[AntonioLao]

The curl and spiral of physical reality could never be overemphasized. In its mathematical classification the spiral has 7 major varieties. The studies of special plane curves classified them as transcendental with the followings: (1) logarithmic spirals are defined as plane curves that cuts radius vectors at a constant angle f. In polar notation the equation is given as r=exp(aq) where a=cotf. These were discovered by Descartes in 1638. (2) The involute spiral of a circle discovered by Huygens in 1693 given by the parametric equations: x=a(cost+tsint) and y=a(sint-tcost) where -¥ < t < ¥. (3) Sinusoidal spiral defined as the locus of the equation r^n = a^n cosnq where n is rational and discovered by Maclaurin in 1718. (4) Euler’s spiral discovered in 1744 is defined by the parametric equations: x=±aò(sint/Öt)dt and y=±aò(cost/Öt)dt where 0 ≤ t < ¥. (5) Archimedean spirals general form was discovered in 1854 by Sacchi given by r^m = a^mq. The special case where m=1 was discovered by Archimedes in 225 B.C. Fermat discovered the case for m=2 in 1636. For m=-1 called hyperbolic spiral was discovered by Varignon in 1704. The Lituus spiral for m=-2 was discovered by Cotés in 1722. (6) Epi spirals defined by the polar equation rcosnq=a. (7) Poinsot’s spirals: rcoshnq=a and rsinhnq=a.

On the other hand, the curl is a vector function operator defined as the vector or cross or outer product of the gradient operator: Ñ=i¶/¶x+j¶/¶y+k¶/¶z and a vector V given by Vxi+Vyj+Vzk: curlV=Ñ´V= i´¶V/¶x+j´¶V/¶y+k´¶V/¶z. Together with the divergence operator Ñ× they describe the spacetime symmetry between the electric field vector and the magnetic field vector. However, this symmetry is not complete unless magnetic monopoles are also discovered.