Today I must conclude these lectures on the fourth dimension
of space, though I actually would like to present a
complicated system in greater detail, which would require
demonstrating many more of Hinton's models. All I can do is refer
you to his three thorough and insightful books. [Note 47] Of course, no
one who is unwilling to use analogies such as those presented in
the previous lectures will be able to acquire a mental image of four-dimensional
space. A new way of developing thoughts is needed.
Now I would like to develop a real image (parallel projection)
of a tessaract. We saw that a square in two-dimensional space has
four edges. Its counterpart in three dimensions is the cube, which
has six square sides (Figure 42).
Figure 42
The four-dimensional counterpart is the tessaract, which is
bounded by eight cubes. Consequently, the projection of a tessaract
into three-dimensional space consists of eight interpenetrating
cubes. We saw how these eight cubes can coincide in
three-dimensional space. I will now construct a different projection
of a tessaract. [Note 48]
Imagine holding a cube up to the light so that it casts a shadow
on the board. We can then trace the shadow with chalk
(Figure 43). As you see, the result is a hexagon. If you imagine
the cube as transparent, you can see that in its projection onto a
plane, the three front faces coincide with the three rear faces to
form the hexagonal figure.
Figure 43
To get a projection that we can apply to a tessaract, please imagine
that the cube in front of you is positioned so that the front
point \(A\) exactly covers the rear point \(C\). If you then eliminate the
third dimension, the result is once again a hexagonal shadow. Let
me draw this for you (Figure 44).
Figure 44
When you imagine the cube in this position, you see only its
three front faces, — the three other faces are concealed behind
them. The faces of the cube appear foreshortened, and its angles
no longer look like right angles. Seen from this perspective, the
cube looks like a regular hexagon. Thus, we have created an
image of a three-dimensional cube in two-dimensional space.
Because this projection shortens the edges and alters the angles,
we must imagine the six square faces of the cube as rhombuses. [Note 49]
Now let's repeat the operation of projecting a three-dimensional
cube onto a plane with a four-dimensional figure that we
project into three-dimensional space. We will use parallel projection
to depict a tessaract, a figure composed of eight cubes, in
the third dimension. Performing this operation on a cube results
in three visible and three invisible edges, — in reality, they jut into
space and do not lie flat in the plane of projection. Now imagine
a cube distorted into a rhombic parallelepiped. [Note 50] If you take eight
such figures, you can assemble the structures defining a tessaract
so that they interpenetrate and doubly coincide with the rhombic
cubes in a rhombic dodecahedron (Figure 45).
Figure 45
This figure has one more axis than a three-dimensional cube.
Naturally, a four-dimensional figure has four axes. Even when its
components interpenetrate, four axes remain. Thus, this projection
contains eight interpenetrating cubes, shown as rhombic
cubes. A rhombic dodecahedron is a symmetrical image or shadow
of a tessaract projected into three-dimensional space. [Note 51]
Although we have arrived at these relationships by analogy,
the analogy is totally valid. Just as a cube can be projected onto
a plane, — a tessaract also can be depicted by projecting it into
three-dimensional space. The resulting projection is to the tessaract as the cube's shadow is to the cube. I believe this operation
is readily understandable.
I would like to link what we have just done to the wonderful
image supplied by Plato and Schopenhauer in the metaphor of
the cave. [Note 52] Plato asks us to imagine people chained in a cave so
that they cannot turn their heads and can see only the rear wall.
Behind them, other people carry various objects past the mouth
of the cave. These people and objects are three dimensional, but
the prisoners see only shadows cast on the wall. Everything in
this room, for example, would appear only as two-dimensional
shadow images on the opposite wall.
Then Plato tells us that our situation in the world is similar. We
are the people chained in the cave. Although we ourselves are
four-dimensional, as is everything else, all that we see appears
only in the form of images in three-dimensional space. [Note 53]
According to Plato, we are dependent on seeing only the three-dimensional
shadow images of things instead of their realities. I
see my own hand only as a shadow image, — in reality, it is four-dimensional.
We see only images of four-dimensional reality,
images like that of the tessaract that I showed you.
