137 - Arthur I. Miller [110]
Because “every ‘correct’ solution (i.e., meeting the demands of nature) must contain 4 as well as 3,” Pauli tried different approaches including 16 (4 × 4) which can be mathematically tied to 12 (the number of signs in the zodiac). He explained why 12 and 16 were related: “The professor further calculates numbers. In the sign of the 12. 12:16 = 3:4—a problem with thorns and horns.” Dividing each term in the ratio 12:16 by four gives the ratio 3:4.
But there was still something missing in the number twelve and its relationship with three and four, something Pauli could not quite put his finger on. He became weary and set aside the problem.
That night he had a dream. In his dream three others are present. One is a Chinese woman—now elevated to a “Sophia,” a seer or wise woman. She tells Pauli, “You must play every conceivable combination of chess.” Chess involves the opposition of black and white and thus of dark and light, the two animas. Dark is compassion and feeling, light is rationalism. It is like the teacher’s advice in The Piano Lesson, that every possible melody can be played on the black and white keys: “It is only a question of knowing how to play.” In his dream Pauli works on the problem of how to make four emerge from two, three, and six, but still can’t solve it. He suddenly wakes up.
Then, in a waking vision, he hit on the solution. “This I will never forget. The experience had a numinostic character (obviously of archetypal nature),” he wrote in wonder, caught up in the moment of enlightenment. “In my youth I often had such experiences with physics problems,” he added.
In the vision, a voice—possibly the Chinese woman’s—wakes him from his dream. The voice tells him the solution in clear tones:
“In your drawings [the six-sided stars in Pauli’s previous letter to von Franz] one element is perfectly correct and another transitory and false. It is correct that the lines number six, but it is false to draw six points. See here”—and I saw a square with clearly marked diagonals. “Can you now see finally the 4 and the 6? Namely, 4 points and 6 lines—or 6 pairs* out of 4 points. They are the same six lines that exist in the I Ching. There the 6, containing 3 as a latent factor, is correct. Now observe the square more closely: 4 of the lines are of equal length, the other two are longer—they are irrationally related, as you know from mathematics. There is no figure with 4 points and 6 equal lines. For this reason symmetry cannot be statically produced and a dance results. The coniunctio refers to the exchange of places during this dance. One can also speak of a game of rhythms and rotations.† Therefore the 3, already contained in a latent form in the square, must be dynamically expressed.”
The square where the dance of diagonals occurred in Pauli’s “waking vision.”
Finally here was the solution to the problem that had been nagging at Pauli for so long. The four, or quaternity, emerged intact, while the three was also already contained “as a latent factor” in the square. This is the way it is in the I Ching, in which each hexagram used for divination is made up of six lines subdivided into two three-line segments (trigrams).
In mathematics the diagonals of a square cannot be expressed as whole numbers. Mathematicians refer to the relationship of the diagonals to the sides of a square as “irrational.” Pauli extended the meaning of this term to psychology. In the dance of the diagonals, three and four are both present in a coniunctio resulting from “a game of rhythms and rotations” in which diagonals and sides transform into one another. Pauli recognized the influence of the I Ching—with