Rose (mathematics)

In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728.
A rose is the set of points in polar coordinates specified by the polar equation
or in Cartesian coordinates using the parametric equations
Roses can also be specified using the sine function. Since
Thus, the rose specified by r = a sin(kθ) is identical to that specified by r = a cos(kθ) rotated counter-clockwise by ⁠π/2k⁠ radians, which is one-quarter the period of either sinusoid.
Since they are specified using the cosine or sine function, roses are usually expressed as polar coordinate (rather than Cartesian coordinate) graphs of sinusoids that have angular frequency of k and an amplitude of a that determine the radial coordinate r given the polar angle θ (though when k is a rational number, a rose curve can be expressed in Cartesian coordinates since those can be specified as algebraic curves).
Roses are directly related to the properties of the sinusoids that specify them.
All roses display one or more forms of symmetry due to the underlying symmetric and periodic properties of sinusoids.
When k is a non-zero integer, the curve will be rose-shaped with 2k petals if k is even, and k petals when k is odd. The properties of these roses are a special case of roses with angular frequencies k that are rational numbers discussed in the next section of this article.
A rose with k = 1 is a circle that lies on the pole with a diameter that lies on the polar axis when r = a cos(θ). The circle is the curve's single petal. (See the circle being formed at the end of the next section.) In Cartesian coordinates, the equivalent cosine and sine specifications are
and
respectively.
A rose with k = 2 is called a quadrifolium because it has 2k = 4 petals and will form a square. In Cartesian coordinates the cosine and sine specifications are
and
respectively.
A rose with k = 3 is called a trifolium because it has k = 3 petals and will form an equilateral triangle. The curve is also called the Paquerette de Mélibée. In Cartesian Coordinates the cosine and sine specifications are
and
respectively. (See the trifolium being formed at the end of the next section.)
A rose with k = 4 is called an octafolium because it has 2k = 8 petals and will form an octagon. In Cartesian Coordinates the cosine and sine specifications are
and
respectively.
A rose with k = 5 is called a pentafolium because it has k = 5 petals and will form a regular pentagon. In Cartesian Coordinates the cosine and sine specifications are
and
respectively.
A rose with k = 6 is called a dodecafolium because it has 2k = 12 petals and will form a dodecagon. In Cartesian Coordinates the cosine and sine specifications are
and
respectively.
The total area of a rose with polar equation of the form r = a cos(kθ) or r = a sin(kθ), where k is a non-zero integer, is
When k is even, there are 2k petals; and when k is odd, there are k petals, so the area of each petal is ⁠πa/4k⁠.
As a consequence, if someone wanted to play the popular game He loves me... he loves me not on a rose like above, instead of counting the petals they could calculate the area of the rose to determine the result of the game.
In general, when k is a rational number in the irreducible fraction form k = ⁠n/d⁠, where n and d are non-zero integers, the number of petals is the denominator of the expression ⁠1/2⁠ − ⁠1/2k⁠ = ⁠n − d/2n⁠. This means that the number of petals is n if both n and d are odd, and 2n otherwise.
A rose with k = ⁠1/2⁠ is called the Dürer folium, named after the German painter and engraver Albrecht Dürer. The roses specified by r = a cos(⁠θ/2⁠) and r = a sin(⁠θ/2⁠) are coincident even though a cos(⁠θ/2⁠) ≠ a sin(⁠θ/2⁠). In Cartesian coordinates the rose is specified as
The Dürer folium is also a trisectrix, a curve that can be used to trisect angles.
A rose with k = ⁠1/3⁠ is a limaçon trisectrix that has the property of trisectrix curves that can be used to trisect angles. The rose has a single petal with two loops. (See the animation below.)
A rose curve specified with an irrational number for k has an infinite number of petals and will never complete. For example, the sinusoid r = a cos(πθ) has a period T = 2, so, it has a petal in the polar angle interval −⁠1/2⁠ ≤ θ ≤ ⁠1/2⁠ with a crest on the polar axis; however there is no other polar angle in the domain of the polar equation that will plot at the coordinates (a,0). Overall, roses specified by sinusoids with angular frequencies that are irrational constants form a dense set (that is, they come arbitrarily close to specifying every point in the disk r ≤ a).