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.
Thus the models can be used to answer any question about sections. For
we have simply to take, instead of the whole cube, a plane, and the
relation of the whole tessaract to that plane can be told by looking at
the model, which, starting with that plane, stretches from it in the
unknown direction.
We have not as yet settled the colour of the interior of Model 9. It is
that part of the tessaract which is traced out by the interior of Cube
1. The unknown direction starts equally and simultaneously from every
point of every part of Cube 1, just as the up direction starts equally
and simultaneously from every point of a square. Let us suppose that the
cube, which is Light-buff, changes to a Wood-colour directly it begins
to trace the tessaract. Then the internal part of the section between 1
and 2 will be a Wood-colour. The sides of the Model 9 are of the
greatest importance. They are the colour of the six cubes, 3, 4, 5, 6,
7, and 8. The colours of 1 and 2 are wanting, viz. Light-buff and
Sage-green. Thus the section between 1 and 2 can be found by its wanting
the colours of the Cubes 1 and 2.
Looking at Models 10, 11, and 12 in a similar manner, the reader will
find they represent the sections between Cubes 3 and 4, Cubes 5 and 6,
and Cubes 7 and 8 respectively.