REPRESENTATION OF FOUR-SPACE BY NAME. STUDY OF TESSARACTS.
We have now surveyed all the preliminary ground, and can study the
masses of tessaracts without obscurity.
We require a scaffold or framework for this purpose, which in three
dimensions will consist of eight cubic spaces or octants assembled round
one point, as in two dimensions it consisted of four squares or
quadrants round a point.
These eight octants lie between the three axes Z, X, Y, which intersect
at the given point, and can be named according to their positions
between the positive and negative directions of those axes. Thus the
octant Z, X, Y, is that which is contained by the positive portions of
all three axes; the octant Z, [=X], Y, that which is to the left of Z,
X, Y, and between the positive parts of Z and Y and the negative of X.
To illustrate this quite clearly, let us take the eight cubes--Urna,
Moles, Plebs, Frenum, Uncus, Pallor, Bidens, Ostrum--and place them in
the eight octants. Let them be placed round the point of intersection of
the axes; Pallor Corvus, Plebs Ilex, etc., will be at that point. Their
positions will then be:--
Urna in the Octant [=Z] [=X] [=Y]
Moles „ „ [=Z] X [=Y]
Plebs „ „ [=Z] X Y
Frenum „ „ [=Z] [=X] Y
Uncus „ „ Z [=X] Y
Pallor „ „ Z X Y
Bidens „ „ Z X [=Y]
Ostrum „ „ Z [=X] [=Y]
The names used for the cubes, as they are before us, are as follows:--
THIRD BLOCK.
Third { Arcus Mala Ovis Mala Portio Mala
Floor. { Laurus Mala Tigris Mala Segmen Mala
{ Axis Mala Troja Mala Aries Mala
Second { Postis Mala Clipeus Mala Tabula Mala
Floor. { Orcus Mala Lacerta Mala Testudo Mala
{ Verbum Mala Luctus Mala Anguis Mala
First { Telum Mala Nepos Mala Angusta Mala
Floor. { Polus Mala Penates Mala Vulcan Mala
{ Cervix Mala Securis Mala Vinculum Mala
SECOND BLOCK.
Third { Ara Mala Vomer Mala Pluma Mala
Floor. { Praeda Mala Sacerdos Mala Hydra Mala
{ Cortex Mala Mica Mala Flagellum Mala
Second { Pilum Mala Glans Mala Colus Mala
Floor. { Ocrea Mala Tessera Mala Domitor Mala
{ Cardo Mala Cudo Mala Malleus Mala
First { Agmen Mala Lacus Mala Arvus Mala
Floor. { Crates Mala Cura Mala Limen Mala
{ Thyrsus Mala Vitta Mala Sceptrum Mala
FIRST BLOCK.
Third { Mars Mala Merces Mala Tyro Mala
Floor. { Spicula Mala Mora Mala Oliva Mala
{ Comes Mala Tibicen Mala Vestis Mala
Second { Ala Mala Cortis Mala Aer Mala
Floor. { Uncus Mala Pallor Mala Tergum Mala
{ Ostrum Mala Bidens Mala Scena Mala
First { Sector Mala Hama Mala Remus Mala
Floor. { Frenum Mala Plebs Mala Sypho Mala
{ Urna Mala Moles Mala Saltus Mala
Their colours can be found by reference to the Models 1, 9, 2, which
correspond respectively to the First, Second, and Third Blocks. Thus,
Urna Mala is Gold; Moles, Orange; Saltus, Fawn; Thyrsus, Stone; Cervix,
Silver. The cubes whose colours are not shown in the Models, are Pallor
Mala, Tessera Mala, and Lacerta Mala, which are equivalent to the
interiors of the Model cubes, and are respectively Light-buff, Wooden,
and Sage-green. These 81 cubes are the cubic sides and sections of the
tessaracts of an 81 tessaractic Set, which measures three inches in
every direction. We suppose it to pass through our space. Let us call
the positive unknown direction Ana (_i.e._, +W) and the negative unknown
direction Kata (-W). Then, as the whole tessaract moves Kata at the rate
of an inch a minute, we see first the First Block of 27 cubes for one
minute, then the Second, and lastly the Third, each lasting one minute.
Now, when the First Block stands in the normal position, the edges of
the tessaract that run from the Corvus corner of Urna Mala, are: Arctos
in Z, Cuspis in X, Dos in Y, Ops in W. Hence, we denote this position by
the following symbol:--
Z X Y W
_a_ _c_ _d_ _o_
where _a_ stands for Arctos, _c_ for Cuspis, _d_ for Dos, and _o_ for
Ops, and the other letters for the four axes in space. _a_, _c_, _d_,
_o_ are the axes of the tessaract, and can take up different directions
in space with regard to us.
