## Tuesday, October 4, 2011

### Pretty Curve with Mathematics: Rose

The rose curve looks like a petalled flower! Sometimes it is also called the rhodonea curve. It was discovered by Guido Grandi, an Italian mathematician. It is a really simple curve. It is just a regular sine curve (what a wave looks like) plotted on a polar graph.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com
Now what's a polar graph? A regular graph has two axes - one horizontal (X) axis and a vertical (Y) axis. A point on this (XY) graph is represented as the combination of two values (x, y) value. To reach such a point you walk x distance on the horizontal (X) axis and then walk y distance parallel to Y axis. A polar graph, on the other hand, describes points as the combination of an angle (θ) and distance. (r). How to reach a point on a polar graph? You start facing east, turn anti clockwise by angle θ and then walk r distance.

On a polar curve, if we change the distance (r) for every angle, we get some interesting plots. Rose curve is one such plot where the distance for any angle is related to the angle through a wave function. That is, as the angle changes, the distance increases gradually to a maximum value and then decreases to a minimum. Depending on how fast the wave function oscillates (pitch or frequency), we get different numbers of rise and fall.

Below is an interactive applet illustrating the rose curve (r = cos(slider1 * θ / slider2)). It does create some beautiful flower patterns. Pause the animation (button at bottom left corner) and change the sliders yourself to observe how the flower changes pattern. Particularly observe the following two curious behaviors:
• Set the second slider to 1. Change the first slider and count the number of petals. Do you see a pattern? Number of petals is same as the number when the number is odd. But it is double of the number when the number is even.
• Set both the sliders to the same value. It's a circle!
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

We can do many more interesting patterns by playing around with this. In another example below, the value is determined by applying the wave function twice (r = cos(k * sin(θ))
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Another interesting pattern made using the Rose curve is the Maurer Rose. In addition to the plain rose curve, it has straight lines joining two points on the curve at a certain distance. It gives an interesting 3D pattern to the curve. Play with the interactive applet below to create some interesting Maurer Roses. Set the thickness to a larger value to see the Rose curve on which the Maurer Rose is based on.

Rose curves are used in practice to describe shapes of various electrical and magnetic fields.

Few other points to ponder on:
• Imagine why the number of petals changes with change of wave frequency.
• Experiment with the sliders of the Maurer Rose applet above, particularly the one marked D. Looking at the pattern, can you tell when D is a prime number?