The Fourth Dimension

GA 324a — 7 November 1905, Berlin

Four-dimensional Space

Our ordinary space has three dimensions: length, width and height. A line extends in one dimension, it has only length. A table is a surface, so it has two dimensions: length and width. A body extends in three dimensions. How does a body of three dimensions come about?

Imagine a shape that has no dimensionality at all: that is the point. It has zero dimensions. When a point moves in one direction, a straight line is created, a one-dimensional shape. If you imagine the line continuing, a surface with length and width is created. Finally, if you imagine the surface moving, it describes a three-dimensional shape. But we cannot use the same method to create a fourth dimension from a three-dimensional object [through movement]. We must try to visualize how we can arrive at the concept of a fourth dimension. [Certain] mathematicians [and natural scientists] have felt compelled to harmonize the spiritual world with our sensual world [by placing the spiritual world in a four-dimensional space], for example, Zöllner.

Imagine a circle. It is closed on all sides in the plane. If someone demands that a coin should come into the circle from outside, we have to cross the circle line (Figure 46). But if you do not want to touch the circle line, you have to lift the coin [into the space] and then put it in. You must necessarily go from the second to the third dimension. If we wanted to conjure a coin into a cube [or into a sphere], we would have to go [out of the third dimension and] through the fourth dimension.'

In this life, the first time I began to grasp what space actually is was when I started to study recent [synthetic projective] geometry. Then I realized what it means to go from a circle to a line (Figure 47). In the most intimate thinking of the soul, the world opens up.

Now let us imagine a circle. If we follow the circle line, we can walk around it and return to the original point. Now let us imagine the circle getting bigger and bigger [holding a tangent line]. In the end, it must merge into a straight line because it flattens out more and more. [When I go through the enlarging circles, I always go down on one side and then come up on the other side and back to the starting point. If I finally move on the straight line, for example to the right into infinity, I have to return from the other side of infinity, since the straight line behaves like a circle in terms of the arrangement of its points. From this we see that space has no end [in the same sense that the straight line has no end, that is, the arrangement of its points is the same as in a closed circle. Accordingly, we must think of infinitely extended space as closed in itself, just as the surface of a sphere is closed in itself]. Thus you have represented infinite space [in the sense of] a circle [or] a sphere. This concept leads us to imagine space in its reality.

If I now imagine that I do not simply disappear [into infinity] and then return [unchanged from the other side], but think to myself that I have a radiating light, this will become weaker and weaker as I move away (seen from a stationary point on the line) and stronger and stronger when I return (with the light from infinity). And if we consider that this light not only has a positive effect, but, as it approaches from the other side here, shines all the more strongly, then you have [here the qualities] positive and negative.

In all natural effects, you will find these two poles, which represent nothing other than the opposite effects of space. From this you get the idea that space is something powerful, and that the forces that work in it are nothing other than the outflow of the power itself. Then we will have no doubt that within our three-dimensional space there could be a force that works from within. You will realize that everything that occurs in space is based on real relationships in space.

If we were to intertwine two dimensions, we would have brought these two into relation. If you want to entwine two [closed] rings, you have to unravel one of them to get the other inside. But now I will demonstrate the inner diversity of space by entwining this structure [a rectangular paper band] twice around itself [that is, holding one end and twisting the other end 360° and then holding the two ends together]. I pin the paper tape together tightly with pins and cut it in half. Now one tape is firmly stuck inside the other. Before that, it was just one tape. So here, by merely intertwining the tapes within the three dimensions, I have created the same thing that I would otherwise have to reach out into the [fourth] dimension to achieve."

This is not a gimmick, but reality. If we have the sun here, and the earth's orbit around the sun here, and the moon's orbit around the earth here (Figure 48), we have to imagine that the earth moves around the sun and therefore the moon's orbit and the earth's orbit are intertwined exactly [like our two paper ribbons]. Now the moon has branched off from the earth [in the course of the earth's development]. This is an internal bifurcation that has occurred in the same way [as the intertwining of our two paper ribbons]. [Through such a way of looking at it] space comes alive in itself.

Now consider a square. Imagine it moving through space in such a way that it forms a cube. Then it must progress within itself.

A cube is composed of six squares, which together form the surface of the cube. To put the cube together [in a clear way], I first place the six squares next to each other [in a plane] (Figure 49). I get the cube again when I put these squares on top of each other. I then have to place the sixth on top by going through the third dimension. Thus I have now laid the cube out in two dimensions. I have transformed a three-dimensional structure by laying it out in two dimensions.

Now imagine that the boundaries of a cube are squares. If I have a three-dimensional cube here, it is bounded by two-dimensional squares. Let's just take a single square. It is two-dimensional and is bounded by four one-dimensional lines. I can expand the four lines into a single dimension (Figure 50). What appears in the one dimension, I will now paint in red [solid line] and the other dimension in blue [dotted line]. Now, instead of saying length and width, I can speak of the red and blue dimensions.

