I find a lot of motivation from topology. If you plot a smooth degree d curve over the complex numbers, it forms a surface of degree g=(d-1)(d-2)/2. In the case of a cubic, we get genus 1, i.e. a torus. Now tori admit a very natural group action, namely addition in (R/Z)^2. And sure enough, if you pick the right homeomorphism, this corresponds to the group action given by the elliptical curve.
Of course, the homeomorphism to (R/Z)^2 does not respect the geometry (it is not conformal). If we want the map to preserve angles, we need our fundamental domain to be a parallelogram instead of a rigid square. The shape of the parallelogram depends on the coefficients of the cubic, and the isomorphism is uniquely defined up to choice of a base point O (mapping to the identity element; for elliptic curves, this is normally taken to be the point at infinity). You still get a group law on the parallelogram from vector addition in the same way, and this pulls back to the precise group action on the elliptic curve.
The real magic is that the resulting group law is algebraic, meaning that a*b can be written as an algebraic function of a and b. This means you can do the same arithmetic over any field, not just the complex numbers, and still get a group action.
Rooks don't have taxicab geometry. Their metric space is compact even on an infinite board. I think you're thinking of the wazir: https://en.wikipedia.org/wiki/Wazir_(chess)
Sure, but who's reading the conversation to determine whether it "looks suspicious"? A regex? A neural network? Who decides the algorithm, and do you really can believe they won't ever change it to serve other more nefarious purposes like suppressing dissent?
> Who decides the algorithm, and do you really can believe they won't ever change it to serve other more nefarious purposes like suppressing dissent?
YES. That's the problem. Whoever controls it has that power. We don't want that. That's the argument against ChatControl: "imagine that those who get in power are against you, and imagine what they can do if they abuse this tool".
But saying that "a law enforcement officer may see a false-positive between a parent and their child and I find this insufferable" won't convince many people, IMHO.
I am confused how we can define a rotation number of the map from S^1 to R^3 defined at the end of the second paragraph. R^3 is nullhomotopic, after all...
My preference is semi-compatible with both conventions:
First = 0
Second = 1
Toward = 2
Third = 3
…
This way, the semantic meaning of the words “first” (prior to all others) and “second” (prior to all but one) are preserved, but we get sensical indexing as well.
1. N4 N5
2. Nx6+ K7
3. R4 N3+!
4. K2 N5
5. N8! Kx8
6. Rx5#
reply