The symmetries of the hypercube
Orbits, palindromes and the nuclear map
Tap a hexagram to walk its orbit under reversal (fan) and complement (dui).
Two millennia-old operations are involutions: the flip (fan 反, reversing the order of the lines) and the opposite (dui 对, complementing each line). They commute, so together with the identity they form the Klein group V₄. That group splits the 64 hexagrams into 20 orbits: families of hexagrams equivalent under symmetry.
Orbit of Dǐng: 4 hexagrams · 4 · 50 · 3 · 49.
A hexagram is a palindrome when the flip leaves it unchanged (1 = 6, 2 = 5, 3 = 4). There are exactly 8. And it turns out they are, one to one, the hexagrams of the 4 King Wen pairs that (being indistinguishable by flipping) are paired by opposite (dui) instead of by fan. Verified: all 8 coincide.
The nuclear hexagram (hu gua) takes the inner lines to form a new one. Unlike fan and dui, it is not reversible: applied again and again, every hexagram ends up falling into one of 3 attractors: 2 fixed points and a cycle of 2 (4 attractor hexagrams), in at most 2 steps.
As a graph, the hypercube has 7 eigenvalues: 6 − 2k with multiplicity C(6,k). They are its harmonic modes, from the global vibration (+6) to the most alternating one (−6), and their multiplicities add up to the 64 hexagrams.
eigenvalue (above: multiplicity) · Σ multiplicities = 64
(There are also 8 antipalindromes, where fan = dui: the other fixed points, this time of flip-plus-opposite.)
The Klein group generated by the flip (fan) and the opposite (dui) splits the 64 into 20 orbits; the 8 palindromes are the special King Wen pairs; the nuclear map falls into 3 attractors.
verificar_simetriasThe types of claim and the full bibliography (APA) are in Foundations.