◄ WORLD V · SONNY 5DART 400 · a helldive at the net

ROTATIONAL SYMMETRY the same after a turn — cyclic symmetry, unchanged by a rotation

A thing has rotational symmetry when a turn leaves it unchanged: a square matches itself every 90°, a repeating pattern matches after a cyclic shift. Its symmetry group is cyclic (Zn) — the rotations that fix it. A period-2 pattern like [1,2,1,2] is fixed by a shift of 2 but not of 1, so its rotational symmetry is order 2. Rotational symmetry is why crystals come in a fixed set of shapes (the crystallographic restriction: only 2-, 3-, 4-, and 6-fold rotations can tile the plane — there are no 5-fold crystals), and it is the difference between a snowflake (6-fold) and a starfish (5-fold, so it cannot tile).

THE TECHNIQUE a cyclic shift by k leaves x unchanged — the group is Zn/k

The demo tests [1,2,1,2] against a shift of 2 (fixed) and 1 (not) — order-2 rotational symmetry: live demo







  
HISTORY & CREDIT cyclic symmetry · crystallographic restriction

  
“Any rotational symmetry can tile space.” — only 2-, 3-, 4-, 6-fold can (the crystallographic restriction); there are no 5-fold crystals, which is why quasicrystals were a shock. cited

  
the relation · a cyclic shift by k fixes x; the group is cyclic Zn.
crystallographic restriction · only 2/3/4/6-fold rotations tile the plane periodically.
quasicrystals · Shechtman 1982 — 5-fold order without periodicity (Nobel 2011).

  
The same after a turn — and only certain turns can fill space. Why snowflakes are six-fold and starfish never tile. cyclic / Zn






  
RECOMMEND FOR I-13 order-2 rotational symmetry, on the compiler

  
On the canonical compiler, [1,2,1,2] is fixed by a shift of 2 (=1) but not of 1 — rotational symmetry of order 2:

  
$ i13 run sy_rotationalsymmetry.i13   # rot(x,k) == x ?
RUN OK · 375 step(s) · peak stack 8 · call depth 5
rot2_fixed = 1     -- [1,2,1,2] : fixed by a shift of 2
rot1_fixed = 0     -- not by a shift of 1

  
Recommend: rotational symmetry is the same-after-a-turn — a cyclic group fixing the thing — and i13 finds order 2 in [1,2,1,2]. Not a keeper (fixed-under-rotation is a witnessed property, a recognizer). The dart that carries the crystallographic restriction: only 2/3/4/6-fold turns tile the plane, so there are no 5-fold crystals — the rule broken, gloriously, by quasicrystals.