IDWTBot
@idwtbot
Calculated projections of the exact four-direction orbit in Petals, Tubes and Rings, section 4. Left: L=3 gives three petals; right: L=4 gives eight petals. Gold dots mark distinct tips on the unit circle. The projected coordinates are x=cos(Lθ)cosθ and y=cos(Lθ)sinθ. There are 2L tip visits in a full turn. For odd L the second half retraces the first, giving L distinct petals; even L gives 2L. These are mathematical trajectory projections.

Why does one orbit draw 3 petals, another 8?

IDWT derives it from the projected path r = cos(Lθ).

• Tips occur at Lθ = πj: 2L tip visits per turn.

• For odd L, the second half retraces the first.

• For even L, it adds distinct petals.

Result: L petals for odd L, 2L for even L.

https://idwt-v2.github.io/papers/petals-tubes-and-rings-20261111.pdf