalogue cubes involves the expenditure of a considerable amount of time. It is advantageous to use them, but they can be replaced by the drawing of the views of the tesseract or by a reference to figs.103,104,105,106of the text.
The slabs are coloured like the twenty-seven cubes of the first cubic block infig. 101, the one with red, white, yellow axes.
The colours of the three sets of twenty-seven cubes are those of the cubes shown infig. 101.
The slabs are used to form the representation of a cube in a plane, and can well be dispensed with by any one who is accustomed to deal with solid figures. But the whole theory depends on a careful observation of how the cube would be represented by these slabs.
In the first step, that of forming a clear idea how aplane being would represent three-dimensional space, only one of the catalogue cubes and one of the three blocks is needed.
Look atfig. 1of the views of the tesseract, or, what comes to the same thing, take catalogue cube No. 1 and place it before you with the red line running up, the white line running to the right, the yellow line running away. The three dimensions of space are then marked out by these lines or axes. Now take a piece of cardboard, or a book, and place it so that it forms a wall extending up and down not opposite to you, but running away parallel to the wall of the room on your left hand.
Placing the catalogue cube against this wall we see that it comes into contact with it by the red and yellow lines, and by the included orange face.
In the plane being’s world the aspect he has of the cube would be a square surrounded by red and yellow lines with grey points.
Now, keeping the red line fixed, turn the cube about it so that the yellow line goes out to the right, and the white line comes into contact with the plane.
In this case a different aspect is presented to the plane being, a square, namely, surrounded by red and white lines and grey points. You should particularly notice that when the yellow line goes out, at right angles to the plane, and the white comes in, the latter does not run in the same sense that the yellow did.
From the fixed grey point at the base of the red line the yellow line ran away from you. The white line now runs towards you. This turning at right angles makes the line which was out of the plane before, come into itin an opposite sense to that in which the line ran which has just left the plane. If the cube does not break through the plane this is always the rule.
Again turn the cube back to the normal position with red running up, white to the right, and yellow away, and try another turning.
You can keep the yellow line fixed, and turn the cube about it. In this case the red line going out to the right the white line will come in pointing downwards.
You will be obliged to elevate the cube from the table in order to carry out this turning. It is always necessary when a vertical axis goes out of a space to imagine a movable support which will allow the line which ran out before to come in below.
Having looked at the three ways of turning the cube so as to present different faces to the plane, examine what would be the appearance if a square hole were cut in the piece of cardboard, and the cube were to pass through it. A hole can be actually cut, and it will be seen that in the normal position, with red axis running up, yellow away, and white to the right, the square first perceived by the plane being—the one contained by red and yellow lines—would be replaced by another square of which the line towards you is pink—the section line of the pink face. The line above is light yellow, below is light yellow and on the opposite side away from you is pink.
In the same way the cube can be pushed through a square opening in the plane from any of the positions which you have already turned it into. In each case the plane being will perceive a different set of contour lines.
Having observed these facts about the catalogue cube, turn now to the first block of twenty-seven cubes.
You notice that the colour scheme on the catalogue cube and that of this set of blocks is the same.
Place them before you, a grey or null cube on the table, above it a red cube, and on the top a null cube again. Then away from you place a yellow cube, and beyond it a null cube. Then to the right place a white cube and beyond it another null. Then complete the block, according to the scheme of the catalogue cube, putting in the centre of all an ochre cube.
You have now a cube like that which is described in the text. For the sake of simplicity, in some cases, this cubic block can be reduced to one of eight cubes, by leaving out the terminations in each direction. Thus, instead of null, red, null, three cubes, you can take null, red, two cubes, and so on.
It is useful, however, to practise the representation in a plane of a block of twenty-seven cubes. For this purpose take the slabs, and build them up against the piece of cardboard, or the book in such a way as to represent the different aspects of the cube.
Proceed as follows:—
First, cube in normal position.
Place nine slabs against the cardboard to represent the nine cubes in the wall of the red and yellow axes, facing the cardboard; these represent the aspect of the cube as it touches the plane.
Now push these along the cardboard and make a different set of nine slabs to represent the appearance which the cube would present to a plane being, if it were to pass half way through the plane.
