The Fourth Dimension
GA 324a
Mathematical considerations of multidimensional spaces can be found as early as the mid-19th century. These only came to the attention of the general public through the combination of spiritualist experiments with the question of the existence of a four-dimensional space. Easy-to-read introductions to the geometry of four-dimensional structures, some of which were written in a novel-like style, helped to spread knowledge of the related problem.
These lectures address the problem of the real existence of a fourth dimension, which was much discussed in the cultural world at the time. This topic was of particular interest to members of the Theosophical Society since the 1880s and 1890s in connection with reports of spiritualist experiments conducted by some renowned scholars with more or less reputable mediums. Steiner does not, however, address these spiritualist components but develops the theme of the fourth and higher dimensions from a very fundamental point of view. A great deal of space is devoted to the geometry of four-dimensional bodies. This serves primarily as a preparatory training for the development of spiritual knowledge. Whether such mathematical concepts are realistic can only be decided by the methods of spiritual science. He describes how a fourth, fifth and sixth dimension and their projection into the physical world is to be thought of from this point of view.
The exact circumstances of the lectures summarized in this volume are not known. However, it can be assumed that Rudolf Steiner was approached from theosophical circles with the request to comment on the problem of the fourth dimension. The lectures given to members of the Theosophical Society were not intended for a scientifically or mathematically trained audience, but nevertheless for an interested audience.
First Lecture | March 24, 1905 | |
Mathematical thinking and reality. The dimensions of space. Movement as a means of passing from lower to higher dimensions. Mirror-image symmetry. Analogies for the relationship between the outer world and internal sensation: bending ever larger line segments into circles, seal and sealing wax. The fourth dimension as a conceptual possibility and as reality. Enlivening our visualization of space: Oscar Simony's models. | ||
Second Lecture | March 31, 1905 | |
Thoughts on four-dimensional space in connection with Hinton's work. Symmetrical relationships. "Loops" in space as real natural processes and forces, for example, the movement of the Moon and Earth around the Sun. Building up the dimensions. Human beings as four-dimensional beings,- at earlier stages of our evolution, we were three-dimensional. The astral world. Point and periphery,- a point radiating light outward is the opposite of a sphere radiating darkness toward the center. The cube and its opposite. Radiant ability as an additional dimension,- its application to squares and cubes. | ||
Third Lecture | May 17, 1905 | |
Studying four-dimensional space as preparation for understanding the astral world and other forms of higher existence. Characteristic attributes of the astral world-, numbers, spatial figures, and time sequences must be read symmetrically, that is, in reverse or as their own mirror images. Morality also appears in inverted or mirror-image form. The periphery is the center. Human life as a backup in the two streams of time running from the past and the future. The threshold, as an astral experience of the panorama a future evolution, includes the question, Do you choose to enter? Kamaloka reveals the unpurified animal nature of the human being; this is the deeper meaning of the doctrine of transmigration of souls. A square on the physical and mental level. The physical square as a stoppage in two pairs of opposing streams. A cube on the physical and mental level. Positive and negative dimensions. The astral world is four dimensional. The animal as a stoppage in the opposing streams represented by the plant and the human being. | ||
Fourth Lecture | May 24, 1905 | |
Exercises in representing three-dimensional figures in two dimensions, with reference to Hinton. Unfolding the three dimensions of a cube,- using colors to represent them. Representing the third dimension in a plane by moving a bi-colored square through a third color. Transferring this operation to the representation of a four dimensional figure, the tessaract. Unfolding a tessaract compared with unfolding a cube. The alchemical mystery and a true view of four-dimensional space. Meditative visualization of mercury, sulfur. Astral substance. | ||
Fifth Lecture | May 31, 1905 | |
Unfolding a cube as a new analogy for three-dimensional representation of a tessaract (four-dimensional cube). Analogy as a method of developing a mental image of four-dimensional figures. Halving the number of surfaces in an octahedron produces a tetrahedron,- this operation cannot be performed on a cube. The geometric properties of the rhombic dodecahedron compared with the cube and the tetrahedron/octahedron. The cube as the counterpart of three-dimensional space. Curved figures bounding two- and three-dimensional figures: curved squares and curved cubes. An ordinary cube is the result of flattening a curved cube. Reversing this operation and curving a three-dimensional figure can result in a four-dimensional figure. | ||
Sixth Lecture | June 7, 1905 | |
A hexagon is the projection of a cube into two-dimensional space, — a rhombic dodecahedron is the projection of a tessaract into three-dimensional space. The axes of the cube and rhombic dodecahedron. Plato's cave metaphor as an image of the relationship between four-dimensional reality and three-dimensional space. Movement or time as the expression and manifestation of life, the fourth dimension. Crystals have planes for boundaries, while living things have spherical boundaries. Destroying a living thing's fourth dimension results in a static three-dimensional image. The fifth dimension, which results from encounters between four-dimensional beings, manifests in the third dimension as sensory activity. Self-awareness is the projection of the sixth dimension into the three-dimensional physical world. What Moses experienced on Mount Sinai is an example of a real four-dimensional being with two ordinary dimensions plus the two higher dimensions of time and sensory activity. The develop | ||
Four-Dimensional Space | November 7, 1905 | |
Creating dimensions through movement. Transforming a circle into a straight line. The importance of recent synthetic, projective geometry in considering space correctly. Space is self-contained. Twists in closed curves (strips of paper) as an example of intertwining dimensions. In reality, the movements of the Moon and Earth around the Sun are similarly intertwined. Enlivening our view of space. Unfolding the cube in a plane and the tessaract in three-dimensional space. A hexagon is a projection of a cube,- a rhombic dodecahedron is a projection of a tessaract. The transition to reality. Time, movement, and development as expressions of the fourth dimension in plants. When time itself becomes alive, sensory ability emerges as the expression of the fifth dimension in animals. Human beings are six-dimensional beings. | ||
On Higher-Dimensional Space | October 22, 1908 | |
Mathematicians can do no more than discuss the possibility of higher-dimensional space. The cube's three dimensions of length, width, and height. What is a plane surface? A merely arithmetical transition to higher dimensions does not lead to reality. Attempting to understand space — for example, infinity — on the basis of numbers leads to confusion. Numbers have no relationship or a neutral relationship to space. Observing repeated disappearances and reappearances points to the existence of a fourth dimension. Refuting a materialistic argument. Unfolding the boundaries of a square and a cube. Unfolding the eight cubes of a tessaract. |