The Fourth Dimension

GA 324a — 31 March 1905, Berlin

Second Lecture

Today I want to discuss some elementary aspects of the idea of multidimensional space [among other things, in connection with the] spirited Hinton.

You will recall how we arrived at the concept of multi-dimensional space, having considered the zeroth dimension [last time]. I would like to briefly repeat the ideas of how we can move from two- to three-dimensional space.

What do we mean by a symmetrical behavior? How do I align a red and a blue [flat figure, which are mirror images of each other]?

With two halves of a circle, I can do this relatively easily by sliding the red [half] circle into the blue one (Figure 10).

This is not so easy in the following [mirror]symmetrical figure (Figure 11). I cannot make the red and blue parts coincide [in the plane], no matter how I try to slide the red into the blue.

But there is a way [to achieve this anyway]: if you step out of the board, that is, out of the second dimension [and use the third dimension, in other words, if you] place the blue figure on the red one [by rotating it through the space around the mirror axis].

The same applies to a pair of gloves: I cannot match one with the other without stepping out of [three-dimensional] space. You have to go through the fourth dimension.

Last time I said that in order to develop an understanding of the fourth dimension, you have to make [the relationships in] space fluid, thereby creating conditions similar to those you have when moving from the second to the third dimension. In the last lesson, we created spatial structures out of paper strips that intertwined. Such interweaving causes certain complications. This is not a game, but such inter-weavings occur in nature all the time. Anyone who reflects on natural processes knows that such inter-weavings really do occur in nature. Material bodies move in such intertwined spatial structures. These movements are endowed with forces, so that the forces also intertwine. Take the movement of the earth around the sun and then the movement of the moon around the earth. The moon moves in an orbit that is itself wound around the earth's orbit around the sun. It thus describes a spiral around a circular line. Because of the movement of the sun, the moon describes another spiral around this. The result is very complicated lines of force that extend through the whole space.

The heavenly bodies behave in relation to each other like the intertwined strips of paper [by Simony, which we looked at last time]. We have to keep in mind that we are dealing with complicated spatial concepts that we can only understand if we do not let them become rigid. If we want to grasp space [in its essence], [we must first conceive it as rigid, but then] make it completely fluid again. [You have to go as far as zero]; the [living] point can be found in it.

Let us once again visualize the structure of the dimensions]. The point is zero-dimensional, the line is one-dimensional, the surface is two-dimensional and the body is three-dimensional. The cube has the three dimensions: height, width and depth. How do the spatial structures [of different dimensions] relate to each other? Imagine that you are a straight line, that you have only one dimension, that you can only move along a straight line. If such beings existed, what would their concept of space be like? Such beings would not perceive one-dimensionality in themselves, but would only be able to imagine points wherever they went. Because in a straight line, if we want to draw something in it, there are only points. A two-dimensional being would only encounter lines, so it would only perceive one-dimensional beings. [A three-dimensional being like] the cube would perceive two-dimensional beings, but could not perceive its [own] three dimensions.

Now, humans can perceive their three dimensions. If we reason correctly, we must say to ourselves: Just as a one-dimensional being can only perceive points, a two-dimensional being only straight lines, and a three-dimensional being only surfaces, so a being that perceives three dimensions must itself be a four-dimensional being. The fact that humans can define external beings in terms of three dimensions, can [deal with] spaces of three dimensions, means that they must be four-dimensional. And just as a cube can perceive only two dimensions and not its third, so it is clear that man cannot perceive the fourth dimension in which he lives. Thus we have shown [that man must be a four-dimensional being]. We swim in the sea [of the fourth dimension, like ice in water].

Let us return once more to the consideration of mirror images (Figure 11). This vertical line represents the cross-section of a mirror. The mirror reflects an image [of the figure on the left]. The process of reflection points beyond the two dimensions into the third dimension. [To understand the direct and continuous connection between the mirror image and the original, we have to add a third dimension to the two.

[Now let us consider the relationship between external space and internal representation.] The cube here apart from me [appears as] an idea in me (Figure 12). The idea [of the cube] is related to the cube like a' mirror image to the original. Our sensory apparatus [creates an imagined image of the cube. If you want to align this with the original cube, you have to go through the fourth dimension. Just as the third dimension has to be transitioned to (during the continuous execution of the two-dimensional) mirroring process, our sensory apparatus has to be four-dimensional if it is to be able to establish a [direct] connection [between the imagined image and the external object].