I can reassemble the cube from six squares. So now I go from the number four [the number of side lines of the square] to the number six [the number of the side surfaces of the cube]. If I go one step further, I get from the number six [the number of the side surfaces of the cube] to the number eight [the number of “side cubes” of a four-dimensional structure]. I now arrange the eight cubes in such a way that the corresponding structure is created in three-dimensional space to that which was previously constructed in two-dimensional space (Figure 51) from six squares.

Imagine that I could turn this structure inside out so that I could turn it right way up and put it together in such a way that I could cover the whole structure with the eighth cube. Then I would get a four-dimensional structure in a four-dimensional space from the eight cubes. This figure is called [by Hinton] the tessaract. Its limiting figure is eight cubes, just as the ordinary cube has six squares as its limiting figure. The [four-dimensional] tessaract is therefore bounded by [eight] three-dimensional cubes.

Imagine a creature that can only see in two dimensions, and this creature would now look at the squares laid out separately, it would only see the squares 1, 2, 3, 4 and 6, but never the hatched square 5 in the middle (Figure 52). It is quite the same for you with the four-dimensional structure. [Since you can only see three-dimensional objects, you] cannot see the hidden cube in the middle.

Now imagine the cube drawn on the board like this [so that the outline forms a regular hexagon]. The other side is hidden behind it. This is a kind of silhouette, a projection of the cube into two-dimensional space (Figure 53). This two-dimensional silhouette of a three-dimensional cube consists of rhombi, oblique rectangles [parallelograms]. If you imagine the cube made of wire, you would also be able to see the rhomboid squares at the back. So here you have six interlocking rhomboid squares in the projection. In this way you can project the whole cube into two-dimensional space.

Now imagine our tetrahedron formed in four-dimensional space. If you project this figure into three-dimensional space, you should get four non-intersecting rhombic parallelpipeds. One of these rhombic parallelpipeds should be drawn as follows (Figure 54).

Eight such shifted rhombic cubes would have to be inserted into each other in order to obtain a three-dimensional image of the four-dimensional tessaract in three-dimensional space. Thus, we can represent the three-dimensional shadow image of such a tessaract with the help of eight rhombic cubes that are suitably inserted into each other. The spatial structure that results is a rhombic dodecahedron with four spatial diagonals (Figure 55). Just as in the rhombus representation of the cube, three directly neighboring rhombuses are shifted into each other, so that only three of the six cube surfaces are seen in the projection, only four non-intersecting rhombic cubes appear in the rhombic dodecahedron only four non-intersecting rhombic cubes appear as projections of the eight boundary cubes, since four of the directly neighboring rhombic cubes completely cover the remaining four.'>

We can construct the three-dimensional shadow of a four-dimensional body, but not the tessaract itself. In the same sense, we are the shadows of four-dimensional beings. Thus, as man rises from the physical to the astral, he must develop his powers of visualization. Let us imagine a two-dimensional being who makes an [intense and repeated] effort to vividly imagine such a [three-dimensional] shadow image. When it then surrenders to the dream, then (...).

When you mentally build up the relationship between the third and fourth dimensions, the forces at work within you allow you to see into [real, not mathematical] four-dimensional space.

We will always be powerless in the higher world if we do not acquire the abilities [to see in the higher world] here [in the world of ordinary consciousness]. Just as a person in the womb develops eyes to see in the physical-sensual world, so must a person in the womb of the earth develop [supernatural] organs, then he will be born in the higher world [as a seer]. The development of the eyes in the womb is an [illuminating] example [of this process].

The cube would have to be constructed from the dimensions of length, width and height. The tessaract would have to be constructed from the dimensions of length, width, height and a fourth dimension.

As the plant grows, it breaks through three-dimensional space. Every being that lives in time breaks through the three [ordinary] dimensions. Time is the fourth dimension. It is invisibly contained in the three dimensions of ordinary space. However, you can only perceive it through clairvoyant power.

A moving point creates a line; when a line moves, a surface is created; and when a surface moves, a three-dimensional body is created. If we now let the three-dimensional space move, we have growth [and development]. This gives you four-dimensional space, time [projected into three-dimensional space as movement, growth, development].

[The geometric consideration of the structure of the three ordinary dimensions] can be found in real life. Time is perpendicular to the three dimensions, it is the fourth, and it grows. When you bring time to life within you, sensation arises. If you increase the time within you, move it within yourself, you have the sentient animal being, which in truth has five dimensions. The human being actually has six dimensions.

We have four dimensions in the etheric realm [astral plane], five dimensions in the astral realm [lower devachan] and six dimensions in the [upper] devachan.

Thus the [spiritual] manifoldness swells up to you. The devachan, as a shadow cast into the astral realm, gives us the astral body; the astral realm, as a shadow cast into the etheric realm, gives us the etheric body, and so on.

Time flows in one direction, which is the withering away of nature, and in the other direction it is the revival. The two points where they merge are birth and death.

The future is constantly coming towards us. If life only went in one direction, nothing new would ever come into being. Man also has genius – that is his future, his intuitions, which flow towards him. The processed past is [the stream coming from the other side; it determines] the essence [of how it has become so far].