CHAPTER IV.
TESSARACT MOVING THROUGH THREE-SPACE. MODELS OF THE SECTIONS.
In order to obtain a clear conception of the higher solid, a certain amount of familiarity with the facts shown in these models is necessary. But the best way of obtaining a systematic knowledge is shown hereafter. What these models enable us to do, is to take a general review of the subject. In all of them we see simply the boundaries of the tessaract in our space; we can no more see or touch the tessaract’s solidity than a plane-being can touch the cube’s solidity.
There remain the four models 9, 10, 11, 12. Model 9 represents what lies between 1 and 2. If 1 be moved an inch in the unknown direction, it traces out the tessaract and ends in 2. But, obviously, between 1 and 2 there must be an infinite number of exactly similar solid sections; these are all like Model 9.
Take the case of a plane-being on the table. He sees the Black square,—that is, he sees the lines round it,—and he knows that, if it moves an inch in some mysterious direction, it traces a new kind of figure, the opposite boundary whereof is the White square. If, then, he has models of the White and Black squares, he has before him the end and beginning of our cube. But between these squares are any number of others, the plane sections of the cube. We can see what they are. The interior of each is a Light-buff (the colour of the substance of the cube), the sides are of the colours of the vertical faces of the cube, and the points of the colours of the vertical lines of the cube, viz., Dark-blue, Blue-green, Light-yellow, Vermilion lines, and Brown, French-grey, Dark-slate, Green points. Thus, the square, in moving in the unknown direction, traces out a succession of squares, the assemblage of which makes the cube in layers. So also the cube, moving in the unknown direction, will at any point of its motion, still be a cube; and the assemblage of cubes thus placed constitutes the tessaract in layers. We suppose the cube to change its colour directly it begins to move. Its colour between 1 and 2 we can easily determine by finding what colours its different parts assume, as they move in the unknown direction. The Gold point immediately begins to trace a Stone-line. We will look at Cube 5 to see what the Vermilion face becomes; we know the interior of that cube is Pale-green (v. Table, [p. 122]). Hence, as it moves in the unknown direction, the Vermilion square forms in its course a series of Pale-green squares. The Brown line gives rise to a Yellow square; hence, at every point of its course in the fourth direction, it is a Yellow line, until, on taking its final position, it becomes a Dull-blue line. Looking at Cube 5, we see that the Deep yellow line becomes a Light-red line, the Green line a Deep Crimson one, the Gold point a Stone one, the Light-blue point a Rich-red one, the Red point an Emerald one, and the Buff point a Light-green one. Now, take the Model 9. Looking at the left side of it, we see exactly that into which the Vermilion square is transformed, as it moves in the unknown direction. The left side is an exact copy of a section of Cube 5, parallel to the Vermilion face.
But we have only accounted for one side of our Model 9. There are five other sides. Take the near side corresponding to the Dark-blue square on Cube 1. When the Dark-blue square moves, it traces a Dark-stone cube, of which we have a copy in Cube 7. Looking at 7 (v. Table, [p. 124]), we see that, as soon as the Dark-blue square begins to move, it becomes of a Dark-stone colour, and has Yellow, Ochre, Yellow-green, and Azure sides, and Stone, Rich-red, Green-blue, Smoke lines running in the unknown direction from it. Now, the side of Model 9, which faces us, has these colours the squares being seen as lines, and the lines as points. Hence Model 9 is a copy of what the cube becomes, so far as the Vermilion and Dark-blue sides are concerned, when, moving in the unknown direction, it traces the tessaract.
We will now look at the lower square of our model. It is a Brick-red square, with Azure, Rose, Sea-blue, and Light-brown lines, and with Stone, Smoke, Magenta, and Light-green points. This, then, is what the Black square should change into, as it moves in the unknown direction. Let us look at Model 3. Here the Stone line, which is the line in the unknown direction, runs downwards. It is turned into the downwards direction, so that the cube traced by the Black square may be in our space. The colour of this cube is Brick-red; the Orange line has traced an Azure, the Blue line a Light-brown, the Crimson line a Rose, and the Green-grey line a Sea-blue square. Hence, the lower square of Model 9 shows what the Black square becomes, as it traces the tessaract; or, in other words, the section of Model 3 between the Black and Bright-green squares exactly corresponds to the lower face of Model 9.
Therefore, it appears that Model 9 is a model of a section of the tessaract, that it is to the tessaract what a square between the Black and White squares is to the cube.
To prove the other sides correct, we have to see what the White, Blue-green, and Light-yellow squares of Cube 1 become, as the cube moves in the unknown direction. This can be effected by means of the Models 4, 6, 8. Each cube can be used as an index for showing the changes through which any side of the first model passes, as it moves in the unknown direction till it becomes Cube 2. Thus, what becomes of the White square? Look at Cube 4. From the Light-blue corner of its White square runs downwards the Rich-red line in the unknown direction. If we take a parallel section below the White square, we have a square bounded by Ochre, Deep-brown, Deep-green, and Light-red lines; and by Rich-red, Green-blue, Sea-green, and Emerald points. The colour of the cube is Chocolate, and therefore its section is Chocolate. This description is exactly true of the upper surface of Model 9.
There still remain two sides, those corresponding to the Light-yellow and Blue-green of Cube 1. What the Blue-green square becomes midway between Cubes 1 and 2 can be seen on Model 6. The colour of the last-named is Oak-yellow, and a section parallel to its Blue-green side is surrounded by Yellow-green, Deep-brown, Dark-grey and Rose lines and by Green-blue, Smoke, Magenta, and Sea-green points. This is exactly similar to the right side of Model 9. Lastly, that which becomes of the Light-yellow side can be seen on Model 8. The section of the cube is a Salmon square bounded by Deep-crimson, Deep-green, Dark-grey and Sea-blue lines and by Emerald, Sea-green, Magenta, and Light-green points.