There would be a white slab, above it a pink one, above that another white one, and six others, representing what would be the nature of a section across the middle of the block of cubes. The section can be thought of as a thin slice cut out by two parallel cuts across the cube. Having arranged these nine slabs, push them along the plane, and make another set of nine to represent whatwould be the appearance of the cube when it had almost completely gone through. This set of nine will be the same as the first set of nine.
Now we have in the plane three sets of nine slabs each, which represent three sections of the twenty-seven block.
They are put alongside one another. We see that it does not matter in what order the sets of nine are put. As the cube passes through the plane they represent appearances which follow the one after the other. If they were what they represented, they could not exist in the same plane together.
This is a rather important point, namely, to notice that they should not co-exist on the plane, and that the order in which they are placed is indifferent. When we represent a four-dimensional body our solid cubes are to us in the same position that the slabs are to the plane being. You should also notice that each of these slabs represents only the very thinnest slice of a cube. The set of nine slabs first set up represents the side surface of the block. It is, as it were, a kind of tray—a beginning from which the solid cube goes off. The slabs as we use them have thickness, but this thickness is a necessity of construction. They are to be thought of as merely of the thickness of a line.
If now the block of cubes passed through the plane at the rate of an inch a minute the appearance to a plane being would be represented by:—
1. The first set of nine slabs lasting for one minute.
2. The second set of nine slabs lasting for one minute.
3. The third set of nine slabs lasting for one minute.
Now the appearances which the cube would present to the plane being in other positions can be shown by means of these slabs. The use of such slabs would be the means by which a plane being could acquire afamiliarity with our cube. Turn the catalogue cube (or imagine the coloured figure turned) so that the red line runs up, the yellow line out to the right, and the white line towards you. Then turn the block of cubes to occupy a similar position.
The block has now a different wall in contact with the plane. Its appearance to a plane being will not be the same as before. He has, however, enough slabs to represent this new set of appearances. But he must remodel his former arrangement of them.
He must take a null, a red, and a null slab from the first of his sets of slabs, then a white, a pink, and a white from the second, and then a null, a red, and a null from the third set of slabs.
He takes the first column from the first set, the first column from the second set, and the first column from the third set.
To represent the half-way-through appearance, which is as if a very thin slice were cut out half way through the block, he must take the second column of each of his sets of slabs, and to represent the final appearance, the third column of each set.
Now turn the catalogue cube back to the normal position, and also the block of cubes.
There is another turning—a turning about the yellow line, in which the white axis comes below the support.
You cannot break through the surface of the table, so you must imagine the old support to be raised. Then the top of the block of cubes in its new position is at the level at which the base of it was before.
Now representing the appearance on the plane, we must draw a horizontal line to represent the old base. The line should be drawn three inches high on the cardboard.
Below this the representative slabs can be arranged.
It is easy to see what they are. The old arrangementshave to be broken up, and the layers taken in order, the first layer of each for the representation of the aspect of the block as it touches the plane.
Then the second layers will represent the appearance half way through, and the third layers will represent the final appearance.
It is evident that the slabs individually do not represent the same portion of the cube in these different presentations.
In the first case each slab represents a section or a face perpendicular to the white axis, in the second case a face or a section which runs perpendicularly to the yellow axis, and in the third case a section or a face perpendicular to the red axis.
But by means of these nine slabs the plane being can represent the whole of the cubic block. He can touch and handle each portion of the cubic block, there is no part of it which he cannot observe. Taking it bit by bit, two axes at a time, he can examine the whole of it.
Look at the views of the tesseract 1, 2, 3, or take the catalogue cubes 1, 2, 3, and place them in front of you, in any order, say running from left to right, placing 1 in the normal position, the red axis running up, the white to the right, and yellow away.
Now notice that in catalogue cube 2 the colours of each region are derived from those of the corresponding region of cube 1 by the addition of blue. Thus null + blue = blue, and the corners of number 2 are blue. Again, red + blue = purple, and the vertical lines of 2 are purple. Blue + yellow = green, and the line which runs away is coloured green.
By means of these observations you may be sure thatcatalogue cube 2 is rightly placed. Catalogue cube 3 is just like number 1.
Having these cubes in what we may call their normal position, proceed to build up the three sets of blocks.
This is easily done in accordance with the colour scheme on the catalogue cubes.
