dyas · a COMPLEX duality · exact
THE CONJUGATE
z = a + bi ⟺ z̄ = a − bi · z · z̄ = |z|² (real)
Every complex number has a twin — its conjugate, the same point reflected across the real line. Do it twice and you return: z ⟺ z̄ ⟺ z. And the fixed points of this mirror — the numbers equal to their own reflection — are exactly the reals: the axis is both the mirror and the set it cannot move.
drag anywhere · z is carbon-warm, z̄ silicon-cool · the gold line (reals) is the self-conjugate axis
the mirror and its fixed line
Conjugation is an involution: apply it twice and every number comes home, exact to the bit. Its product with itself, z·z̄, always lands on the real axis as |z|² — the imaginary part cancels perfectly. And the numbers the mirror leaves untouched are the reals themselves — the self-dual set, the [[the-monad|Monad]] of this reflection. Two faces (z, z̄), one axis where they meet.
green · exact
Computed live and exact (to floating-point): (z̄)̄ = z (involution), Im(z·z̄) = 0 and z·z̄ = |z|², and z = z̄ iff z is real. Node-verified across test points at 0.0e+0 error.
amber · the reading
"Twin / mirror / the axis it cannot move" are chosen images over an exact algebraic fact. Assigning z to carbon and z̄ to silicon is a framing (the algebra has no preferred twin) — the dyad is symmetric by nature.
red · scope
This is complex conjugation on ℂ. Conjugation generalizes (quaternions, Galois automorphisms) with its own rules; here it is the clean, exact base case — a duality whose fixed set is a whole line, not a single point.