Last time we attempted to visualize a four-dimensional
spatial figure by reducing it to three dimensions. First we
converted a three-dimensional figure into a two-dimensional
one. We substituted colors for dimensions, constructing
our image using three colors to represent the three dimensions of
a cube. Then we unfolded the cube so that all of its surfaces lay in
a plane, resulting in six squares whose differently colored edges
represented the three dimensions in two-dimensional space.
We then envisioned transferring each square of the cube's surface
into the third dimension as moving the square through a colored
fog and allowing it to reappear on the other side. We imagined
all the surface squares moving through and being tinted by
transition squares. Thus, we used colors to attempt to picture a
three-dimensional cube in two dimensions. To represent squares
in one dimension, we used two different colors for their edge
pairs, — to represent a cube in two dimensions, we used three colors.
Depicting a four-dimensional figure in three-dimensional
space required a fourth color.
Then we imagined a cube with three different surface colors as
analogous to our square with two different edge colors. Each
such cube moved through a cube of the fourth color, — that is, it
disappeared into the fourth dimension or color. In accordance
with Hinton's analogy, we made each boundary cube move
through the new fourth color and reappear on the other side in
its own original color.
Figure 31
Now I would like to give you another analogy. We will begin
once again by reducing three dimensions to two in preparation
for reducing four dimensions to three. We must envision constructing
our cube out of its six square sides, but instead of leaving
all six squares attached when we spread them out, we will
arrange them differently, as shown here (Figure 31). As you see,
we have split the cube into two groups of three squares each. Both
groups lie in the same plane. We must understand the location of
each group when we reassemble the cube. To complete the cube,
I must place one group above the other so that square 6 lies over
square 5. Once square 5 is in position, I must fold squares 1 and 2
upward, while squares 3 and 4 must be folded downward (Figure
32). The corresponding pairs of line segments — that is, the ones
of the same color (here, with the same number and weight of
slashes as shown in Figure 31) — will then coincide. These lines
that are spread out in two-dimensional space coincide when we
make the transition to three-dimensional space.
Figure 32
A square consists of four edges, a cube of six squares, and a
four-dimensional figure of eight cubes. [Note 36] Hinton calls this four-dimensional
figure a tessaract. Our task is not simply to put these
eight cubes together into a single cube, but to do so by making
each one pass through the fourth dimension. When I do to a tessaract
what I just did to a cube, I must observe the same law. We
must use the analogy of the relationship of a three-dimensional
figure to its two-dimensional counterpart to discover the relationship
of a four-dimensional figure to its three-dimensional counterpart.
In the case of an unfolded cube, I had two groups of three
squares. Similarly, unfolding a four-dimensional tessaract in three-dimensional
space results in two groups of four cubes, which look
like this (Figure 33). This eight-cube method is very ingenious.
Figure 33
We must handle these four cubes in three-dimensional space
exactly as we handled the squares in two-dimensional space.
Look closely at what I have done here. Unfolding a cube so that
it lies flat in two-dimensional space results in a grouping of six
squares. Performing the corresponding operation on a tessaract
results in a system of eight cubes (Figure 34). We have transferred
our reflections on three-dimensional space to four-dimensional
space. Folding up the squares and making their edges coincide
in three-dimensional space corresponds to folding up the
cubes and making their surfaces coincide in four-dimensional
space. Laying the cube flat in two-dimensional space resulted in
corresponding lines that coincided when we reconstructed the
cube. Something similar happens to the surfaces of individual
cubes in the tessaract. Laying out a tessaract in three-dimensional
space results in corresponding surfaces that will later coincide.
Thus, in a tessaract, the upper horizontal surface of cube 1 lies in
the same plane as the front surface of cube 5 when we move into
the fourth dimension.
Figure 34
Similarly, the right surface of cube 1 coincides with the front surface
of cube 4, the left square in cube 1 coincides with the front
square in cube 3, and the lower square in cube 1 coincides with
the front square in cube 6. Similar correspondences exist
between the remaining surfaces. When the operation is completed,
the cube that remains is cube 7, the interior cube that was surrounded
by the other six. [Note 37]
As you see, we are concerned once more with finding analogies
between the third and fourth dimensions. As we saw in one
of the illustrations from the last lecture (Figure 29), just as a fifth
square surrounded by four others remains invisible to any being
who can see only in two dimensions, the same applies to the seventh
cube in this instance. It remains hidden from three-dimensional
vision. In a tessaract, this seventh cube corresponds to an
eighth cube, its counterpart in the fourth dimension.
