From the King Wen sequence to the wave — the derivation, step by step

The I Ching contains 64 hexagrams — every possible stack of six lines, each line either solid (yang) or broken (yin). The oldest known ordering of these 64 figures is the King Wen sequence, and it has a strange internal discipline: the hexagrams come in pairs, where the second of each pair is the first turned upside down. Eight hexagrams look the same upside down; those are instead paired with their opposite — every line flipped.

McKenna’s starting intuition was that this sequence is not decorative. If the ancient arrangers encoded something deliberate in the ordering, then the transitions between hexagrams — how much changes from one to the next — might carry information. That question is the seed of everything that follows.

Walk the sequence and count, at each step, how many of the six lines change to turn each hexagram into the next (the sequence wraps: hexagram 64 steps back to hexagram 1). The result is a series of 64 small integers — the first order of difference. Graphed, it looks random. It is not.

The properties McKenna found here convinced him the sequence was engineered:

In his words, these regularities are not coincidences but the artifacts of conscious intent — the fingerprints, he believed, of the sequence’s ancient designers.

Now the first strange move. Take the graph of Figure 1, rotate a copy of it 180° in the plane, and lay it back over the original. McKenna found that the two curves touch at both ends — they close, like a sealed vessel — and that hexagrams facing each other across the closed figure always have position numbers summing to 64.

This closed, bi-directional figure — 384 lines of change — is the object everything else is built from. McKenna treated it the way the I Ching treats a single line: as a unit that can itself be stacked and scaled.

A hexagram is six lines — but it is also two trigrams, and it is also one whole figure. McKenna rebuilt that anatomy out of his double wave: six copies end-to-end at the line scale, two copies (each three times larger) at the trigram scale, and one copy (six times larger) spanning the whole. All share one origin and one end point.

He also noticed the arithmetic of the object rhymed with the sky: 384 subunits is almost exactly 13 lunar months (29.53 × 13 = 383.89 days) — suggesting, to him, that the figure had once been a lunar calendar.

To turn geometry into numbers, every feature of the tri-level wave was scored: the skew of each line segment (thirteen discrete types, signed by direction), the divergence between the forward and backward curves (seven types of gap and overlap), and the scale of the level it sits on (trigram values ×3, hexagram values ×6). Collapsing all of it into a single line of numbers yields the data set at the heart of the theory: 384 integer values.

for 1 ≤ j ≤ 32:
    angle_lin[j] = −angle_lin[j]      // footnote 22: the "half twist"

One thing remains to make the object a map: pinning it to the calendar. The wave’s mathematics fix its shape but not its position — some date must be chosen as the zero point where the final cycle ends, and every other date follows from it. McKenna screened the wave against history and chose the fit he found most convincing: a 67.3-year cycle beginning at Hiroshima lands in mid-November 2012, which he then nudged to 21 December 2012 to meet the end of the Maya Long Count.

That is the honest shape of the final step: the endpoint was selected, not derived — a best fit, chosen by eye, on a wave whose own numbers rest on the half-twist. The finished object is the one you can drag through on the node page: novelty rising as the wave falls, all the way down to a chosen zero.