Materials and Methods: Knitted and sewn

Designer & Artist: Susan Schmoyer

Year: 2013--2014

Complex elliptic curves admit a both a description as an
equation of the form *y*^{2} = 4*x*^{3}-*ax*-*b* and as a torus. On the equation
side, we can consider solutions in rational numbers *x*, *y*.
This piece shows the generators for all rational solutions
for the elliptic curve, *y*^{2} = 4*x*^{3} - (6750/49)*x* - (-13500/49), as points sewn onto
the corresponding torus. The set of all rational solutions is isomorphic to **Z**_{2} + **Z** + **Z**.

Let *A*
represent the generator of order 2; let *B* and *C* denote the generators of infinite order.
We invert the Weierstrass parametrization, *wp*(*z*),
to convert *A, B*, and *C* into points on the torus, called *A'*, *B'*, and *C'* (see below).

The point *A'* is sewn onto the torus as a dark grey point (see below-er).

The point *B'* and several of its integer multiples are sewn on as magenta points.

The point *C'* and several of its integer multiples are sewn on as purple points.

The origin is depicted as a light grey point.