1 · Observe

The curve is traced as θ sweeps 0 → 4π. Watch how the radius grows and shrinks.

2 · Manipulate

3 · Predict

Set the rose with n = 4. Before tracing: how many petals do you expect? Now change n to 5 — does the petal count follow the same rule for odd n?

4 · See

5 · Explain

In polar coordinates each angle θ gets its own radius r. A rose r = a·cos(nθ) has n petals if n is odd, 2n petals if n is even — because the curve must retrace itself before closing. A limaçon r = a + b·cosθ grows a dimple (a < 2b) or an inner loop (a < b) as b overtakes a.