FOUR-SPACE. GENESIS OF A TESSARACT. ITS REPRESENTATION IN THREE-SPACE.
Hitherto we have only looked at Model 1. This, with the next seven,
represent what we can observe of the simplest body in Higher Space. A
few words will explain their construction. A point by its motion traces
a line. A line by its motion traces either a longer line or an area; if
it moves at right angles to its own direction, it traces a rectangle.
For the sake of simplicity, we will suppose all movements to be an inch
in length and at right angles to each other. Thus, a point moving traces
a line an inch long; a line moving traces a square inch; a square moving
traces a cubic inch. In these cases each of these movements produces
something intrinsically different from what we had before. A square is
not a longer line, nor a cube a larger square. When the cube moves, we
are unable to see any new direction in which it can move, and are
compelled to make it move in a direction which has previously been used.
Let us suppose there is an unknown direction at right angles to all our
known directions, just as a third direction would be unknown to a being
confined to the surface of the table. And let the cube move in this
unknown direction for an inch. We call the figure it traces a Tessaract.
The models are representations of the appearances a Tessaract would
present to us if shown in various ways. Consider for a moment what
happens to a square when moved to form a cube. Each of its lines, moved
in the new direction, traces a square; the square itself traces a new
figure, a cube, which ends in another square. Now, our cube, moved in a
new direction, will trace a tessaract, whereof the cube itself is the
beginning, and another cube the end. These two cubes are to the
tessaract as the Black square and White square are to the cube. A
plane-being could not see both those squares at once, but he could make
models of them and let them both rest in his plane at once. So also we
can make models of the beginning and end of the tessaract. Model 1 is
the cube, which is its beginning; Model 2 is the cube which is its end.
It will be noticed that there are no two colours alike in the two
models. The Silver point corresponds to the Gold point, that is, the
Silver point is the termination of the line traced by the Gold point
moving in the new direction. The sides correspond in the following
manner:--
SIDES.
_Model 1._ _Model 2._
Black corresponds to Bright-green
White „ „ Light-grey
Vermilion „ „ Indian-red
Blue-green „ „ Yellow-ochre
Dark-blue „ „ Burnt-sienna
Light-yellow „ „ Dun
The two cubes should be looked at and compared long enough to ensure
that the corresponding sides can be found quickly. Then there are the
following correspondencies in points and lines.
POINTS.
_Model 1._ _Model 2._
Gold corresponds to Silver
Fawn „ „ Turquoise
Terra-cotta „ „ Earthen
Buff „ „ Blue tint
Light-blue „ „ Quaker-green
Dull-purple „ „ Peacock-blue
Deep-blue „ „ Orange-vermilion
Red „ „ Purple
LINES.
_Model 1._ _Model 2._
Orange corresponds to Leaf-green
Crimson „ „ Dull-green
Green-grey „ „ Dark-purple
Blue „ „ Purple-brown
Brown „ „ Dull-blue
French-grey „ „ Dark-pink
Dark-slate „ „ Pale-pink
Green „ „ Indigo
Reddish „ „ Brown-green
Bright-blue „ „ Dark-green
Leaden „ „ Pale-yellow
Deep-yellow „ „ Dark
The colour of the cube itself is invisible, as it is covered by its
boundaries. We suppose it to be Sage-green.
These two cubes are just as disconnected when looked at by us as the
black and white squares would be to a plane-being if placed side by side
on his plane. He cannot see the squares in their right position with
regard to each other, nor can we see the cubes in theirs.
Let us now consider the vermilion side of Model 1. If it move in the X
direction, it traces the cube of Model 1. Its Gold point travels along
the Orange line, and itself, after tracing the cube, ends in the
Blue-green square. But if it moves in the new direction, it will also
trace a cube, for the new direction is at right angles to the up and
away directions, in which the Brown and Blue lines run. Let this square,
then, move in the unknown direction, and trace a cube. This cube we
cannot see, because the unknown direction runs out of our space at once,
just as the up direction runs out of the plane of the table. But a
plane-being could see the square, which the Blue line traces when moved
upwards, by the cube being turned round the Blue line, the Orange line
going upwards; then the Brown line comes into the plane of the table in
the -X direction. So also with our cube. As treated above, it runs from
the Vermilion square out of our space. But if the tessaract were turned
so that the line which runs from the Gold point in the unknown direction
lay in our space, and the Orange line lay in the unknown direction, we
could then see the cube which is formed by the movement of the Vermilion
square in the new direction.
