SELF-DUALITY equal to its own opposite — a structure that is its own mirror under duality
Many structures have a dual — swap points for lines, AND for OR, 0 for 1, a shape for the shape of its holes. A structure is self-dual when it equals its own dual: the operation that inverts everything sends it back to itself. The majority function is self-dual (maj(¬a,¬b,¬c) = ¬maj(a,b,c) — flip all inputs and the output flips); a square (as a graph) is self-dual, and so is the tetrahedron; Maxwell's equations in vacuum are self-dual under swapping electric and magnetic fields. Self-duality is a fixed-point of the duality involution — the structures balanced exactly on the mirror between a thing and its opposite.
THE TECHNIQUE structure = its own dual; e.g. maj(¬a,¬b,¬c) = ¬maj(a,b,c)
The demo shows the majority function is self-dual: flipping all inputs flips the output: live demo
HISTORY & CREDIT self-dual codes, functions, polytopes
“A thing and its opposite are different.” — a self-dual structure is its own opposite: invert everything and it returns. The fixed point of duality. cited
duality · points↔lines, AND↔OR, 0↔1, faces↔vertices — an involution on structures. self-dual · equal to its own dual: majority (Boolean), the tetrahedron (polytope), self-dual codes. physics · vacuum Maxwell equations are self-dual under E↔B.
Invert everything — points, values, fields — and it comes back to itself. Balanced on the mirror between a thing and its opposite. self-duality
RECOMMEND FOR I-13 the self-dual majority, on the compiler
On the canonical compiler, maj(1,1,0)=1 and maj(0,0,1)=0=¬1 — flipping all inputs flips the output (self-dual):
$ i13 run sy_selfduality.i13 # maj(~a,~b,~c) == ~maj(a,b,c) ?
RUN OK · 43 step(s) · peak stack 4 · call depth 1
m = 1 md = 0
self_dual = 1 -- flip all inputs -> the output flips: self-dual
Recommend: self-duality is a structure equal to its own opposite — the fixed point of the duality involution — and i13 shows majority is self-dual. Not a keeper (self-duality is a fixed point of the duality map, and B43 established fixed-points collapse; the involution itself is the self-inverse axis). The dart that closes the batch's symmetry circle: the deepest symmetry is being your own mirror under the operation that inverts everything.