Wavelace

Rational Rose

Polar · r = f(θ, t)

r = cos((n/d) · θ)

Open in the app The dials and keys named below are the app's.

What it draws

A rose is r = cos(k·θ), and the usual ones give k a whole number. Here k is the fraction n/d, and both parts are sliders. Nothing depends on the clock, so the figure stands still.

At the opening values the fraction is 3/2. The cosine reaches 1 whenever 1.5·θ is a multiple of 2π, which is every 240°, and it returns to 0 a quarter of that either side, where the curve passes through the origin. Three of the six petals are drawn where the cosine is positive. The other three come from the angles where it is negative, since a negative radius is plotted in the opposite direction.

How many petals

Put the fraction in lowest terms first. The count is n when n·d is odd and 2n otherwise, and the formula repeats after d turns of θ. So 5/3 draws five petals over three turns, and 7/4 draws fourteen over four.

That rule is why the fraction is worth having. A whole number is the case d = 1, which draws k petals for odd k and 2k for even. The counts it reaches are 1, 3, 4, 5, 7, 8, 9, 11, 12 and onward, every even one a multiple of four. Six, ten and fourteen petals are out of reach of any whole number, and halves draw all three. Turns of θ opens at 12 so that most fractions the two sliders can make have room to close; one with a larger denominator leaves an unfinished sweep instead.

History

Guido Grandi studied these curves between 1723 and 1728 and called them rhodonea, after the rose.

Try

  • Drag d to 1 for the ordinary whole number rose: three petals, closing in a single turn. Take n to 10 from there and the figure is the preset Rose 20.
  • Drag n to 5: ten petals over two turns, another count no whole number can draw.
  • Drag n to 3.1. In lowest terms that is 31/20, sixty two petals needing twenty turns, so the rose stops unfinished until Turns of θ is raised to 20.
  • Drag n to 0: the cosine is 1 at every angle and the rose collapses to a circle.
  • Turn on trace and watch the pen cross the origin between one petal and the next.

Read more

Polar · r = f(θ, t)