FOUR-SPACE: ITS REPRESENTATION IN THREE-SPACE.
We now come to the essential difficulty of our task. All that has gone
before is preliminary. We have now to frame the method by which we shall
introduce through our space-figures the figures of a higher space. When
a plane-being studies our shapes of cubes, he has to use squares. He is
limited at the outset. A cube appears to him as a square. On Model 1 we
see the various squares as which the cube can appear to him. We suppose
the plane-being to look from the extremity of the Z axis down a vertical
plane. First, there is the Moena square. Then there is the square given
by a section parallel to Moena, which he recognises by the variation of
the bounding lines as soon as the cube begins to pass through his plane.
Then comes the Murex square. Next, if the cube be turned round the Z
axis and passed through, he sees the Alvus and Proes squares and the
intermediate section. So too with the Syce and Mel squares and the
section between them.
Now, dealing with figures in higher space, we are in an analogous
position. We cannot grasp the element of which they are composed. We can
conceive a cube; but that which corresponds to a cube in higher space is
beyond our grasp. But the plane-being was obliged to use two-dimensional
figures, squares, in arriving at a notion of a three-dimensional figure;
so also must we use three-dimensional figures to arrive at the notion
of a four-dimensional. Let us call the figure which corresponds to a
square in a plane and a cube in our space, a tessaract. Model 1 is a
cube. Let us assume a tessaract generated from it. Let us call the
tessaract Urna. The generating cube may then be aptly called Urna Mala.
We may use cubes to represent parts of four-space, but we must always
remember that they are to us, in our study, only what squares are to a
plane-being with respect to a cube.
Let us again examine the mode in which a plane-being represents a Block
of cubes with slabs. Take Block 1 of the 81 Set of cubes. The
plane-being represents this by nine slabs, which represent the Moena
face of the block. Then, omitting the solidity of these first nine
cubes, he takes another set of nine slabs to represent the next wall of
cubes. Lastly, he represents the third wall by a third set, omitting the
solidity of both second and third walls. In this manner, he evidently
represents the extension of the Block upwards and sideways, in the Z and
X directions; but in the Y direction he is powerless, and is compelled
to represent extension in that direction by setting the three sets of
slabs alongside in his plane. The second and third sets denote the
height and breadth of the respective walls, but not their depth or
thickness. Now, note that the Block extends three inches in each of the
three directions. The plane-being can represent two of these dimensions
on his plane; but the unknown direction he has to represent by a
repetition of his plane figures. The cube extends three inches in the Y
direction. He has to use 3 sets of slabs.
The Block is built up arbitrarily in this manner: Starting from Urna
Mala and going up, we come to a Brown cube, and then to a Light-blue
cube. Starting from Urna Mala and going right, we come to an Orange and
a Fawn cube. Starting from Urna Mala and going away from us, we come to
a Blue and a Buff cube. Now, the plane-being represents the Brown and
Orange cubes by squares lying next to the square which represents Urna
Mala. The Blue cube is as close as the Brown cube to Urna Mala, but he
can find no place in the plane where he can place a Blue square so as to
show this co-equal proximity of both cubes to the first. So he is forced
to put a Blue square anywhere in his plane and say of it: “This Blue
square represents what I should arrive at, if I started from Urna Mala
and moved away, that is in the Y or unknown direction.” Now, just as
there are three cubes going up, so there are three going away. Hence,
besides the Blue square placed anywhere on the plane, he must also place
a Buff square beyond it, to show that the Block extends as far away as
it does upwards and sideways. (Each cube being a different colour, there
will be as many different colours of squares as of cubes.) It will
easily be seen that not only the Gold square, but also the Orange and
every other square in the first set of slabs must have two other squares
set somewhere beyond it on the plane to represent the extension of the
Block away, or in the unknown Y direction.