* * * * *
Let us now take a smaller four-dimensional set. Of the 81 Set let us
take the following:--
Z X Y W
_a_ _c_ _d_ _o_
SECOND BLOCK.
Second Floor. { Ocrea Mala Tessera Mala
{ Cardo Mala Cudo Mala
First Floor. { Crates Mala Cura Mala
{ Thyrsus Mala Vitta Mala
FIRST BLOCK.
Second Floor. { Uncus Mala Pallor Mala
{ Ostrum Mala Bidens Mala
First Floor. { Frenum Mala Plebs Mala
{ Urna Mala Moles Mala
Let the First Block be put up before us in Z X Y, (Urna Corvus is at the
junction of our axes Z X Y). The Second Block is now one inch distant in
the unknown direction; and, if we suppose the tessaractic Set to move
through our space at the rate of one inch a minute, the Second will
enter in one minute, and replace the first. But, instead of this, let us
suppose the tessaracts to turn so that Ops, which now goes W, shall go
-X. Then we can see in our space that cubic side of each tessaract which
is contained by the lines Arctos, Dos, and Ops, the cube Vesper; and we
shall no longer have the Mala sides but the Vesper sides of the
tessaractic Set in our space. We will now build it up in its Vesper view
(as we built up the cubic Block in its Alvus view). Take the Gold cube,
which now means Urna Vesper, and place it on the left hand of its former
position as Urna Mala, that is, in the octant Z [=X] Y. Thyrsus Vesper,
which previously lay just beyond Urna Vesper in the unknown direction,
will now lie just beyond it in the -X direction, that is, to the left of
it. The tessaractic Set is now in the position
Z X Y W
_a_ _ō_ _d_ _c_
(the minus sign over the _o_ meaning that Ops runs in the negative
direction), and its Vespers lie in the following order:--
SECOND BLOCK.
Second Floor. { Tessara Pallor
{ Cudo Bidens
First Floor. { Cura Plebs
{ Vitta Moles
FIRST BLOCK.
Second Floor. { Ocrea Uncus
{ Cardo Ostrum
First Floor. { Crates Frenum
{ Thyrsus Urna
The name Vesper is left out in the above list for the sake of brevity,
but should be used in studying the positions.
[Illustration: Fig. 20.]
On comparing the two lists of the Mala view and Vesper view, it will be
seen that the cubes presented in the Vesper view are new sides of the
tessaract, and that the arrangement of them is different from that in
the Mala view. (This is analogous to the changes in the slabs from the
Moena to Alvus view of the cubic Block.) Of course, the Vespers of all
these tessaracts are not visible at once in our space, any more than are
the Moenas of all three walls of a cubic Block to a plane-being. But if
the tessaractic Set be supposed to move through space in the unknown
direction at the rate of an inch a minute, the Second Block will present
its Vespers after the First Block has lasted a minute. The relative
position of the Mala Block and the Vesper Block may be represented in
our space as in the diagram, Fig. 20. But it must be distinctly
remembered that this arrangement is quite conventional, no more real
than a plane-being’s symbolization of the Moena Wall and the Alvus Wall
of the cubic Block by the arrangement of their Moena and Alvus faces,
with the solidity omitted, along one of his known directions.
The Vespers of the First and Second Blocks cannot be in our space
simultaneously, any more than the Moenas of all three walls in plane
space. To render their simultaneous presence possible, the cubic or
tessaractic Block or Set must be broken up, and its parts no longer
retain their relations. This fact is of supreme importance in
considering higher space. Endless fallacies creep in as soon as it is
forgotten that the cubes are merely representative as the slabs were,
and the positions in our space merely conventional and symbolical, like
those of the slabs along the plane. And these fallacies are so much
fostered by again symbolizing the cubic symbols and their symbolical
positions in perspective drawings or diagrams, that the reader should
surrender all hope of learning space from this book or the drawings
alone, and work every thought out with the cubes themselves.
If we want to see what each individual cube of the tessaractic faces
presented to us in the last example is like, we have only to consider
each of the Malas similar in its parts to Model 1, and each of the
Vespers to Model 5. And it must always be remembered that the cubes,
though used to represent both Mala and Vesper faces of the tessaract,
mean as great a difference as the slabs used for the Moena and Alvus
faces of the cube.
If the tessaractic Set move Kata through our space, when the Vesper
faces are presented to us, we see the following parts of the tessaract
Urna (and, therefore, also the same parts of the other tessaracts):
(1) Urna Vesper, which is Model 5.
(2) A parallel section between Urna Vesper and Urna Idus, which is Model
11.
(3) Urna Idus, which is Model 6.
When Urna Idus has passed Kata our space, Moles Vesper enters it; then a
section between Moles Vesper and Moles Idus, and then Moles Idus. Here
we have evidently observed the tessaract more minutely; as it passes
Kata through our space, starting on its Vesper side, we have seen the
parts which would be generated by Vesper moving along Cuspis--that is
Ana.