All of these analogies serve to prepare us for the fourth dimension,
since nothing in our ordinary view of space forces us to add
other dimensions to the three familiar ones. Following Hinton's
example, we might also use colors here and think of cubes put
together so that the corresponding colors coincide. Other than
through such analogies, it is almost impossible to give any guidance
in how to conceive of a four-dimensional figure.
I would now like to talk about another way of representing
four-dimensional bodies in three-dimensional space that may
make it easier for you to understand what is actually at issue.
Here we have an octahedron, which has eight triangular surfaces
that meet in obtuse angles (Figure 35).
Figure 35
Please imagine this figure and then follow this train of thought
with me. You see, these edges are where two surfaces intersect.
Two intersect at \(AB\), for example, and two at \(EB\). The only difference
between an octahedron and a cube is the angle at which the
surfaces intersect. Whenever surfaces intersect at right angles, as
they do in a cube, the figure that is formed must be a cube. But
when they intersect at an obtuse angle, as they do here, an octahedron
is formed. By making the surfaces intersect at different
angles, we construct different geometric figures. [Note 38]
Figure 36
Next, envision a different way of making the surfaces of an
octahedron intersect. Picture that one of these surfaces here,
such as \(AEB\), is extended on all sides and that the lower surface,
\(BCE\), and the surfaces \(ADF\) and \(EDC\), at the back of the figure, are
similarly extended. These extended surfaces must also intersect.
There is a two-fold symmetry at this line of reflection also called
"half-turn symmetry." When these surfaces are extended, the
other four original surfaces of the octahedron, \(ABF\), \(EBC\), \(EAD\),
and \(DCF\), are eliminated. Out of eight original surfaces, four
remain, and these four form a tetrahedron, which also can be
called half an octahedron because it causes half of the surfaces of
the octahedron to intersect. It is not half an octahedron in the
sense of cutting the octahedron in half in the middle. When the
other four surfaces of the octahedron are extended until they
intersect, they also form a tetrahedron. The original octahedron
is the intersection of these two tetrahedrons. In stereometry or
geometric crystallography, what is called half a figure is the result
of halving the number of surfaces rather than of dividing the
original figure in two. This is very easy to visualize in the case of
an octahedron. [Note 39] If you imagine a cubed halved in the same way,
by making one surface intersect with another surface, you will
always get a cube. Half of a cube is always another cube. There
is an important conclusion to be drawn from this phenomenon,
but first I would like to use another example. [Note 40]
Figure 37
Here we have a rhombic dodecahedron (Figure 37). As you
see, its surfaces meet at specific angles. Here we also have a system of four wires — I will call them axial wires — that run in different
directions, that is, they are diagonals connecting specific
opposite corners of the rhombic dodecahedron. These wires represent
the system of axes in the rhombic dodecahedron, similar
to the system of axes you can imagine in a cube. [Note 41]
In a system of three perpendicular axes, a cube results when
stoppage occurs in each of these axes, producing intersecting
surfaces. Causing the axes to intersect at different angles results
in different geometric solids. The axes of a rhombic dodecahedron,
for example, intersect at angles that are not right angles.
Halving a cube results in a cube. [Note 42] This is true only of a cube.
When the number of surfaces in a rhombic dodecahedron is
halved, a totally different geometric figure results. [Note 43]
Figure 38
Now let's consider how an octahedron relates to a tetrahedron.
Let me show you what I mean. The relationship is clearly apparent
if we gradually transform a tetrahedron into an octahedron.
For this purpose, let's take a tetrahedron and cut off its vertices,
as shown here (Figure 38). We continue to cut off larger portions
until the cut surfaces meet on the edges of the tetrahedron. The
form that remains is an octahedron. By cutting off the vertices at
the appropriate angle, we have transformed a spatial figure
bounded by four planes into an eight-sided figure.
