## Leaves from the scrapbook-2

As described here these entries are from the scrapbook of Somakhya.

Pinakasena was also doing stuff in preparation for that famous mathematics competition that Mandara was studying for. He raised the question of describing all the following curves with single equation in a one parameter space: straight line, circle, parabola, hyperbola, cardioid, lemniscate, tri-lemniscate, tri-hyperbola, 3-flower etc. Because of the hours we had spent pondering about these curves we were able to give him that right away as the following polar equation with a single parameter a:
$\rho=\left(\cos \left(a\cdot \theta \right)\right)^{\frac{1}{a}}$

When a=1 it is a circle; a=-1, a line; a= 1/2, a cardioid; a=-1/2, a parabola; a= 2, a lemniscate; a=-2, a hyperbola; a=3/2, a 3-flower; a=3, a tri-lemniscate; a=-3/2 a tri-hyperbola with $120^o$ asymptotes; a=-3, a tri-hyperbola with $60^o$ asymptotes and so on (the n-gon conics). In general if $a=\frac{p}{q}$ where p and q are mutually prime integers then it is a curve of p lobes or branches. If $\frac{p}{q}$ is negative then it is a diverging curve and if it is positive it closes with maximal radius of a unit. If $\frac{p}{q}; p=1, q>1$ then the curve internally loops with the number of crossovers being the floor of the square root of q.

Both Indrasena and Pinakasena were sort of cursing themselves that it could be so easy. I pointed out that it was indeed the easy part and pulled out the theorem of the intersection of three ellipses with shared foci on a triangle and informed the upātreya that it was the least of the questions that the mahārathin-s were supposed to surmount in such competitions.

Prove the lines passing through the points of intersections of the three ellipses with foci on a triangle are concurrent.

Another was to double the cube as the yavana-s did to Apollo with ellipses. Then I assuaged him that there was no point struggling for such competitions – if one was truly a mahārathin one would know it and competitions should not matter. If one were not a mahārathin one should study such things just for mental entertainment or knowledge acquisition and play those games as a professional of which one is a master.

Indrasena then revealed to me how he had figured out a way to find genes that had really undergone selection in different Hindu populations. We looked at the genes he identified for sometime and thought about what they might imply.