Two other arrangements of the tessaracts have to be learnt besides those
from the Mala and Vesper aspect. One of them is the Pluvium aspect.
Build up the Set in Z X [=Y], letting Arctos run Z, Cuspis X, and Ops
[=Y]. In the common plane Moena, Urna Pluvium coincides with Urna Mala,
though they cannot be in our space together; so too Moles Pluvium with
Moles Mala, Ostrum Pluvium with Ostrum Mala. And lying towards us, or
[=Y], is now that tessaract which before lay in the W direction from
Urna, viz., Thyrsus. The order will therefore be the following (a star
denotes the cube whose corner is at point of intersection of the axes,
and the name Pluvium must be understood to follow each of the names):
Z X Y W
_a_ _c_ _ō_ _d_
SECOND BLOCK.
Second Floor. { Uncus Pallor
{ Ocrea Tessera
First Floor. { Frenum Plebs
{ Crates Cura
FIRST BLOCK.
Second Floor. { Ostrum Bidens
{ Cardo Cudo
First Floor. {*Urna Moles
{ Thyrsus Vitta
Thus the wall of cubes in contact with that wall of the Mala position
which contains the Urna, Moles, Ostrum, and Bidens Malas, is a wall
composed of the Pluviums of Urna, Moles, Ostrum, and Bidens. The wall
next to this, and nearer to us, is of Thyrsus, Vitta, Cardo, Cudo,
Pluviums. The Second Block is one inch out of our Space, and only enters
it if the Block moves Kata. Model 7 shows the Pluvium cube; and each of
the cubes of the tessaracts seen in the Pluvium position is a Pluvium.
If the tessaractic Set moved Kata, we would see the Section between
Pluvium and Tela for all but a minute; and then Tela would enter our
space, and the Tela of each tessaract would be seen. Model 12 shows the
Section from Pluvium to Tela. Model 8 is Tela. Tela only lasts for a
flash, as it has only the minutest magnitude in the unknown or Ana
direction. Then, Frenum Pluvium takes the place of Urna Tela; and, when
it passes through, we see a similar section between Frenum Pluvium and
Frenum Tela, and lastly Frenum Tela. Then the tessaractic Set passes
out, or Kata, our space. A similar process takes place with every other
tessaract, when the Set of tessaracts moves through our space.
There is still one more arrangement to be learnt. If the line of the
tessaract, which in the Mala position goes Ana, or W, be changed into
the [=Z] or downwards direction, the tessaract will then appear in our
space below the Mala position; and the side presented to us will not be
Mala, but that which contains the lines Dos, Cuspis, and Ops. This side
is Model 3, and is called Lar. Underneath the place which was occupied
by Urna Mala, will come Urna Lar; under the place of Moles Mala, Moles
Lar; under the place of Frenum Mala, Frenum Lar. The tessaract, which in
the Mala position was an inch out of our space Ana, or W, from Urna
Mala, will now come into it an inch downwards, or [=Z], below Urna
Mala, with its Lar presented to us; that is, Thyrsus Lar will be below
Urna Lar. In the whole arrangement of them written below, the highest
floors are written first, for now they stretch downwards instead of
upwards. The name Lar is understood after each.
Z X Y W
_ō_ _c_ _d_ _a_
SECOND BLOCK.
Second Floor. { Uncus Pallor
{ Ostrum Bidens
First Floor. { Ocrea Tessera
{ Cardo Cudo
FIRST BLOCK.
Second Floor. { Frenum Plebs
{ *Urna Moles
First Floor. { Crates Cura
{ Thyrsus Vitta
Here it is evident that what was the lower floor of Malas, Urna, Moles,
Plebs, Frenum, now appears as if carried downwards instead of upwards,
Lars being presented in our space instead of Malas. This Block of Lars
is what we see of the tessaract Set when the Arctos line, which in the
Mala position goes up, is turned into the Ana, or W, direction, and the
Ops line comes in downwards.
The rest of the tessaracts, which consists of the cubes opposite to the
four treated above, and of the tessaractic space between them, is all
Ana in our space. If the tessaract be moved through our space--for
instance, when the Lars are present in it--we see, taking Urna alone,
first the section between Urna Lar and Urna Velum (Model 10), and then
Urna Velum (Model 4), and similarly the sections and Velums of each
tessaract in the Set. When the First Block has passed Kata our space,
Ostrum Lar enters; and the Lars of the Second Block of tessaracts occupy
the places just vacated by the Velums of the First Block. Then, as the
tessaractic Set moves on Kata, the sections between Velums and Lars of
the Second Block of tessaracts enter our space, and finally their
Velums. Then the whole tessaractic Set disappears from our space.
When we have learnt all these aspects and passages, we have experienced
some of the principal features of this small Set of tessaracts.