Rose (mathematics)
From Free net encyclopedia
In mathematics, a rose is a sinusoid plotted in polar coordinates. Up to similarity,
- <math>\!\,r=cos(k\theta)</math>
One obtains a rose-like graph with <math>2k</math> petals if <math>k</math> is even and <math>k</math> petals if <math>k</math> 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 <math>\frac{\pi}{2}</math> radians—or that <math>\sin(kt + \frac{\pi}{2}) = \cos(kt)</math>, and the graphs coincide.
More interesting results arise when <math>k</math> is a rational. If <math>k</math> is irrational, without bounds on <math>\,\!\theta</math>, a disc results. In more detail, if <math>k</math> is irrational, the number of petals is irrational, and the only thing preventing you from a solid-appearing disc is the upper limit on <math>\,\!\theta</math>. However, if <math>k</math> can be approximated closely by a rational number with small denominator, then the curve will appear to nearly close after not too many revolutions. For example, assuming a <math>k</math> of <math>\pi</math>, a <math>\,\!\theta</math> limit of 2520 degrees (14<math>\pi</math> radians) will give a nearly closed circle, as <math>\pi</math> is approximately 22/7, and so at <math>\theta = 14\pi</math>, r evaluates to <math>cos(14\pi^2)</math>; we have <math>14\pi^2 \approx 44\pi</math> and so <math>cos(14\pi^2)</math> is very near <math>cos(0) = 1</math>.