[hutchisonc]

For what it’s worth, in the math context (not thinking about applications), the group law is extremely natural. Every variety has an associated group called the Picard group tells you something about geometry of the variety. But for elliptic curves, it turns out there is a bijection between the complex points on the curve and the elements of (the degree 0 subgroup of) its Picard group, so it inherits the group structure this way. This is the same group structure as the usual one defined explicitly. I might write more about this when I get home.

[xyzzyz]

To expand a little on top of that, you can think of Picard group of a variety as a group of linear combinations of codimension 1 subvarieties. For a 9-dimensional variety, these will be a dimension 8 subvarieties. To connect this huge free group to geometry of the original variety, we introduce certain constraint equations, i.e. we set certain linear combinations to equal to zero. These zero combinations are set to be those combinations that are obtained by intersecting the variety with a (codimension 1) hyperplane: if you take an embedded n-variety, and intersect it with n-1 dimensional plane, you typically get a result that’s a finite union of n-1 subvarieties. One can specify assignment of intersection numbers to each of these varieties, so that eg if a hyperplane intersects a variety “cleanly across” (transversally), the intersection number (and so the coefficient in the constrain equation) is 1, whereas if the intersecting hyperplane is tangent along the intersection, the intersection number will correspond to the degree of tangency (eg. the line y = 0 intersects the parabola y = x^2 at the origin with the intersection number (tangency degree) equal to 2).

Now, if the variety in question is a curve, the codimension 1 subvarieties will be of dimension 0, that is, finite sets of points. Moreover, if the curve is of degree 3, then hyperplanes (lines) will intersect it in exactly 3 points (counting the tangent intersections properly). Thus, we will get a bunch of constraints of the form:

P + Q + R = 0

This makes our huge group of linear combinations into rather simple group of points: take two points, P and Q. Run line across them, take third point of intersection: this is the negative of the sum P + Q. This is the procedure shown on the animations of the OP.

The point of this is that none of this is arbitrary: it’s just a lucky coincidence that happens only in dimension 1 and degree 3. One can introduce group structure on some other complex curves (which will actually look to us like surfaces), but it is not nearly as straightforward.

[tptacek]

Yes please.

[jacobolus]

If elliptic curves seem tricky, you can follow the same idea with conics https://arxiv.org/pdf/math/0311306.pdf https://www.researchgate.net/profile/Shailesh-Shirali/public...