The Fourth Dimension
Questions and Answers IV
22 April 1909, Düsseldorf
(The wording of the Question has not been preserved.)
We are able to visualize three-dimensional space. An important theorem of the Platonic school is "God geometrizes." [Note 12] Basic geometric concepts awaken clairvoyant abilities. [Note 13] Positional geometry proves that the same point is everywhere on the circumference — the infinitely distant point on the right is the same as the starting point on the left. Thus, ultimately, the universe is a sphere, and we return to our starting point. [Note 14] Whenever I use geometric theorems, they turn into concepts at the borderline of normal conceptuality. [Note 15] Here, three-dimensional space returns us to our starting point. That is how in astral space, point A can work on point B without any connection between them. [Note 16]
We introduce materialism into theosophy when we make the mistake of assuming that matter becomes increasingly less dense as we move toward the spirit. This kind of thinking does not lead to the spirit, but ideas about the connection between point A and point B allow us to visualize the fourth dimension. As an example, we can think of the narrow waist of the gall wasp (Figure 63). [Note 17] What if the physical connection in the middle were absent and the two parts moved around together, connected only by astral activity? Now extend this concept to many spheres of activity (Figure 64) in higher-dimensional space.
"This statement cannot be found in Plato's works. It comes from the table conversations recounted by Plutarch that form one section of his Moralia. There, one participant in the conversations says, "God is constantly doing geometry — if this statement actually can be ascribed to Plato." Plutarch adds, This statement is nowhere to be found in Plato's writings, but there is sufficient evidence that it is his, and it is in harmony with his character" (Plutarch, Moralia, "Quaestiones convivales," book VIII, question 2; Stephanus 718c).
l3See also Rudolf Steiner's essay "Mathematik und Okkultismus (1904) in Philosophic und Anthroposophie (GA 35).
HSee the notes to the questions and answers of September 2, 1906, and June 28, 1908. The term positional geometry is an outdated name for synthetic projective geometry.
,5From the perspective of projective geometry, all theorems in Euclidean geometry having only to do with the position and arrangements of points, lines, and planes (and not with any measurements) are seen as special or borderline" instances of general projective theorems.
l6Two points A and B of a projective straight line s separate the line into two segments (Figure 91), one of which includes the distant point of line s. In projective geometry, both segments are considered to connect points A and B. In Euclidean geometry, however, only the segment that does not include the distant point of the straight line g is considered a connection between A and B.
,7Gall wasp: Similar discussions about the possibility of individual parts of a whole affecting each other without being spatially connected also are found in Rudolf Steiners lectures of October 22, 1906, in Berlin (in GA 96) and March 22, 1922, in Dornach (in GA 222). None of the many subspecies of gall wasps described in the scientific literature match Rudolf Steiner's description, but a long, stem-like connection between the head and the abdomen occurs in several species of grave wasps, especially in the sand wasp subspecies. The note taker may have misheard the name of this insect.