Coming now to the representation of a four-dimensional block, we see
that we can show only three dimensions by cubic blocks, and that the
fourth can only be represented by repetitions of such blocks. There must
be a certain amount of arbitrary naming and colouring. The colours have
been chosen as now stated. Take the first Block of the 81 Set. We are
familiar with its colours, and they can be found at any time by
reference to Model 1. Now, suppose the Gold cube to represent what we
can see in our space of a Gold tessaract; the other cubes of Block 1
give the colours of the tessaracts which lie in the three directions X,
Y, and Z from the Gold one. But what is the colour of the tessaract
which lies next to the Gold in the unknown direction, W? Let us suppose
it to be Stone in colour. Taking out Block 2 of the 81 Set and arranging
it on the pattern of Model 9, we find in it a Stone cube. But, just as
there are three tessaracts in the X, Y, and Z directions, as shown by
the cubes in Block 1, so also must there be three tessaracts in the
unknown direction, W. Take Block 3 of the 81 Set. This Block can be
arranged on the pattern of Model 2. In it there is a Silver cube where
the Gold cube lies in Block 1. Hence, we may say, the tessaract which
comes next to the Stone one in the unknown direction from the Gold, is
of a Silver colour. Now, a cube in all these cases represents a
tessaract. Between the Gold and Stone cubes there is an inch in the
unknown direction. The Gold tessaract is supposed to be Gold throughout
in all four directions, and so also is the Stone. We may imagine it in
this way. Suppose the set of three tessaracts, the Gold, the Stone, and
the Silver to move through our space at the rate of an inch a minute. We
should first see the Gold cube which would last a minute, then the Stone
cube for a minute, and lastly the Silver cube a minute. (This is
precisely analogous to the appearance of passing cubes to the
plane-being as successive squares lasting a minute.) After that, nothing
would be visible.
Now, just as we must suppose that there are three tessaracts proceeding
from the Gold cube in the unknown direction, so there must be three
tessaracts extending in the unknown direction from every one of the 27
cubes of the first Block. The Block of 27 cubes is not a Block of 27
tessaracts, but it represents as much of them as we can see at once in
our space; and they form the first portion or layer (like the first
wall of cubes to the plane-being) of a set of eighty-one tessaracts,
extending to equal distances in all four directions. Thus, to represent
the whole Block of tessaracts there are 81 cubes, or three Blocks of 27
each.
Now, it is obvious that, just as a cube has various plane boundaries, so
a tessaract has various cube boundaries. The cubes of the tessaract,
which we have been regarding, have been those containing the X, Y, and Z
directions, just as the plane-being regarded the Moena faces containing
the X and Z directions. And, as long as the tessaract is unchanged in
its position with regard to our space, we can never see any portion of
it which is in the unknown direction. Similarly, we saw that a
plane-being could not see the parts of a cube which went in the third
direction, until the cube was turned round one of its edges. In order to
make it quite clear what parts of a cube came into the plane, we gave
distinct names to them. Thus, the squares containing the Z and X
directions were called Moena and Murex; those containing the Z and Y,
Alvus and Proes; and those the X and Y, Syce and Mel. Now, similarly
with our four axes, any three will determine a cube. Let the tessaract
in its normal position have the cube Mala determined by the axes Z, X,
Y. Let the cube Lar be that which is determined by X, Y, W, that is, the
cube which, starting from the X Y plane, stretches one inch in the
unknown or W direction. Let Vesper be the cube determined by Z, Y, W,
and Pluvium by Z, X, W. And let these cubes have opposite cubes of the
following names:
Mala has Margo
Lar „ Velum
Vesper „ Idus
Pluvium „ Tela
Another way of looking at the matter is this. When a cube is generated
from a square, each of the lines bounding the square becomes a square,
and the square itself becomes a cube, giving two squares in its initial
and final positions. When a cube moves in the new and unknown direction,
each of its planes traces a cube and it generates a tessaract, giving in
its initial and final positions two cubes. Thus there are eight cubes
bounding the tessaract, six of them from the six plane sides and two
from the cube itself. These latter two are Mala and Margo. The cubes
from the six sides are: Lar from Syce, Velum from Mel, Vesper from
Alvus, Idus from Proes, Pluvium from Moena, Tela from Murex. And just as
a plane-being can only see the squares of a cube, so we can only see the
cubes of a tessaract. It may be said that the cube can be pushed partly
through the plane, so that the plane-being sees a section between Moena
and Murex. Similarly, the tessaract can be pushed through our space so
that we can see a section between Mala and Margo.