Figure 39
What I have just done to a tetrahedron cannot be done to a
cube. [Note 44] A cube is unique in that it is the counterpart of three-dimensional
space. Imagine that all the space in the universe is
structured by three axes that are perpendicular to each other.
Inserting planes perpendicular to these three axes always produces
a cube (Figure 39). Thus, whenever we use the term cube to mean a
theoretical cube rather than a specific one, we are talking about
the cube as the counterpart of three-dimensional space. Just as the
tetrahedron can be shown to be the counterpart of an octahedron
by extending half of the octahedrons sides until they intersect, an
individual cube is also the counterpart of all of space. [Note 45] If you imagine
all of space as positive, the cube is negative. The cube is the
polar opposite of space in its entirety. The physical cube is the
geometric figure that actually corresponds to all of space.
Suppose that instead of a three-dimensional space bounded by
two-dimensional planes, we have a space bounded by six spheres,
which are three-dimensional figures. I start by defining a two-dimensional
space with four intersecting circles, i.e., two-dimensional
figures. Now imagine these circles growing bigger and
bigger, — that is, the radius grows ever longer and the midpoint
becomes increasingly distant. With time, the circles will be transformed
into straight lines (Figure 40). Then, instead of four circles,
we have four intersecting straight lines and a square.
Figure 40
Now instead of circles, imagine six spheres, forming a mulberry-
like shape (Figure 41). Picture the spheres growing ever larger,
just as the circles did. Ultimately, these spheres will become
the planes defining a cube, just as the circles became the lines
defining a square. This cube is the result of six spheres that have
become flat. The cube, therefore, is only a special instance of the
intersection of six spheres, just as the square is simply a special
instance of four intersecting circles.
Figure 41
When you clearly realize that these six spheres flattening into
planes correspond to the squares we used earlier to define a
cube — that is, when you visualize a spherical figure being transformed
into a flat one — the result is the simplest possible three-dimensional
figure. A cube can be imagined as the result of flattening
six intersecting spheres.
We can say that a point on a circle must pass through the second
dimension to get to another point on the circle. But if the
circle has become so large that it forms a straight line, any point
on the circle can get to any other point by moving only through
the first dimension. Let's consider a square that is bounded by
two-dimensional figures. As long as the four figures defining a
square are circles, they are two-dimensional. Once they become
straight lines, however, they are one-dimensional.
The planes defining a cube develop out of three-dimensional
figures (spheres) when one dimension is removed from each of
the six spheres. These defining surfaces come about by being
bent straight, through reducing their dimensions from three to
two. They have sacrificed a dimension. They enter the second
dimension by sacrificing the dimension of depth. Thus, we could
say that each dimension of space comes about by sacrificing the
next higher dimension.
If we have a three-dimensional form with two-dimensional
boundaries, and so reduce three-dimensional forms to two
dimensions, you must conclude from this that, if we consider
three-dimensional space, we have to think of each direction as
the flattened version of an infinite circle. Then if we move in one
direction, we would ultimately return to the same point from the
opposite direction. Thus each ordinary dimension of space has
come about through the loss of the next higher dimension. A triaxial
system is inherent in our three-dimensional space. Each of
its three perpendicular axes has sacrificed the next dimension to
become straight.
In this way, we achieve three-dimensional space by straightening
each of its three axial directions. Reversing the process,
each element of space also could be curved again, resulting in
this train of thought: When you curve a one-dimensional figure,
the resulting figure is two-dimensional. A curved two-dimensional
figure becomes three-dimensional. And, finally, curving a
three-dimensional figure produces a four-dimensional figure.
Thus, four-dimensional space can be imagined as curved three-dimensional
space. [Note 46]
At this point, we can make the transition from the dead to the
living. In this bending you can find spatial figures that reveal this
transition from death to life. At the transition to three-dimensionality,
we find a special instance of four-dimensional space, — it
has become flat. To human consciousness, death is nothing more
than bending three dimensions into four dimensions. With
regard to the physical body taken by itself, the opposite is true:
death is the flattening of four dimensions into three.
See note 30 from the Fourth Lecture.
