The curves with polar equation ρ = cos(nθ) (0 ≤ θ ≤ 2π) are also known as Grandi’s roses, in honor of Guido Grandi who communicated his discovery to Gottfried Wilhelm Leibniz in 1713. Curves with polar equation ρ = sin(nθ) (0 ≤ θ ≤ 2π) are equivalent to the preceding ones, up to a rotation of π/(2n) radians.

As can be seen in Figure 10, Grandi’s roses display n petals if n is odd and 2n petals if n is even. By using these polar equations it is impossible to obtain roses with 4n + 2 (n ∈ N ∪ {0}) petals. Roses with 4n + 2 petals can be obtained by using the Bernoulli Lemniscate and its extensions. More precisely:

• The trigonometric function y=cos2x  (−π4+kπ≤x≤π4+kπ) becomes the so-called Bernoulli Lemniscate ρ=cos1/2(2θ) (−π4+kπ≤θ≤π4+kπ)   (k ∈ N) that is a rose with two petals (Figure 11).

• The functions y=cos(4n+2)x  (n>1)  (−π4(2n+1)+kπ2n+1≤x≤π4(2n+1)+kπ2n+1) become the polar equations ρ=cos1/2[(4n+2)θ]  (−π4(2n+1)+kπ2n+1≤θ≤π4(2n+1)+kπ2n+1) (k ∈ N) which give roses with 4n + 2 petals.

Figure 10

Rhodonea cos(2θ) and cos(5θ).

Figure 11

Bernoulli Lemniscate.

A few graphs of Rhodonea curves with fractional indices are shown in Figures 12–14.

Figure 12

Rhodonea cos(pq θ).

Figure 13

Rhodonea cos(14 θ) and cos(54 θ).

Figure 14

Rhodonea cos(18 θ) and cos(38 θ) .