The square and the circle
Shao Yong's diagram, the one Leibniz saw
Tap a hexagram in the square or the circle, and apply the eight symmetries to follow it.
Shao Yong laid the 64 hexagrams out at once in a 8×8 square and in a circle, both in binary order. The Jesuit Joachim Bouvet sent this diagram to Leibniz (who had already invented binary arithmetic) in a letter of 4 November 1701 that Leibniz received on 1 April 1703; he recognised the identity at once: reading the square in order is counting from 0 to 63. He cited it in the Explication of that same year.
The column fixes the lower trigram and the row the upper one; the value of each cell is (inferior × 8) + superior. Walking the cells in order 0, 1, 2, … 63 lights up, line by line, the binary count. Kun (0) bottom left, Qian (63) top right.
On the hexagrams: changes nothing.
fixed points: 64
Derived from the real layout of the square (column = lower trigram, row = upper inverted) and verified over the 64 hexagrams: the geometry of the paper and the algebra of the trigrams are the same group. The 8 symmetries split the 64 hexagrams into 10 orbits (Burnside: the average number of fixed points).
In the circle, the opposite of each hexagram (dui, its six lines inverted) is its exact reflection over the vertical Qian-Kun axis. Its antipode (the diametrically opposite point) is, instead, only flipping the bottom line.
The 64 cells of the 8x8 square are exactly 0 to 63; the 8 geometric symmetries of the square are 8 trigram operations.
verificar_shao_yongShao Yong's diagram reached Europe through Bouvet: his letter is dated 4 November 1701, Leibniz received it on 1 April 1703 and published the Explication that same year.
The types of claim and the full bibliography (APA) are in Foundations.