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.

All roses display one or more forms of symmetry due to the underlying symmetric and periodic properties of sinusoids.

A rose with k = 3 is called a trifolium[9] because it has k = 3 petals and will form an equilateral triangle.

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 rose is specified as[17] The Dürer folium is also a trisectrix, a curve that can be used to trisect angles.

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).

Roses specified by the sinusoid r = cos( kθ ) for various rational numbered values of the angular frequency k = ⁠ n / d ⁠ .
Roses specified by r = sin( kθ ) are rotations of these roses by one-quarter period of the sinusoid in a counter-clockwise direction about the pole (origin). For proper mathematical analysis, k must be expressed in irreducible form.
Artistic depiction of roses with different parameter settings
The rose r = cos(4 θ ) . Since k = 4 is an even number, the rose has 2 k = 8 petals. Line segments connecting successive peaks lie on the circle r = 1 and will form an octagon . Since one peak is at (1,0) the octagon makes sketching the graph relatively easy after the half-cycle boundaries (corresponding to apothems) are drawn.
The rose specified by r = cos(7 θ ) . Since k = 7 is an odd number, the rose has k = 7 petals. Line segments connecting successive peaks lie on the circle r = 1 and will form a heptagon . The rose is inscribed in the circle r = 1 .