The first block we already know. Build up the second block, beginning with a blue corner cube, placing a purple on it, and so on.
Having these three blocks we have the means of representing the appearances of a group of eighty-one tesseracts.
Let us consider a moment what the analogy in the case of the plane being is.
He has his three sets of nine slabs each. We have our three sets of twenty-seven cubes each.
Our cubes are like his slabs. As his slabs are not the things which they represent to him, so our cubes are not the things they represent to us.
The plane being’s slabs are to him the faces of cubes.
Our cubes then are the faces of tesseracts, the cubes by which they are in contact with our space.
As each set of slabs in the case of the plane being might be considered as a sort of tray from which the solid contents of the cubes came out, so our three blocks of cubes may be considered as three-space trays, each of which is the beginning of an inch of the solid contents of the four-dimensional solids starting from them.
We want now to use the names null, red, white, etc., for tesseracts. The cubes we use are only tesseract faces. Let us denote that fact by calling the cube of null colour, null face; or, shortly, null f., meaning that it is the face of a tesseract.
To determine which face it is let us look at the catalogue cube 1 or the first of the views of the tesseract, whichcan be used instead of the models. It has three axes, red, white, yellow, in our space. Hence the cube determined by these axes is the face of the tesseract which we now have before us. It is the ochre face. It is enough, however, simply to say null f., red f. for the cubes which we use.
To impress this in your mind, imagine that tesseracts do actually run from each cube. Then, when you move the cubes about, you move the tesseracts about with them. You move the face but the tesseract follows with it, as the cube follows when its face is shifted in a plane.
The cube null in the normal position is the cube which has in it the red, yellow, white axes. It is the face having these, but wanting the blue. In this way you can define which face it is you are handling. I will write an “f.” after the name of each tesseract just as the plane being might call each of his slabs null slab, yellow slab, etc., to denote that they were representations.
We have then in the first block of twenty-seven cubes, the following—null f., red f., null f., going up; white f., null f., lying to the right, and so on. Starting from the null point and travelling up one inch we are in the null region, the same for the away and the right-hand directions. And if we were to travel in the fourth dimension for an inch we should still be in a null region. The tesseract stretches equally all four ways. Hence the appearance we have in this first block would do equally well if the tesseract block were to move across our space for a certain distance. For anything less than an inch of their transverse motion we should still have the same appearance. You must notice, however, that we should not have null face after the motion had begun.
When the tesseract, null for instance, had moved ever so little we should not have a face of null but a section of null in our space. Hence, when we think of the motionacross our space we must call our cubes tesseract sections. Thus on null passing across we should see first null f., then null s., and then, finally, null f. again.
Imagine now the whole first block of twenty-seven tesseracts to have moved tranverse to our space a distance of one inch. Then the second set of tesseracts, which originally were an inch distant from our space, would be ready to come in.
Their colours are shown in the second block of twenty-seven cubes which you have before you. These represent the tesseract faces of the set of tesseracts that lay before an inch away from our space. They are ready now to come in, and we can observe their colours. In the place which null f. occupied before we have blue f., in place of red f. we have purple f., and so on. Each tesseract is coloured like the one whose place it takes in this motion with the addition of blue.
Now if the tesseract block goes on moving at the rate of an inch a minute, this next set of tesseracts will occupy a minute in passing across. We shall see, to take the null one for instance, first of all null face, then null section, then null face again.
At the end of the second minute the second set of tesseracts has gone through, and the third set comes in. This, as you see, is coloured just like the first. Altogether, these three sets extend three inches in the fourth dimension, making the tesseract block of equal magnitude in all dimensions.
We have now before us a complete catalogue of all the tesseracts in our group. We have seen them all, and we shall refer to this arrangement of the blocks as the “normal position.” We have seen as much of each tesseract at a time as could be done in a three-dimensional space. Each part of each tesseract has been in our space, and we could have touched it.
The fourth dimension appeared to us as the duration of the block.
If a bit of our matter were to be subjected to the same motion it would be instantly removed out of our space. Being thin in the fourth dimension it is at once taken out of our space by a motion in the fourth dimension.
But the tesseract block we represent having length in the fourth dimension remains steadily before our eyes for three minutes, when it is subjected to this transverse motion.
We have now to form representations of the other views of the same tesseract group which are possible in our space.