In ancient Greece, Plato attempted to explain that the bodies we
know are actually four-dimensional and that we see only their
shadow images in three-dimensional space. This statement is not
completely arbitrary, as I will explain shortly. Initially, of course,
we can say that it is mere speculation. How can we possibly imagine
that there is any reality to these figures that appear on the wall?
But now imagine yourselves sitting here in a row, unable to move.
Suddenly, the shadows begin moving. You cannot possibly conclude
that the images on the wall could move without leaving the
second dimension. When an image moves on the wall, something
must have caused movement of the actual object, which is not on
the wall. Objects in three-dimensional space can move past each
other, something their two-dimensional shadow images cannot do
if you imagine them as impenetrable — that is, as consisting of substance.
If we imagine these images to be substantial, they cannot
move past each other without leaving the second dimension.
As long as the images on the wall remain motionless, I have no
reason to conclude that anything is happening away from the
wall, outside the realm of two-dimensional shadow images. As
soon as they begin to move, however, I am forced to investigate
the source of the movement and to conclude that the change can
originate only in a movement outside the wall, in a third dimension.
Thus, the change in the images has informed us that there
is a third dimension in addition to the second.
Although a mere image possesses a certain reality and very
specific attributes, it is essentially different from the real object.
A mirror image, too, is undeniably a mere image. You see yourself
in the mirror, but you are also present out here. Without the
presence of a third element — that is, a being that moves — you
cannot really know which one is you. The mirror image makes
the same movements as the original, — it has no ability to move
itself but is dependent on the real object, the being. In this way,
we can distinguish between an image and a being by saying that
only a being can produce change or movement out of itself. I
realize that the shadow images on the wall cannot make themselves
move, — therefore, they are not beings. I must transcend the
images in order to discover the beings.
Now apply this train of thought to the world in general. The
world is three-dimensional, but if you consider it by itself, grasping
it in thought, you will discover that it is essentially immobile.
Even if you imagine it frozen at a certain point in time, however,
the world is still three-dimensional. In reality, the world is not
the same at any two points in time. It changes. Now imagine the
absence of these different moments — what is, remains. If there
were no time, the world would never change, but even without
time or changes it would still be three-dimensional. Similarly, the
images on the wall remain two dimensional, but the fact that
they change suggests the existence of a third dimension. That
the world is constantly changing but would remain three-dimensional
even without change suggests that we need to look for the
change in a fourth dimension. The reason for change, the cause
of change, the activity of change, must be sought outside the
third dimension. At this point you grasp the existence of the
fourth dimension and the justification for Plato's metaphor. We
can understand the entire three-dimensional world as the shadow
projection of a four-dimensional world. The only question is
how to grasp the reality of this fourth dimension.
Of course, we must understand that it is impossible for the
fourth dimension to enter the third directly. It cannot. The
fourth dimension cannot simply fall into the third dimension.
Now I would like to show you how to acquire a concept of transcending
the third dimension. (In one of my earlier lectures here,
I attempted to awaken a similar idea in you.) [Note 54] Imagine that we
have a circle. If you picture this circle getting bigger and bigger,
so that any specific segment becomes flatter and fatter, the diameter
eventually becomes so large that the circle is transformed
into a straight line. A line has only one dimension, but a circle
has two. How do we get back into two dimensions? By bending
the straight line to form a circle again.
When you imagine curving a circular surface, it first becomes
a bowl and eventually, if you continue to curve it, a sphere. A
curved line acquires a second dimension and a curved plane a
third. And if you could still make a cube curve, it would have to
curve into the fourth dimension, and the result would be a spherical
tessaract. [Note 55] A spherical surface can be considered a curved
two-dimensional figure. In nature, the sphere appears in the form
of the cell, the smallest living being. The boundaries of a cell are
spherical. Here we have the difference between the living and
the lifeless. Minerals in their crystalline form are always bounded
by planes, by flat surfaces, while life is built up out of cells and
bounded by spherical surfaces. Just as crystals are built up out of
flattened spheres, or planes, life is built up out of cells, or abutting
spheres. The difference between the living and the lifeless
lies in the character of their boundaries. An octahedron is bounded
by eight triangles. When we imagine its eight sides as spheres,
the result is an eight-celled living thing.