There is a method of approaching the matter, which settles all
difficulties, and provides us with a nomenclature for every part of the
tessaract. We have seen how by writing down the names of the cubes of a
block, and then supposing that their number increases, certain sets of
the names come to denote points, lines, planes, and solid. Similarly, if
we write down a set of names of tessaracts in a block, it will be found
that, when their number is increased, certain sets of the names come to
denote the various parts of a tessaract.
For this purpose, let us take the 81 Set, and use the cubes to represent
tessaracts. The whole of the 81 cubes make one single tessaractic set
extending three inches in each of the four directions. The names must be
remembered to denote tessaracts. Thus, Corvus is a tessaract which has
the tessaracts Cuspis and Nugæ to the right, Arctos and Ilex above it,
Dos and Cista away from it, and Ops and Spira in the fourth or unknown
direction from it. It will be evident at once, that to write these names
in any representative order we must adopt an arbitrary system. We
require them running in four directions; we have only two on paper. The
X direction (from left to right) and the Y (from the bottom towards the
top of the page) will be assumed to be truly represented. The Z
direction will be symbolized by writing the names in floors, the upper
floors always preceding the lower. Lastly, the fourth, or W, direction
(which has to be symbolized in three-dimensional space by setting the
solids in an arbitrary position) will be signified by writing the names
in blocks, the name which stands in any one place in any block being
next in the W direction to that which occupies the same position in the
block before or after it. Thus, Ops is written in the same place in the
Second Block, Spira in the Third Block, as Corvus occupies in the First
Block.
Since there are an equal number of tessaracts in each of the four
directions, there will be three floors Z when there are three X and Y.
Also, there will be three Blocks W. If there be four tessaracts in each
direction, there will be four floors Z, and four blocks W. Thus, when
the number in each direction is enlarged, the number of blocks W is
equal to the number of tessaracts in each known direction.
On pp. 136, 137 were given the names as used for a cubic block of 27 or
64. Using the same and more names for a tessaractic Set, and remembering
that each name now represents, not a cube, but a tessaract, we obtain
the following nomenclature, the order in which the names are written
being that stated above:
THIRD BLOCK.
Upper { Solia Livor Talus
Floor. { Lensa Lares Calor
{ Felis Tholus Passer
-----------------
Middle { Lixa Portica Vena
Floor. { Crux Margo Sal
{ Pagus Silex Onager
-----------------
Lower { Panax Mensura Mugil
Floor. { Opex Lappa Mappa
{ Spira Luca Ancilla
SECOND BLOCK.
Upper { Orsa Mango Libera
Floor. { Creta Velum Meatus
{ Lucta Limbus Pator
-----------------
Middle { Camoena Tela Orca
Floor. { Vesper Tessaract Idus
{ Pagina Pluvium Pactum
-----------------
Lower { Lis Lorica Offex
Floor. { Lua Lar Olla
{ Ops Lotus Limus
FIRST BLOCK.
Upper { Olus Semita Lama
Floor. { Via Mel Iter
{ Ilex Callis Sors
-----------------
Middle { Bucina Murex Daps
Floor. { Alvus Mala Proes
{ Arctos Moena Far
-----------------
Lower { Cista Cadus Crus
Floor. { Dos Syce Bolus
{ Corvus Cuspis Nugæ
It is evident that this set of tessaracts could be increased to the
number of four in each direction, the names being used as before for the
cubic blocks on pp. 136, 137, and in that case the Second Block would be
duplicated to make the four blocks required in the unknown direction.
Comparing such an 81 Set and 256 Set, we should find that Cuspis, which
was a single tessaract in the 81 Set became two tessaracts in the 256
Set. And, if we introduced a larger number, it would simply become
longer, and not increase in any other dimension. Thus, Cuspis would
become the name of an edge. Similarly, Dos would become the name of an
edge, and also Arctos. Ops, which is found in the Middle Block of the 81
Set, occurs both in the Second and Third Blocks of the 256 Set; that is,
it also tends to elongate and not extend in any other direction, and may
therefore be used as the name of an edge of a tessaract.