Let us then turn the block of tesseracts so that another face of it comes into contact with our space, and then by observing what we have, and what changes come when the block traverses our space, we shall have another view of it. The dimension which appeared as duration before will become extension in one of our known dimensions, and a dimension which coincided with one of our space dimensions will appear as duration.
Leaving catalogue cube 1 in the normal position, remove the other two, or suppose them removed. We have in space the red, the yellow, and the white axes. Let the white axis go out into the unknown, and occupy the position the blue axis holds. Then the blue axis, which runs in that direction now will come into space. But it will not come in pointing in the same way that the white axis does now. It will point in the opposite sense. It will come in running to the left instead of running to the right as the white axis does now.
When this turning takes place every part of the cube 1 will disappear except the left-hand face—the orange face.
And the new cube that appears in our space will run to the left from this orange face, having axes, red, yellow, blue.
Take models 4, 5, 6. Place 4, or suppose No. 4 of the tesseract views placed, with its orange face coincident with the orange face of 1, red line to red line, and yellow line to yellow line, with the blue line pointing to the left. Then remove cube 1 and we have the tesseract face which comes in when the white axis runs in the positive unknown, and the blue axis comes into our space.
Now place catalogue cube 5 in some position, it does not matter which, say to the left; and place it so that there is a correspondence of colour corresponding to the colour of the line that runs out of space. The line that runs out of space is white, hence, every part of this cube 5 should differ from the corresponding part of 4 by an alteration in the direction of white.
Thus we have white points in 5 corresponding to the null points in 4. We have a pink line corresponding to a red line, a light yellow line corresponding to a yellow line, an ochre face corresponding to an orange face. This cube section is completely named in Chapter XI. Finally cube 6 is a replica of 1.
These catalogue cubes will enable us to set up our models of the block of tesseracts.
First of all for the set of tesseracts, which beginning in our space reach out one inch in the unknown, we have the pattern of catalogue cube 4.
We see that we can build up a block of twenty-seven tesseract faces after the colour scheme of cube 4, by taking the left-hand wall of block 1, then the left-hand wall of block 2, and finally that of block 3. We take, that is, the three first walls of our previous arrangement to form the first cubic block of this new one.
This will represent the cubic faces by which the group of tesseracts in its new position touches our space. We have running up, null f., red f., null f. In the next vertical line, on the side remote from us, we have yellow f.,orange f., yellow f., and then the first colours over again. Then the three following columns are, blue f., purple f., blue f.; green f., brown f., green f.; blue f., purple f., blue f. The last three columns are like the first.
These tesseracts touch our space, and none of them are by any part of them distant more than an inch from it. What lies beyond them in the unknown?
This can be told by looking at catalogue cube 5. According to its scheme of colour we see that the second wall of each of our old arrangements must be taken. Putting them together we have, as the corner, white f. above it, pink f. above it, white f. The column next to this remote from us is as follows:—light yellow f., ochre f., light yellow f., and beyond this a column like the first. Then for the middle of the block, light blue f., above it light purple, then light blue. The centre column has, at the bottom, light green f., light brown f. in the centre and at the top light green f. The last wall is like the first.
The third block is made by taking the third walls of our previous arrangement, which we called the normal one.
You may ask what faces and what sections our cubes represent. To answer this question look at what axes you have in our space. You have red, yellow, blue. Now these determine brown. The colours red, yellow, blue are supposed by us when mixed to produce a brown colour. And that cube which is determined by the red, yellow, blue axes we call the brown cube.
When the tesseract block in its new position begins to move across our space each tesseract in it gives a section in our space. This section is transverse to the white axis, which now runs in the unknown.
As the tesseract in its present position passes across our space, we should see first of all the first of the blocksof cubic faces we have put up—these would last for a minute, then would come the second block and then the third. At first we should have a cube of tesseract faces, each of which would be brown. Directly the movement began, we should have tesseract sections transverse to the white line.
There are two more analogous positions in which the block of tesseracts can be placed. To find the third position, restore the blocks to the normal arrangement.
Let us make the yellow axis go out into the positive unknown, and let the blue axis, consequently, come in running towards us. The yellow ran away, so the blue will come in running towards us.