When you "curve" a cube, which is a three-dimensional figure,
the result is a four-dimensional figure, the spherical tessaract. But
if you curve all of space, the resulting figure relates to three-dimensional
space as a sphere relates to a plane. [Note 56] As a three-dimensional
object, a cube, like any crystal, is bounded by planes. The
essence of a crystal is that it is constructed of flat boundary planes.
The essence of life is that is constructed of curved surfaces, namely,
cells, while a figure on a still higher level of existence would be
bounded by four-dimensional structures. A three-dimensional figure
is bounded by two-dimensional figures. A four-dimensional
being — that is, a living thing — is bounded by three-dimensional
beings, namely, spheres and cells. A five-dimensional being is
bounded by four-dimensional beings, namely, spherical tessaracts.
Thus, we see the need to move from three-dimensional beings to
four-dimensional and then five-dimensional beings.
What needs to happen with a four-dimensional being? [Note 57] A
change must take place within the third dimension. In other
words, when you hang pictures, which are two-dimensional, on
the wall, they generally remain immobile. When you see two-dimensional
images moving, you must conclude that the cause of
the movement can lie only outside the surface of the wall — that
is, that the third dimension of space prompts the change. When
you find changes taking place within the third dimension, you
must conclude that a fourth dimension has an effect on beings
who experience changes within their three dimensions of space.
We have not truly recognized a plant when we know it only in
its three dimensions. Plants are constantly changing. Change is an
essential aspect of plants, a token of a higher form of existence. A
cube remains the same, — its form changes only when you break it.
A plant changes shape by itself, which means that the change
must be caused by some factor that exists outside the third dimension
and is expressed in the fourth dimension. What is this factor?
You see, if you draw this cube at different points in time, you will
find that it always remains the same. But when you draw a plant
and compare the original to your copy three weeks later, the original
will have changed. Our analogy, therefore, is fully valid. Every
living thing points to a higher element in which its true being
dwells, and time is the expression of this higher element. Time is
the symptomatic expression or manifestation of life (or the fourth
dimension) in the three dimensions of physical space. In other
words, all beings for whom time is intrinsically meaningful are
images of four-dimensional beings. After three years or six years,
this cube will still be the same. A lily seedling changes, however,
because time has real meaning for it. What we see in the lily is
merely the three-dimensional image of the four-dimensional lily
being. Time is an image or projection of the fourth dimension, of
organic life, into the three spatial dimensions of the physical world.
To clarify how each successive dimension relates to the preceding
one, please follow this line of thought: A cube has three
dimensions. To imagine the third, you tell yourself that it is perpendicular
to the second and that the second is perpendicular to
the first. It is characteristic of the three dimensions that they are
perpendicular to each other. We also can conceive of the third
dimension as arising out of the next dimension, the fourth. Envision
coloring the faces of a cube and manipulating the colors in a specific
way, as Hinton did. The changes you induce correspond
exactly to the change undergone by a three-dimensional being
when it develops over time, thus passing into the fourth dimension.
When you cut through a four-dimensional being at any point — that
is, when you take away its fourth dimension — you destroy the
being. Doing this to a plant is just like taking an impression of the
plant and casting it in plaster. You hold it fast by destroying its
fourth dimension, the time factor, and the result is a three-dimensional
figure. When time, the fourth dimension, is critically important
to any three-dimensional being, that being must be alive.
And now we come to the fifth dimension. You might say that
this dimension must have another boundary that is perpendicular
to the fourth dimension. We saw that the relationship between the
fourth dimension and the third is similar to the relationship
between the third and second dimensions. It is more difficult to
imagine the fifth dimension, but once again we can use an analogy
to give us some idea about it. How does any dimension come
about? When you draw a line, no further dimensions emerge as
long as the line simply continues in the same direction. Another
dimension is added only when you imagine two opposing directions
or forces that meet and neutralize at a point. The new dimension
arises only as an expression of the neutralization of forces. We
must be able to see the new dimension as the addition of a line in
which two streams of forces are neutralized. We can imagine the
dimension as coming either from the right or from the left, as positive
in the first instance and negative in the second. Thus I grasp
each independent dimension as a polar stream of forces with both
a positive and a negative component. The neutralization of the
polar component forces is the new dimension.