Take Model 5. There is on it a Vermilion square. Place this so that it
touches the Vermilion square on Model 1. All the marks of the two
squares are identical. This Cube 5, is the one traced by the Vermilion
square moving in the unknown direction. In Model 5, the whole figure,
the tessaract, produced by the movement of the cube in the unknown
direction, is supposed to be so turned that the Orange line passes into
the unknown direction, and that the line which went in the unknown
direction, runs opposite to the old direction of the Orange line.
Looking at this new cube, we see that there is a Stone line running to
the left from the Gold point. This line is that which the Gold point
traces when moving in the unknown direction.
It is obvious that, if the Tessaract turns so as to show us the side, of
which Cube 5 is a model, then Cube 1 will no longer be visible. The
Orange line will run in the unknown or fourth direction, and be out of
our sight, together with the whole cube which the Vermilion square
generates, when the Gold point moves along the Orange line. Hence, if we
consider these models as real portions of the tessaract, we must not
have more than one before us at once. When we look at one, the others
must necessarily be beyond our sight and touch. But we may consider them
simply as models, and, as such, we may let them lie alongside of each
other. In this case, we must remember that their real relationships are
not those in which we see them.
We now enumerate the sides of the new Cube 5, so that, when we refer to
it, any colour may be recognised by name.
The square Vermilion traces a Pale-green cube, and ends in an Indian-red
square.
(The colour Pale-green of this cube is not seen, as it is entirely
surrounded by squares and lines of colour.)
Each Line traces a Square and ends in a Line.
The Blue line} {Light-brown square} and{Purple-brown line
„ Brown „ }traces{Yellow „ }ends{Dull-blue „
„ Deep-yellow „ } a {Light-red „ } in {Dark „
„ Green „ } {Deep-crimson „ } a {Indigo „.
Each Point traces a Line and ends in a Point.
The Gold point} {Stone line} and{Silver point
„ Buff „ }traces{Light-green „ }ends{Blue-tint „
„ Light-blue „ } a {Rich-red „ } in {Quaker-green „
„ Red „ } {Emerald „ } a {Purple „.
It will be noticed that besides the Vermilion square of this cube
another square of it has been seen before. A moment’s comparison with
the experience of a plane-being will make this more clear. If a
plane-being has before him models of the Black and White squares of the
Cube, he sees that all the colours of the one are different from all the
colours of the other. Next, if he looks at a model of the Vermilion
square, he sees that it starts from the Blue line and ends in a line of
the White square, the Deep-yellow line. In this square he has two lines
which he had before, the Blue line with Gold and Buff points, the
Deep-yellow line with Light-blue and Red points. To him the Black and
White squares are his Models 1 and 2, and the Vermilion square is to him
as our Model 5 is to us. The left-hand square of Model 5 is Indian-red,
and is identical with that of the same colour on the left-hand side of
Model 2. In fact, Model 5 shows us what lies between the Vermilion face
of 1, and the Indian-red face of 2.
From the Gold point we suppose four perfectly independent lines to
spring forth, each of them at right angles to all the others. In our
space there is only room for three lines mutually at right angles. It
will be found, if we try to introduce a fourth at right angles to each
of three, that we fail; hence, of these four lines one must go out of
the space we know. The colours of these four lines are Brown, Orange,
Blue, Stone. In Model 1 are shown the Brown, Orange, and Blue. In Model
5 are shown the Brown, Blue, and Stone. These lines might have had any
directions at first, but we chose to begin with the Brown line going up,
or Z, the Orange going X, the Blue going Y, and the Stone line going in
the unknown direction, which we will call W.
Consider for a moment the Stone and the Orange lines. They can be seen
together on Model 7 by looking at the lower face of it. They are at
right angles to each other, and if the Orange line be turned to take the
place of the Stone line, the latter will run into the negative part of
the direction previously occupied by the former. This is the reason that
the Models 3, 5, and 7 are made with the Stone line always running in
the reverse direction of that line of Model 1, which is wanting in each
respectively. It will now be easy to find out Models 3 and 7. All that
has to be done is, to discover what faces they have in common with 1 and
2, and these faces will show from which planes of 1 they are generated
by motion in the unknown direction.