Put catalogue cube 1 in its normal position. Take catalogue cube 7 and place it so that its pink face coincides with the pink face of cube 1, making also its red axis coincide with the red axis of 1 and its white with the white. Moreover, make cube 7 come towards us from cube 1. Looking at it we see in our space, red, white, and blue axes. The yellow runs out. Place catalogue cube 8 in the neighbourhood of 7—observe that every region in 8 has a change in the direction of yellow from the corresponding region in 7. This is because it represents what you come to now in going in the unknown, when the yellow axis runs out of our space. Finally catalogue cube 9, which is like number 7, shows the colours of the third set of tesseracts. Now evidently, starting from the normal position, to make up our three blocks of tesseract faces we have to take the near wall from the first block, the near wall from the second, and then the near wall from the third block. This gives us the cubic block formed by the faces of the twenty-seven tesseracts which are now immediately touching our space.
Following the colour scheme of catalogue cube 8,we make the next set of twenty-seven tesseract faces, representing the tesseracts, each of which begins one inch off from our space, by putting the second walls of our previous arrangement together, and the representation of the third set of tesseracts is the cubic block formed of the remaining three walls.
Since we have red, white, blue axes in our space to begin with, the cubes we see at first are light purple tesseract faces, and after the transverse motion begins we have cubic sections transverse to the yellow line.
Restore the blocks to the normal position, there remains the case in which the red axis turns out of space. In this case the blue axis will come in downwards, opposite to the sense in which the red axis ran.
In this case take catalogue cubes 10, 11, 12. Lift up catalogue cube 1 and put 10 underneath it, imagining that it goes down from the previous position of 1.
We have to keep in space the white and the yellow axes, and let the red go out, the blue come in.
Now, you will find on cube 10 a light yellow face; this should coincide with the base of 1, and the white and yellow lines on the two cubes should coincide. Then the blue axis running down you have the catalogue cube correctly placed, and it forms a guide for putting up the first representative block.
Catalogue cube 11 will represent what lies in the fourth dimension—now the red line runs in the fourth dimension. Thus the change from 10 to 11 should be towards red, corresponding to a null point is a red point, to a white line is a pink line, to a yellow line an orange line, and so on.
Catalogue cube 12 is like 10. Hence we see that to build up our blocks of tesseract faces we must take the bottom layer of the first block, hold that up in the air, underneath it place the bottom layer of the second block,and finally underneath this last the bottom layer of the last of our normal blocks.
Similarly we make the second representative group by taking the middle courses of our three blocks. The last is made by taking the three topmost layers. The three axes in our space before the transverse motion begins are blue, white, yellow, so we have light green tesseract faces, and after the motion begins sections transverse to the red light.
These three blocks represent the appearances as the tesseract group in its new position passes across our space. The cubes of contact in this case are those determinal by the three axes in our space, namely, the white, the yellow, the blue. Hence they are light green.
It follows from this that light green is the interior cube of the first block of representative cubic faces.
Practice in the manipulations described, with a realization in each case of the face or section which is in our space, is one of the best means of a thorough comprehension of the subject.
We have to learn how to get any part of these four-dimensional figures into space, so that we can look at them. We must first learn to swing a tesseract, and a group of tesseracts about in any way.
When these operations have been repeated and the method of arrangement of the set of blocks has become familiar, it is a good plan to rotate the axes of the normal cube 1 about a diagonal, and then repeat the whole series of turnings.
Thus, in the normal position, red goes up, white to the right, yellow away. Make white go up, yellow to the right, and red away. Learn the cube in this position by putting up the set of blocks of the normal cube, over and over again till it becomes as familiar to you as in the normal position. Then when this is learned, and the corresponding changes in the arrangements of the tesseract groups are made, another change should be made: let, in the normal cube, yellow go up, red to the right, and white away.
Learn the normal block of cubes in this new position by arranging them and re-arranging them till you know without thought where each one goes. Then carry out all the tesseract arrangements and turnings.
If you want to understand the subject, but do not see your way clearly, if it does not seem natural and easy to you, practise these turnings. Practise, first of all, the turning of a block of cubes round, so that you know it in every position as well as in the normal one. Practise by gradually putting up the set of cubes in their new arrangements. Then put up the tesseract blocks in their arrangements. This will give you a working conception of higher space, you will gain the feeling of it, whether you take up the mathematical treatment of it or not.