Taking this as our starting point, let's develop a mental image
of the fifth dimension. We must first imagine positive and negative
aspects of the fourth dimension, which we know is the
expression of time. Let's picture a collision between two beings for
whom time is meaningful. The result will have to be similar to the
neutralization of opposing forces that we talked about earlier.
When two four-dimensional beings connect, the result is their
fifth dimension. The fifth dimension is the result or consequence
of an exchange or neutralization of polar forces, in that two living
things who influence each other produce something that they do
not have in common either in the three ordinary dimensions of
space or in the fourth dimension, in time. This new element has
its boundaries outside these dimensions. It is what we call empathy
or sensory activity, the capacity that informs one being about
another. It is the recognition of the inner (soul-spiritual) aspect of
another being. Without the addition of the higher, fifth dimension —
that is, without entering the realm of sensory activity, — no
being would ever be able to know about any aspects of another
being that lie outside time and space. Of course, in this sense we
understand sensory activity simply as the fifth dimension's projection
or expression in the physical world.
It would be too difficult to build up the sixth dimension in the
same way, so for now I will simply tell you what it is. If we continued
along the same line of thinking, we would find that the expression
of the sixth dimension in the three-dimensional world is self-awareness.
As three-dimensional beings, we humans share our
image character with other three-dimensional beings. Plants possess
an additional dimension, the fourth. For this reason, you will never
discover the ultimate being of the plant in the three dimensions of
space. You must ascend to a fourth dimension, to the astral sphere.
If you want to understand a being that possesses sensory ability, you
must ascend to the fifth dimension, lower devachan or the Rupa
sphere, and to understand a being with self-awareness — namely, the
human being — you must ascend to the sixth dimension, upper
devachan or the Arupa sphere. The human beings we encounter at
present are really six-dimensional beings. What we have called sensory
ability (or empathy) and self-awareness are projections of the
fifth and sixth dimensions, respectively, into ordinary three-dimensional
space. Albeit unconsciously for the most part, human beings
extend all the way into these spiritual spheres, — only there can their
essential nature be recognized. As six-dimensional beings, we
understand the higher worlds only when we attempt to relinquish
the characteristic attributes of lower dimensions.
I cannot do more than suggest why we believe the world to be
merely three-dimensional. Our view is based on seeing the world
as a reflection of higher factors. The most you can see in a mirror
is a mirror image of yourself. In fact, the three dimensions of
our physical space are reflections, material images of three higher,
causal, creative dimensions. Thus, our material world has a
polar spiritual counterpart in the group of the three next higher
dimensions, that is, in the fourth, fifth, and sixth. Similarly, the
fourth through sixth dimensions have their polar counterparts in
still more distant spiritual worlds, in dimensions that remain a
matter of conjecture for us.
Consider water and water that has been allowed to freeze. In
both cases, the substance is the same, but water and ice are very
different in form. You can imagine a similar process taking place
with regard to the three higher human dimensions. When you
imagine human beings as purely spiritual beings, you must envision
them as possessing only the three higher dimensions of self-awareness,
feeling, and time and that these dimensions are
reflected in the three ordinary dimensions in the physical world.
When yogis (students of esotericism) want to ascend to
knowledge of the higher worlds, they must gradually replace
reflections with realities. For example, when they consider a
plant, they must learn to replace the lower dimensions with the
higher ones. Learning to disregard one of a plant's spatial dimensions
and substitute the corresponding higher dimension — namely,
time — enables them to understand a two-dimensional being
that is moving. What must students of esotericism do to make this
being correspond to reality rather than remaining a mere image? If
they were simply to disregard the third dimension and add the
fourth, the result would be something imaginary. The following
thought will help us move toward an answer: By filming a living
being, even though we subtract the third dimension from events
that were originally three-dimensional, the succession of images
adds the dimension of time. When we then add sensory ability to
this animated image, we perform an operation similar to the one I
described as curving a three-dimensional figure into the fourth
dimension. The result of this operation is a four-dimensional figure
whose dimensions include two of our spatial dimensions and two
higher ones, namely, time and sensory ability. Such beings do
indeed exist, and now that I have come to the real conclusion of
our study of the dimensions, I would like to name them for you.