Take Model 7. On one side of it there is a Dark-blue square, which is
identical with the Dark-blue square of Model 1. Placing it so that it
coincides with 1 by this square line for line, we see that the square
nearest to us is Burnt-sienna, the same as the near square on Model 2.
Hence this cube is a model of what the Dark-blue square traces on moving
in the unknown direction. Here the unknown direction coincides with the
negative away direction. In fact, to see this cube, we have been obliged
to suppose the Blue line turned into the unknown direction, for we
cannot look at more than three of these rectangular lines at once in our
space, and in this Model 7 we have the Brown, Orange, and Stone lines.
The faces, lines, and points of Cube 7 can be identified by the
following list.
The Dark-blue square traces a Dark-stone cube (whose interior is
rendered invisible by the bounding squares), and ends in a Burnt-sienna
square.
Each Line traces a Square and ends in a Line.
The Orange line} {Azure square} and{Leaf-green line
„ Brown „ }traces{Yellow „ }ends{Dull-blue „
„ French-grey „ } an {Yellow-green „ } in {Dark-pink „
„ Reddish „ } {Ochre „ } a {Brown-green „.
Each Point traces a Line and ends in a Point.
The Gold point } {Stone line } and{Silver point
„ Fawn „ }traces{Smoke „ }ends{Turquoise „
„ Light-blue „ } a {Rich-red „ } in {Quaker-green „
„ Dull-purple „ } {Green-blue „ } a {Peacock-blue „.
If we now take Model 3, we see that it has a Black square uppermost, and
has Blue and Orange lines. Hence, it evidently proceeds from the Black
square in Model 1; and it has in it Blue and Orange lines, which proceed
from the Gold point. But besides these, it has running downwards a Stone
line. The line wanting is the Brown line, and, as in the other cases,
when one of the three lines of Model 1 turns out into the unknown
direction, the Stone line turns into the direction opposite to that from
which the line has turned. Take this Model 3 and place it underneath
Model 1, raising the latter so that the Black squares on the two
coincide line for line. Then we see what would come into our view if the
Brown line were to turn into the unknown direction, and the Stone line
come into our space downwards. Looking at this cube, we see that the
following parts of the tessaract have been generated.
The Black square traces a Brick-red cube (invisible because covered by
its own sides and edges), and ends in a Bright-green square.
Each Line traces a Square and ends in a Line.
The Orange line} {Azure square } and{Leaf-green line
„ Crimson „ }traces{Rose „ }ends{Dull-green „
„ Green-grey „ } an {Sea-blue „ } in {Dark-purple „
„ Blue „ } {Light-brown „ } a {Purple-brown „.
Each Point traces a Line and ends in a Point.
The Gold point} {Stone line} and{Silver point
„ Fawn „ }traces{Smoke „ }ends{Turquoise „
„ Terra-cotta „ } a {Magenta „ } in {Earthen „
„ Buff „ } {Light-green „ } a {Blue-tint „.
This completes the enumeration of the regions of Cube 3. It may seem a
little unnatural that it should come in downwards; but it must be
remembered that the new fourth direction has no more relation to
up-and-down than to right-and-left or to near-and-far.
And if, instead of thinking of a plane-being as living on the surface of
a table, we suppose his world to be the surface of the sheet of paper
touching the Dark-blue square of Cube 1, then we see that a turn round
the Orange line, which makes the Brown line go into the plane-being’s
unknown direction, brings the Blue line into his downwards direction.
There still remain to be described Models 4, 6, and 8. It will be shown
that Model 4 is to Model 3 what Model 2 is to Model 1. That is, if, when
3 is in our space, it be moved so as to trace a tessaract, 4 will be
the opposite cube in which the tessaract ends. There is no colour common
to 3 and 4. Similarly, 6 is the opposite boundary of the tessaract
generated by 5, and 8 of that by 7.
A little closer consideration will reveal several points. Looking at
Cube 5, we see proceeding from the Gold point a Brown, a Blue, and a
Stone line. The Orange line is wanting; therefore, it goes in the
unknown direction. If we want to discover what exists in the unknown
direction from Cube 5, we can get help from Cube 1. For, since the
Orange line lies in the unknown direction from Cube 5, the Gold point
will, if moved along the Orange line, pass in the unknown direction. So
also, the Vermilion square, if moved along in the direction of the
Orange line, will proceed in the unknown direction. Looking at Cube 1 we
see that the Vermilion square thus moved ends in a Blue-green square.
