The Clover and the Trefoil as Groups

An elliptic curve is an algebraic curve of genus one, and, as such, it carries the structure of an abelian group. In this article, we consider the operation on four real algebraic curves: the trihyperbola, a cubic curve; the Kiepert trefoil, a compact curve with six degrees; the clover, a twelfth-degree compact curve; and the inverted clover, which has six degrees. All four curves are members of the family of sinusoidal spirals. Our main theorem is that the four curves are all naturally isomorphic, i.e., birationally equivalent. For each curve, we consider the geometric construction of the group operation and the algebraic computation. The trefoil turns has a geometric structure involving intersections with circles. The inverted clover has an operation involving tangent lines. The operation on the clover involves tangent circles.