◄ WORLD V · SONNY 5DART 398 · a helldive at the net

THE ORBIT everywhere a symmetry can send you — the reach of a group on a point

The orbit of a thing is everywhere the group can send it: all the distinct results of applying every symmetry. Rotate a necklace of distinct beads and you get as many arrangements as beads; rotate one of period-2 pattern and you get only two. The orbit's size is governed by the orbit–stabilizer theorem (|orbit| × |stabilizer| = |group|): the more symmetries fix a thing, the smaller its orbit. Orbits partition a set into classes of things the symmetry treats as equivalent — which is exactly how symmetry defines “the same up to a transformation,” from chemistry's equivalent atoms to a puzzle's identical states.

THE TECHNIQUE orbit = { g·x : g in G } ; size = |G| / |stabilizer|

The demo counts the distinct cyclic rotations (the orbit size = fundamental period) of three lists: live demo







  
HISTORY & CREDIT orbit–stabilizer · Lagrange

  
“A symmetric object has many forms.” — exactly as many as its orbit, and the more symmetries fix it the fewer: |orbit|×|stabilizer|=|group|. Symmetry counts. cited

  
the orbit · {g·x : g∈G} — all images of x.
orbit–stabilizer · |orbit|·|stab| = |G| (a corollary of Lagrange's theorem).
partition · orbits split a set into equivalence classes — “same up to a symmetry.”

  
Everywhere the group can send a thing — more symmetry fixing it, smaller its reach. How symmetry decides what counts as the same. orbit–stabilizer






  
RECOMMEND FOR I-13 orbit sizes, on the compiler

  
On the canonical compiler, the cyclic orbit (fundamental period) is 4 for all-distinct, 2 for a period-2 pattern, 1 for a constant:

  
$ i13 run sy_orbit.i13   # distinct cyclic rotations = fundamental period
RUN OK · 1139 step(s) · peak stack 8 · call depth 8
orbit3 = 4       -- [1,2,3,3] : all rotations distinct
orbit_sq = 2     -- [1,2,1,2] : two distinct
orbit_flat = 1   -- [1,1,1,1] : one (fully symmetric)

  
Recommend: the orbit is the reach of a symmetry — how many forms a thing has — and i13 counts 4/2/1 as the pattern gains symmetry. Not a keeper (orbit size is a counting theorem, witnessed; orbit–stabilizer was batch 38's dart). The dart that shows symmetry partitions: things one orbit apart are “the same up to a transformation,” which is the whole point of caring about symmetry at all.