◄ WORLD V · SONNY 5DART 399 · a helldive at the net

REFLECTION SYMMETRY the same in a mirror — a palindrome is its own reflection

The simplest symmetry: reflection. A thing has mirror symmetry when it equals its own reflection — a palindrome reads the same reversed, a butterfly matches its mirror, an even function satisfies f(−x)=f(x). Reflection is an involution: do it twice and you are back (dart 401), so the symmetry group is the smallest non-trivial one, Z₂. It is the corpus's deepest motif — the | < { [ ~ ~ ] } > | nesting, the A→Z→A palindrome, the whole 5-world mirror — and the reason a mirror-symmetric object carries only half as much information as it looks: the second half is the first, reflected.

THE TECHNIQUE x = reverse(x) — mirror-fixed; the group is Z₂

The demo checks two lists for reflection symmetry: [1,2,1] is a palindrome, [1,2,3] is not: live demo







  
HISTORY & CREDIT reflection · bilateral symmetry

  
“A mirror-symmetric thing is twice the structure.” — it is half: the second side is the first, reflected, so it carries only half the information. cited

  
the relation · x = reverse(x) (a palindrome); f(−x)=f(x) (an even function).
the group · Z₂ — reflect twice = identity (an involution, dart 401).
the corpus · the A→Z→A palindrome, the cell nesting, the 5-world mirror.

  
The same in a mirror — half the structure, doubled by reflection. The corpus's oldest shape. reflection / Z₂






  
RECOMMEND FOR I-13 the palindrome check, on the compiler

  
On the canonical compiler, [1,2,1] is reflection-symmetric (=1), [1,2,3] is not:

  
$ i13 run sy_reflectionsymmetry.i13   # x == reverse(x) ?
RUN OK · 64 step(s) · peak stack 5 · call depth 2
sym = 1     -- [1,2,1] : a palindrome
asym = 0    -- [1,2,3] : not mirror-symmetric

  
Recommend: reflection is the corpus's oldest symmetry — a palindrome is its own mirror — and i13 checks it. Not a keeper (mirror-symmetry is a property a thing has, checked by a predicate — a recognizer, per B41; and the reflection group is the self-inverse axis, dart 401). The dart that names the shape under the whole SONNY 5 palindrome, and the reason a symmetric thing is half its apparent size.