Imagine two spatial dimensions — that is, a plane — and suppose
that this plane is endowed with movement. Picture it curving
to become a sensate being pushing a two-dimensional surface
in front of it. Such a being is very dissimilar to and acts very differently
from a three-dimensional being in our space. The surface
being that we have constructed is completely open in one direction.
It looks two-dimensional, — it comes toward you, and you
cannot get around it. This being is a radiant being, — it is nothing
other than openness in a particular direction. Through such a
being, initiates then become familiar with other beings whom
they describe as divine messengers approaching them in flames
of fire. The description of Moses receiving the Ten
Commandments on Mount Sinai shows simply that he was
approached by such a being and could perceive its dimensions. [Note 58]
This being, which resembled a human being minus the third
dimension, was active in sensation and time.
The abstract images in religious documents are more than
mere outer symbols. They are mighty realities that we can learn
about by taking possession of what we have been attempting to
understand through analogies. The more diligently and energetically
you ponder such analogies, the more eagerly you submerse
yourself in them, the more they work on your spirit to release
higher capacities. This applies, for example, to the explanation of
the analogy between the relationship of a cube to a hexagon and
that of a tessaract to a rhombic dodecahedron. The latter is the
projection of a tessaract into the three-dimensional physical
world. By visualizing these figures as if they possessed independent
life — that is, by allowing a cube to grow out of its projection,
the hexagon, and the tessaract to develop out of its projection,
the rhombic dodecahedron — your lower mental body
learns to grasp the beings I just described. When you not only
have followed my suggestions but also have made this operation
come alive as esoteric students do, in full waking consciousness,
you will notice that four-dimensional figures begin to appear in
your dreams. At that point, you no longer have far to go to be able
to bring them into your waking consciousness. You then will be
able to see the fourth dimension in every four-dimensional being.
The astral sphere is the fourth dimension.
Devachan up to Rupa is the fifth dimension.
Devachan up to Arupa is the sixth dimension. [Note 59]
These three worlds — physical, astral, and heavenly
(devachan) — encompass six dimensions. The still higher worlds
are the polar opposites of these dimensions.
PresumabIy, this reference is to Hinton's books Scientific Romances [1886], A
New Era of Thought [ 1900], and The Fourth Dimension [ 1904].
Strictly speaking, the depiction of a tessaract in the previous lecture (May 31,
1905) is not a projection but simply an unfolded view. In the present lecture,
Steiner proceeds to construct an orthogonal parallel projection of a tessaract
in three-dimensional space, taking one of its diagonals as the direction of projection.
Considering the framework formed by the edges of a cube, an oblique parallel
projection of the cube onto a plane generally consists of two parallel, non-coinciding
cubes and the line segments connecting their corresponding corners
(Figure 88: oblique parallel projections of a cube).
Figure 88
If the diagonal AC is selected as the direction of projection, vertices A and
C coincide, producing an oblique hexagon and its diagonals. The images of
the six individual faces of the cube can be reconstructed from this hexagon
by tracing all the possible parallelograms defined by the existing structure
of lines. Each of these parallelograms overlaps with two others, and the
hexagon's surface is covered twice by the faces of the cube. When the
direction of projection is perpendicular to the plane of projection, the
resulting image of a cube is a regular hexagon (Figure 89: orthogonal parallel
projections of a cube).
Figure 89
Note that the three diagonals of the hexagon also represent the three (zone)
axes of the cube. The zone associations belonging to each of these axes — that
is, the four faces of the cube that parallel it — appear as four parallelograms or
rhombuses with one edge coinciding with the corresponding axis.