Looking at the cubes which represent the Syce tessaracts, we find that,
though they increase in number, they increase only in two directions;
therefore, Syce may be taken to signify a square. But, looking at what
comes from Syce in the W direction, we find in the Middle Block of the
81 Set one Lar, and in the Second and Third Blocks of the 256 Set four
Lars each. Hence, Lar extends in three directions, X, Y, W, and becomes
a cube. Similarly, Moena is a plane; but Pluvium, which proceeds from
it, extends not only sideways and upwards like Moena, but in the unknown
direction also. It occurs in both Middle Blocks of the 256 Set. Hence,
it also is a cube. We have now considered such parts of the Sets as
contain one, two, and three dimensions. But there is one part which
contains four. It is that named Tessaract. In the 256 Set there are
eight such cubes in the Second, and eight in the Third Block; that is,
they extend Z, X, Y, and also W. They may, therefore, be considered to
represent that part of a tessaract or tessaractic Set, which is
analogous to the interior of a cube.
The arrangement of colours corresponding to these names is seen on Model
1 corresponding to Mala, Model 2 to Margo, and Model 9 to the
intermediate block.
When we take the view of the tessaract with which we commenced, and in
which Arctos goes Z, Cuspis X, Dos Y, and Ops W, we see Mala in our
space. But when the tessaract is turned so that the Ops line goes -X,
while Cuspis is turned W, the other two remaining as they were, then we
do not see Mala, but that cube which, in the original position of the
tessaract, contains the Z, Y, W, directions, that is, the Vesper cube.
A plane-being may begin to study a block of cubes by their Syce squares;
but if the block be turned round Dos, he will have Alvus squares in his
space, and he must then use them to represent the cubic Block. So, when
the tessaractic Set is turned round, Mala cubes leave our space, and
Vespers enter.
There are two ways which can be followed in studying the Set of
tessaracts.
I. Each tessaract of one inch every way can be supposed to be of the
same colour throughout, so that, whichever way it be turned, whichever
of its edges coincide with our known axes, it appears to us as a cube of
one uniform colour. Thus, if Urna be the tessaract, Urna Mala would be a
Gold cube, Urna Vesper a Gold cube, and so on. This method is, for the
most part, adopted in the following pages. In this case, a whole Set of
4 × 4 × 4 × 4 tessaracts would in colours resemble a set composed of
four cubes like Models 1, 9, 9, and 2. But, when any question about a
particular tessaract has to be settled, it is advantageous, for the sake
of distinctness, to suppose it coloured in its different regions as the
whole set is coloured.
II. The other plan is, to start with the cubic sides of the inch
tessaract, each coloured according to the scheme of the Models 1 to 8.
In this case, the lines, if shown at all, should be very thin. For, in
fact, only the faces would be seen, as the width of the lines would only
be equal to the thickness of our matter in the fourth dimension, which
is indistinguishable to the senses. If such completely coloured cubes be
used, less error is likely to creep in; but it is a disadvantage that
each cube in the several blocks is exactly like the others in that
block. If the reader make such a set to work with for a time, he will
gain greatly, for the real way of acquiring a sense of higher space is
to obtain those experiences of the senses exactly, which the observation
of a four-dimensional body would give. These Models 1-8 are called sides
of the tessaract.
To make the matter perfectly clear, it is best to suppose that any
tessaract or set of tessaracts which we examine, has a duplicate exactly
similar in shape and arrangement of parts, but different in their
colouring. In the first, let each tessaract have one colour throughout,
so that all its cubes, apprehended in turn in our space, will be of one
and the same colour. In the duplicate, let each tessaract be so coloured
as to show its different cubic sides by their different colours. Then,
when we have it turned to us in different aspects, we shall see
different cubes, and when we try to trace the contacts of the tessaracts
with each other, we shall be helped by realizing each part of every
tessaract in its own colour.