◄ WORLD V · SONNY 5DART 498 · a helldive at the net

THE 90-DEGREE ROTATION (x,y) → (−y, x) — exact rotation on the lattice, order four

A general rotation needs sine and cosine, but the quarter-turn is exact on the integer lattice: (x, y) → (−y, x) rotates a point 90° counter-clockwise with nothing but a negation and a swap. It is multiplication by the Gaussian integer i, and it has order four: apply it four times and you return to the start, cycling a point through four positions. Unlike a reflection (dart 499), it is not an involution — two turns give the opposite point, not the original. The finite rotation group, exact and trig-free.

THE TECHNIQUE (x,y) → (−y,x); four applications = identity (order 4)

The demo rotates (3,1) by 90° four times and returns to (3,1) — while two turns give the opposite point: live demo







  
HISTORY & CREDIT lattice rotation · Gaussian integer i

  
“Rotation always needs trig.” — the 90° turn is a negate-and-swap, exact on the lattice, multiplication by i. cited

  
the map · (x,y) → (−y,x) — 90° CCW, negate and swap.
order four · four turns = identity; two turns = the opposite point (not the original).
the algebra · it is multiplication by the Gaussian integer i (i⁴=1).

  
A quarter-turn done exactly by a negate and a swap — a finite-order rotation, no trig. finite order






  
RECOMMEND FOR I-13 the four positions, on the compiler

  
On the canonical compiler, (3,1) rotated four times returns to (3,1); two rotations give (−3,−1), not the original:

  
$ i13 run cg_rotation.i13   # (x,y) -> (-y,x)
RUN OK · 40 step(s) · peak stack 3 · call depth 0
(3,1) -> (-1,3) -> (-3,-1) -> (1,-3) -> (3,1)
order4 = 1          -- four turns = identity
not_involution = 1  -- two turns != the original

  
Recommend as a NULL — a finite-order group element. The 90° rotation is exact and elegant, but it is a symmetry-group action (order 4), which the corpus files under witnessed group structure (B39) — and it is not the seated self-inverse axis (that is order 2; this is order 4, note the distinction). No new invariant. NULL — the lattice quarter-turn, multiplication by i.