If you only imagined [two-dimensionally], you would [only] have a dream image in front of you, but you would have no idea that there is an object outside. Our imagination is a direct inversion of our ability to imagine [external objects by means of] four-dimensional space.

The human being in the astral state [during earlier stages of human evolution] was only a dreamer, he had only such ascending dream images.” He then passed from the astral realm to physical space. Thus we have mathematically defined the transition from the astral to the [physical-] material being. Before this transition occurred, the astral human being was a three-dimensional being and therefore could not extend his [two-dimensional] ideas to the objective [three-dimensional physical-material] world. But when he [himself] became physical-material, he still acquired the fourth dimension [and could therefore also experience three-dimensionally].

Due to the peculiar design of our sensory apparatus, we are able to align our perceptions with external objects. By relating our perceptions to external things, we pass through four-dimensional space, imposing the perception on the external object.

How would things appear if we could see from the other side, if we could enter into things and see them from there? To do that, we would have to pass through the fourth dimension.

The astral world itself is not a world of four dimensions. But the astral world together with its reflection in the physical world is four-dimensional. Anyone who is able to see the astral world and the physical world at the same time lives in four-dimensional space. The relationship of our physical world to the astral world is a four-dimensional one.

One must learn to understand the difference between a point and a sphere. In reality, this point would not be passive, but a point radiating light in all directions (Figure 13).

What would be the opposite of such a point? Just as there is an opposite to a line that goes from left to right, namely a line that goes from right to left, there is also an opposite to the point. We imagine an enormous sphere, in reality of infinite size, that radiates darkness from all sides, but now inwards (Figure 14). This sphere is the opposite of the point.

These are two real opposites: the point radiating light and infinite space, which is not a neutral dark entity, but one that floods space with darkness from all sides. [As a contrast, this results in] a source of darkness and a source of light. We know that a straight line that extends to infinity returns to the same point from the other side. Likewise, it is with a point that radiates light in all directions. This light comes back [from infinity] as its opposite, as darkness.

Now let us consider the opposite case. Take the point as the source of darkness. The opposite is a space that radiates light from all sides.

As was recently demonstrated [in the previous lecture], the point behaves in this way; it does not disappear [into infinity, it returns from the other side] (Figure 15).

[Similarly, when a point expands or radiates out, it does not lose itself in infinity; it returns from infinity as a sphere.] The sphere, the spherical, is the opposite of the point. Space lives in the point. The point is the opposite of space.

What is the opposite of a cube? Nothing other than the whole of infinite space, except for the piece that is cut out here [by the cube]. So we have to imagine the [total] cube as infinite space plus its opposite. We cannot do without polarities if we want to imagine the world as powerfully dynamic. [Only in this way] do we have things in their life.

If the occultist were to imagine the cube as red, the space around it would be green, because red is the complementary color of green. The occultist not only has simple ideas for himself, he has vivid ideas, not abstract, dead ideas. The occultist must enter into things from within himself. Our ideas are dead, while the things in the world are alive. We do not live with our abstract ideas in the things themselves. So we have to imagine the infinite space in the corresponding complementary color to the radiating star. By doing such exercises, you can train your thinking and gain confidence in how to imagine dimensions.

You know that the square is a two-dimensional spatial quantity. A square composed of four red- and blue-shaded sub-squares is a surface that radiates differently in different directions (Figure 16). The ability to radiate differently in different directions is a three-dimensional ability. So here we have the three dimensions of length, width and radiance.

What we did here with the surface, we also think of as being done for the cube. Just as the square above was made up of four sub-squares, we can imagine the cube as being made up of eight sub-cubes (Figure 17). This initially gives us the three dimensions of height, width and depth. Within each sub-cube, we can then distinguish a specific light-emitting capacity, which results in a further dimension in addition to height, width and depth: the radiation capacity.

You can imagine a square made up of four sub-squares, a cube made up of eight different sub-cubes. And now imagine a body that is not a cube, but has a fourth dimension. We have created the possibility of understanding this through radiative capacity. If each [of the eight partial cubes] has a different radiating power, then if I have only the one cube that radiates only in one direction, if I want to obtain the cube that radiates in all directions, I have to add another one on the left, doubling it with an opposite one, I have to put it together out of 16 cubes.

Next lesson we will have the opportunity to consider how we can think of a multidimensional space.