The situation described here corresponds to Figure 76 in the case of a cube
laid out in a plane:
Figure 76
The location of square 6, directly "above" square 5, cannot be directly depicted
in a plane. The upper edge of square 2, the lower edge of square 4, and the
right and left edges of squares 3 and 1, respectively, must be seen as identical
to the edges of square 6.
Correspondingly, cubes 7 and 8 "coincide" and cannot be distinguished in
three-dimensional space by any direct means. The upper and lower surfaces of
cubes 5 and 6, respectively, the left and right surfaces of 3 and 4, respectively,
and the front and back surfaces of 1 and 2, respectively, also constitute the surfaces
of cube 8. Unfolding a cube makes it easier to note the coincidence
between the edges of the sixth square and those of its neighboring squares
(Figure 77).
Figure 77
Figure 78 shows the corresponding situation in the case of a tessaract. The surfaces
of the eighth cube must be seen as identical to the corresponding surfaces
of neighboring cubes.
Figure 78
In each of the five regular convex polyhedrons — cube, tetrahedron, octahedron,
dodecahedron, and icosahedron — all the angles of surface intersection
are equal. The angle of intersection is unique to each regular polyhedron.
The surfaces of any regular polyhedron are polygons that are both similar
and regular, — that is, all of their edges are of equal length, and all of their angles
are equal. Thus, we simply need to investigate how many polygons can meet
at one vertex in order to gain a complete overview of all possible regular polyhedrons.
Let's begin with equilateral triangles (Figure 79). Two equilateral triangles
cannot be joined together to form one vertex of a polyhedron. Three
such triangles yield a tetrahedron, four form one vertex of an octahedron, and
five form one vertex of an icosahedron. Six triangles lie flat in a plane and cannot
form a vertex.
Figure 79
Three regular rectangular solids (i.e., squares) form one vertex of a cube,
while four lie flat in a plane. Three pentagons form one vertex of a dodecahedron, but four pentagons would overlap (Figure 80).
Figure 80
Three hexagons lie flat in a plane, and three heptagons overlap. Thus, there
cannot possibly be more than the five types of regular polyhedrons mentioned
earlier.
Rudolf Steiner refers here to a standard procedure in geometric crystallography.
The seven classes of crystals are based on the symmetries of the seven
possible crystallographic systems of axes. A symmetry group, which represents
all of the symmetry elements of one class, is called a holohedry. The polyhedrons
belonging to such symmetry groups are called holohedral shapes. They
are simple polyhedrons that can be converted into each other through symmetrical
operations that all belong to a single crystal system. Hemihedral
forms are polyhedrons with half as many surfaces as the corresponding holohedral
forms. Hemihedrons are derived from holohedrons through the extension
of some of the surfaces of the holohedrons and the disappearances of others.
The symmetry group of the hemihedrons is correspondingly reduced (subgroup
of holohedries of index 2). In this sense, a tetrahedron is a hemihedral
variation on an octahedron because it has half the number of surfaces.
Crystallographers also have introduced tetardohedrons, polyhedrons with
one-fourth the number of surfaces of the corresponding holohedral figures and
a correspondingly reduced symmetry group (subgroup of holohedries of index
4). For more information, see Hochstetter/Bisching [1868], pp. 20ff; Schoute
[1905], pp. 190ff; and Niggli [1924], pp. 70ff and 129ff.
In a cube, any two intersecting surfaces meet in a right angle. No matter
which surfaces we choose, extending them always will result in a figure with
90° angles of intersection. In a cube, however, reducing the number of surfaces
no longer results in a closed polyhedron.
"In this case, the axes of a cube are the three perpendicular directions that
intersect in the cube's midpoint, — one pair of surfaces is perpendicular to each
axis. These axes are also the axes of the three zones of a cube (Figure 81). A
zone or zone association is a set of at least three surfaces that are parallel to the
straight line of a zone axis.
Figure 81
A rhombic dodecahedron is easy to construct with the help of a cube. First all
six diagonal planes connecting opposite edges of the cube are constructed
(Figure 82). Then the mirror images of the resulting six internal pyramids are
constructed on the outside of the cube (Figure 83). The four "axes" mentioned
in the lecture are the diagonals of the rhombic dodecahedron that coincide
with the diagonals of the cube.