’“Earlier in this lecture, Steiner called a distorted or oblique square a "rhombus,"
which is a parallelogram with four equal sides. The corresponding solid
figure, Steiner's "rhombic parallelepiped," is an oblique cube — i.e., a parallelepiped
whose edges are all the same length.
If we see the tessaract as the framework formed by its edges, the result of projecting
the tessaract into three-dimensional space generally consists of two
parallel, displaced oblique cubes and the line segments connecting their corresponding
vertices (Figure 90: oblique parallel projections of a tessaract).
Figure 90
When the direction of projection passes through the diagonal A'C, the endpoints
A' and C coincide, resulting in a rhombic dodecahedron with four diagonals.
In the first figure, it is easy to trace the images of the eight cubes defining
the boundaries of a tessaract: they are all the possible parallelepipeds
formed by the edges of the existing framework. These parallelepipeds include
the original cube, the displaced cube, and the six parallelepipeds that share one
face each with the original and displaced cubes. This situation does not change
fundamentally when we make the transition to the rhombic dodecahedron,
except that in this instance all of the "rhombic cubes" (parallelepipeds) interpenetrate
in such a way that they fill the internal space of the rhombic dodecahedron
exactly twice, with each parallelepiped including portions of three
others.
The four diagonals of the rhombic dodecahedron that appear in the projection
of the tessaract are the zone axes of the rhombic dodecahedron's four
associations of six faces each. Each such face association consists of all six surfaces
that are parallel to a single zone axis. (Note that in a rhombic dodecahedron
the axes pass through vertices rather than through the centers of faces, as
in a cube.)
These four axes, however, are also the projections of the tessaract's four perpendicular
axes in four-dimensional space. A cube's three axes pass through the
centers of its square sides. Analogously, a tessaract's axes pass through the middle
of the cubes that form its sides. In parallel projection, the middle of a cube
is transformed into the middle of the corresponding parallelepiped. As we can
ascertain by studying all eight parallelepipeds in a rhombic dodecahedron, the
four axes pass exactly through the middles of these parallelepipeds.
A cube's four perpendicular axes are simultaneously the zone axes of its
three face associations of four faces each. Similarly, a tessaract's four axes are
also the zone axes of four cell associations of six cells each (cell equals the cube
forming a side of the tessaract). In the rhombic dodecahedron, the cells
belonging to each axis are easy to find: they are the six parallelepipeds with
one edge coinciding with that axis.
Plato, The Republic, book 7, 514a-518c. It has not yet been possible to ascertain where Schopenhauer used this metaphor.
Zöllner drew attention to this interpretation of Plato’s cave metaphor in his essay Über Wirkungen in die Ferne [ 1878a], pp. 260ff.
See the lecture of March 24, 1905.
What Steiner seems to mean here by spherical tessaract is not a four-dimensional
cube in the narrower sense but rather its topological equivalent, the
three-dimensional sphere in four-dimensional space, which is produced by
curving and attaching two solid three-dimensional spheres. See Note 11 ,
Lecture 5.
See Note 9 above and Note 11, Lecture 5.
The remaining text of this lecture incorporates fragments of transcripts quoted in Haase's essay [1916], which helped clarify the meaning.
Exodus 19, — also Exodus 33 and 34.
ln theosophical literature, the three upper regions of the land of spirit were
called Arupa regions, in contrast to the four lower, or Rupa, regions. See the
editor's note to Rudolf Steiner's Die Grundelemente der Esoterik,'The Basic Elements
of Esotericism" (GA 93a), p. 281 ff. On the seven regions of the land of spirit,
see Rudolf Steiner, Theosophy (GA 9), 'The Country of Spirit Beings." On the
problem of dimensionality in connection with the planes or regions of the spirit
world, see also Rudolf Steiner's lecture of May 17, 1905, — his response to
questions asked by A. Strakosch on March 11, 1920, — the questions and answers
of April 7, 1921 (GA 76), and April 12, 1922 (GA 82); and the lectures of
August 19, 29, 22, and 26, 1923 (GA 227).
