Application to the Step from Plane to Solid.
Look at [fig. 1] of 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 it in 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 what would 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 a familiarity 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 arrangements have 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.