# Rose (mathematics)

In mathematics, a rose is a sinusoid plotted in polar coordinates. Up to similarity,

[itex]\!\,r=cos(k\theta)[itex]

One obtains a rose-like graph with [itex]2k[itex] petals if [itex]k[itex] is even and [itex]k[itex] petals if [itex]k[itex] is odd. Assuming you use the given form, the whole rose will appear inside a unit circle. Using sine instead of cosine, and vice versa, the graphs differ by a rotation of [itex]\frac{\pi}{2}[itex] radians—or that [itex]\sin(kt + \frac{\pi}{2}) = \cos(kt)[itex], and the graphs coincide.

More interesting results arise when [itex]k[itex] is a rational. If [itex]k[itex] is irrational, without bounds on [itex]\,\!\theta[itex], a disc results. In more detail, if [itex]k[itex] is irrational, the number of petals is irrational, and the only thing preventing you from a solid-appearing disc is the upper limit on [itex]\,\!\theta[itex]. Assuming a [itex]k[itex] of [itex]\pi[itex], a [itex]\,\!\theta[itex] limit of 2520 degrees (14[itex]\pi[itex] radians) will give you the first complete circle.

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