Mathematical reference
- Odd frequency
- r=3 cos(3θ) has 3 petals; θ from 0 to π traces the full rose
- Even frequency
- r=3 cos(2θ) has 4 petals
- Conditions on the count
- For a pure rose r=a cos(nθ), a≠0 and positive integer n: n petals if n is odd, 2n if even
Lesson transcript
This is the text of the narrated lesson. References to turning, highlighting or changing an object describe the linked interactive lesson; its demonstrations are not embedded in this reading page.
Follow one petal
Begin with r equals three cosine of three theta. The constant offset is zero. The first three sets the maximum magnitude of radius; the three inside cosine makes the radius oscillate three times as direction makes one full turn. Start at the rightmost tip, where theta is zero and radius is three.
Turn the direction through thirty degrees. The angle inside cosine has reached ninety degrees, where cosine is zero. The marker returns to the origin. We have followed half of the right-hand petal, from its outer tip inward. The faint curve is a guide; follow the bright trace to see which part has actually been visited.
Continue to sixty degrees. Cosine of three theta is now minus one, making radius minus three. The direction points up and right, but the negative sign sends the marker down and left. That point is another petal tip. Signed radius lets a relatively small turn of direction reach the other side of the picture.
Continue to ninety degrees. The radius reaches zero again and the marker returns to the origin. You have seen a complete lower-left petal between two origin visits. A petal is traced by moving out from the origin to a tip and back; the sign of the radius helps determine where that journey lies.
Three petals in half a turn
Start from the origin at ninety degrees and continue to one hundred twenty degrees. The radius rises to plus three. This time the marker follows the chosen direction to the upper-left tip. Compare this positive-radius tip with the negative-radius tip below it. Different signs can produce similar-looking petals in different directions.
Continue toward one hundred fifty degrees. The marker returns from that upper-left tip to the origin. Two complete petals are now visible in the bright trace, together with the first half of the right-hand petal. Keep track of the path traveled, rather than counting visits to the origin as if each visit were a new petal.
Finish the half-turn at one hundred eighty degrees. The radius becomes minus three. A leftward direction with a negative radius places the marker at the original right-hand tip. The missing half of that petal is now traced, and the entire three-petal rose is complete after only half a turn of direction.
Trace the same first half-turn once more in a single sweep. Watch the sign changes in the small graph while the point visits all three petals. The direction has advanced through only one hundred eighty degrees, yet the signed radius has supplied positions around the whole rose. A full turn is not always the shortest interval needed to draw a polar curve.
The second half retraces
Now continue from half a turn to a complete turn. Watch the marker go over the same three petals. No new petal appears. The coordinates have changed, but the geometric locations repeat. We need to distinguish the path of the moving marker from the set of points making up the finished curve.
Compare the starting tip again. At theta zero, radius plus three gives the point three units to the right. This is the first of two coordinate descriptions of the same tip. Keep its location in mind while we change both the direction and the sign.
At theta equal to pi, radius is minus three. The direction has reversed and the sign has reversed, so the two reversals put the point back at that same right-hand tip. The extra half-turn inside three theta is an odd multiple of pi, which changes cosine’s sign. This cancellation happens for every angle, not only at the tip.
Try frequency five. Follow a half-turn and the bright trace visits five petals. An odd integer frequency again reverses the radius sign when the direction reverses. That is why a pure cosine rose with positive odd integer frequency n has n petals and repeats its positions after a half-turn.
Even frequencies behave differently
Change to r equals three cosine of two theta. Start at the right-hand tip. Now the radius makes two complete oscillations during a full turn. The outline has four petals. We will connect those four petals to the fact that an even frequency does not reverse cosine’s sign after a half-turn.
Trace the first half-turn. At its end, the direction points left and the radius is still plus three, because cosine of two pi is one. The marker therefore reaches the left-hand tip, opposite its starting position. The direction reversed, but the radius sign did not cancel that reversal.
Continue through the second half-turn. The trace now completes the remaining parts of the four-petal rose and returns to its starting point. For an even integer frequency n, the positive and negative radius lobes produce distinct petals. The count is two n, so frequency two gives four petals.
Compare frequency four and trace one full turn. Eight petals appear. The same even-frequency reasoning applies: reversing direction leaves the radius unchanged, so opposite locations remain distinct. Frequency four gives eight petals, while the odd frequency five gave five. Increasing frequency by one does not always increase the petal count by one.
Sine rotates the pattern
Return to the three-petal cosine rose. Trace its half-turn and look at the petal directions: one tip points to the right and two point to the left. We will change cosine to sine while keeping the offset, amplitude and frequency fixed. This isolates orientation from the petal-count question.
Here is the sine version. Trace it through a half-turn. There are still three petals, but their directions have rotated by thirty degrees. Sine of three theta equals cosine of the quantity three theta minus pi over two. A quarter-cycle change inside the function corresponds to only a thirty-degree change in the geometric angle.
Look at the sine rose’s upper-right tip. It occurs at theta equal to thirty degrees, because three theta is ninety degrees and sine is one. Radius is plus three. The inner frequency divides the orientation change: a ninety-degree shift of the argument becomes ninety divided by three degrees of direction.
Compare a sine rose with frequency two. Its four petal directions are diagonal. The corresponding cosine rose pointed along the axes; the sine version is rotated forty-five degrees. The shift is ninety divided by the frequency. Sine changes orientation here while the odd-versus-even rule continues to determine the petal count.
Use the rule with its assumptions
Predict the result for frequency six before following the marker. Six is even, so the pure rose should have twelve petals. Watch one full-turn trace confirm those twelve distinct tips. The equation still has zero offset and a nonzero amplitude, the conditions under which we have been applying this count.
Now predict frequency five again. Five is odd, so a half-turn suffices to trace five petals. Compare that complete rose with the twelve-petal one. The petal rule is a relationship involving the parity of the frequency, not a general claim that larger coefficients always make a more complicated picture.
Add an offset of one while keeping frequency three. Trace the new curve. It no longer has the same simple three-petal form: shifting every radius changes which values are positive and negative. Our pure-rose rule does not apply unchanged. The constant term is a mathematical condition, not a detail we can ignore because the picture still looks flower-like.
Remove the offset and return to the pure three-petal rose. Follow the familiar half-turn. You can now explain its petals through signed radius and repeated positions. For a pure sine or cosine rose with positive integer frequency: odd n gives n petals; even n gives two n. Keep those assumptions attached to the rule when you explore other curves.