Then, looking at Model 6, on it, corresponding to the Vermilion square
on Cube 5, is a Blue-green square.
Cube 6 thus shows what exists an inch beyond 5 in the unknown direction.
Between the right-hand face on 5 and the right-hand face on 6 lies a
cube, the one which is shown in Model 1. Model 1 is traced by the
Vermilion square moving an inch along the direction of the Orange line.
In Model 5, the Orange line goes in the unknown direction; and looking
at Model 6 we see what we should get at the end of a movement of one
inch in that direction. We have still to enumerate the colours of Cubes
4, 6, and 8, and we do so in the following list. In the first column is
designated the part of the cube; in the columns under 4, 6, 8, come the
colours which 4, 6, 8, respectively have in the parts designated in the
corresponding line in the first column.
Cube itself:--
4 6 8
Chocolate Oak-yellow Salmon
Squares:--
Lower face Light-grey Rose Sea-blue
Upper White Deep-brown Deep-green
Left-hand Light-red Yellow-ochre Deep-crimson
Right-hand Deep-brown Blue-green Dark-grey
Near Ochre Yellow-green Dun
Far Deep-green Dark-grey Light-yellow
Lines:--
On ground, going round the square from left to right:--
4 6 8
1. Brown-green Smoke Dark-purple
2. Dark-green Crimson Magenta
3. Pale-yellow Magenta Green-grey
4. Dark Dull-green Light-green
Vertical, going round the sides from left to right:--
1. Rich-red Dark-pink Indigo
2. Green-blue French-grey Pale-pink
3. Sea-green Dark-slate Dark-slate
4. Emerald Pale-pink Green
Round upper face in same order:--
1. Reddish Green-blue Pale-yellow
2. Bright-blue Bright-blue Sea-green
3. Leaden Sea-green Leaden
4. Deep-yellow Dark-green Emerald
Points:--
On lower face, going from left to right:--
1. Quaker-green Turquoise Blue-tint
2. Peacock-blue Fawn Earthen
3. Orange-vermilion Terra-cotta Terra-cotta
4. Purple Earthen Buff
On upper face:--
1. Light-blue Peacock-blue Purple
2. Dull-purple Dull-purple Orange-vermilion
3. Deep-blue Deep-blue Deep-blue
4. Red Orange-vermilion Red
If any one of these cubes be taken at random, it is easy enough to find
out to what part of the Tessaract it belongs. In all of them, except 2,
there will be one face, which is a copy of a face on 1; this face is, in
fact, identical with the face on 1 which it resembles. And the model
shows what lies in the unknown direction from that face. This unknown
direction is turned into our space, so that we can see and touch the
result of moving a square in it. And we have sacrificed one of the three
original directions in order to do this. It will be found that the line,
which in 1 goes in the 4th direction, in the other models always runs in
a negative direction.
Let us take Model 8, for instance. Searching it for a face we know, we
come to a Light-yellow face away from us. We place this face parallel
with the Light-yellow face on Cube 1, and we see that it has a Green
line going up, and a Green-grey line going to the right from the Buff
point. In these respects it is identical with the Light-yellow face on
Cube 1. But instead of a Blue line coming towards us from the Buff
point, there is a Light-green line. This Light-green line, then, is that
which proceeds in the unknown direction from the Buff point. The line is
turned towards us in this Model 8 in the negative Y direction; and
looking at the model, we see exactly what is formed when in the motion
of the whole cube in the unknown direction, the Light-yellow face is
moved an inch in that direction. It traces out a Salmon cube (_v._ Table
on p. 127), and it has Sea-blue and Deep-green sides below and above,
and Deep-crimson and Dark-grey sides left and right, and Dun and
Light-yellow sides near and far. If we want to verify the correctness of
any of these details, we must turn to Models 1 and 2. What lies an inch
from the Light-yellow square in the unknown direction? Model 2 tells
us, a Dun square. Now, looking at 8, we see that towards us lies a Dun
square. This is what lies an inch in the unknown direction from the
Light-yellow square. It is here turned to face us, and we can see what
lies between it and the Light-yellow square.