Figures 82-83
These four axes are the four zone axes of the rhombic dodecahedron — that
is, each of them is parallel to six surfaces of this figure. These four groups of
six planes are called the zones of the rhombic dodecahedron.
Because its vertices are not all similar, a rhombic dodecahedron is not a regular
polyhedron. Three surfaces intersect in each of the vertices that emerge
from the cube, while four surfaces intersect in each of the other vertices. The
zone axes pass through the vertex points where three surfaces meet. Note that
the "axes" described here represent a specific selection from the seven possible
diagonals (straight line segments connecting opposite comer points).
Figure 84
About the drawings: The rhombic dodecahedron, like the other geometric
figures depicted here, is drawn in oblique parallel projection, which is best
suited to freehand drawing on the board. This projection results in slight distortions
of subsequent figures, which must be taken into account.
In addition to the axes described in the previous note, a rhombic dodecahedron
also has axes perpendicular to its surfaces. If a rhombic dodecahedron is
held in place while its four zone axes are rotated 45° around the perpendicular
axis of the underlying cube, the axes then intersect the midpoints of eight of
the rhombic dodecahedrons surfaces. The figure formed by these surfaces is an
octahedron consisting of the four pairs of surfaces that are perpendicular to the
zone axes (rotated 45°) of the rhombic dodecahedron (Figure 85). Adding to
these four axes the two horizontal axes (also rotated 45°) of the cube (see previous
note) results in a system of six "axes"; each surface of the rhombic dodecahedron
is perpendicular to one of them.
Figure 85
Halving the number of surfaces of a cube does not produce any new surface
angles. A rhombic dodecahedron can be "halved" in several different ways (Figures 86 and 87). When this operation produces a closed polyhedron, it is
an oblique parallelepiped.
Figure 86Figure 87
This statement presupposes that the cuts in the tetrahedron or cube are made
parallel to existing surfaces. Successively cutting off the vertices of a cube so
that the cut surfaces are perpendicular to the cube's diagonals results first in a
cube-octahedron and eventually in an octahedron.
See also Steiner's lecture of March 31, 1905. No matter which three of the six
planes defining a cube are selected, the result of extending them into space
results in a "figure" that stretches to infinity. If the three surfaces we select are
perpendicular to each other, the result is a geometric figure consisting of three perpendicular axes and the planes that connect them in pairs. Such a figure can
be seen as representing three-dimensional Euclidean space and is also the geometric
basis of every Euclidean or Cartesian coordinate system.
Here and in the remainder of the lectures, Steiners presentation seems to
have been substantially abridged, and, as a result, various perspectives overlap.
To the series square-cube-tessaract, we can add another series of geometrical
figures where the planes or faces of the figure are curved rather than
straight or flat. We can call the figures of this second series curved squares,
curved cubes, and curved tessaracts. In such a figure, the elements forming its
edges or sides have the same number of dimensions as the total figure.
The circle, the spherical surface (two-dimensional sphere), and the solid
(three-dimensional) sphere are topologically equivalent to the rectilinear elements
defining the boundaries of a square, a cube, and a tessaract respectively.
The disc, ball, and four-dimensional ball are topologically equivalent to the
square, the cube, and the tessaract respectively.
On the other hand, suitable bending of a one-dimensional line segment
results in a two-dimensional segment of a curve or — in a special instance — in
a segment of a circle. Bending a disc produces a three-dimensional figure, a
hollow hemisphere. Bending a solid sphere produces a four-dimensional figure
(in a special instance, a section of a four-dimensional sphere).
In this way, a circle can be constructed from two curved line segments
whose ends are joined. Similarly, in three-dimensional space, a spherical surface
can be constructed from two discs that are first curved and then joined at
their edges. In four-dimensional space, a three-dimensional sphere results
when two curved solid spheres are joined at their surfaces (two-dimensional
spheres). This three-dimensional sphere relates to three-dimensional space as a
ball (the surface of an ordinary sphere) relates to a plane. [Mathematician David
Cooper comments: You are comparing filled-in figures rather than boundaries
in both cases. A sphere (the boundary of a ball) is two-dimensional, so the two-dimensional
sphere's volume means the (three-dimensional) ball.]