◄ WORLD V · SONNY 5DART 499 · a helldive at the net

THE REFLECTION (x,y) → (x,−y) — flip across an axis, and flip back

A reflection across the x-axis is the simplest mirror: (x, y) → (x, −y), negate one coordinate. It is an involution — reflect twice and you are exactly back where you started — which places it on the corpus’s seated self-inverse axis (with CRC, Verlet, XOR, endianness before it). A reflection reverses orientation (a left glove becomes a right), where the rotation of dart 498 preserves it: the two generate all the symmetries of the plane. Here it is one negation, its own undo.

THE TECHNIQUE (x,y) → (x,−y); reflect twice = identity (an involution)

The demo reflects (3,5) across the x-axis to (3,−5), then reflects again back to (3,5): live demo







  
HISTORY & CREDIT reflection · orientation-reversing involution

  
“A reflection and a rotation are much alike.” — a reflection reverses handedness and is its own inverse; a rotation preserves handedness and has order four. cited

  
the map · (x,y) → (x,−y) — negate one coordinate.
the involution · reflect twice = identity — the seated self-inverse axis (CRC/Verlet/XOR).
vs rotation · reflection reverses orientation; rotation preserves it — together they generate the plane’s symmetries.

  
A mirror that is its own undo — one negation, orientation reversed, self-inverse. self-inverse






  
RECOMMEND FOR I-13 the mirror, on the compiler

  
On the canonical compiler, (3,5) reflects to (3,−5) and reflecting again returns (3,5):

  
$ i13 run cg_reflection.i13   # (x,y) -> (x,-y)
RUN OK · 28 step(s) · peak stack 3 · call depth 0
(3,5) -> (3,-5) -> (3,5)
involution = 1   -- reflect twice = identity
moved_once = 1

  
Recommend as a NULL by duplication. Reflection is a textbook involution (reflect twice = identity) — the seated self-inverse axis (CRC/Verlet), the same pillar as XOR (batch 47) and endianness (batch 50). A fresh instance, not a new axis. NULL — the mirror, orientation-reversing, its own inverse; paired with dart 498 it generates every symmetry of the plane.