The permutation $(-2,9,-4,7,-6,5,-8,3,1)$ can be considered magical. With their negative values diametrical to $0$ at $(0,0)$, a placement of integers begins so that all zero-sum triples form straight lines.
The infinite sequence of integers is a series of placements on the given curve: $(1,-1,2,-2,3,-3,3,-6,2,-9,...)$ for values $1$ to $10$.
$(1)$ at place $1$. Always pick the branch with 1. Position $(-35, -21)$.
$(-2,1)$ at place $-1$, indicating negative, position $(-7, -7)$.
$(-2,3,1)$ at place $2$ and position $(-21, -7)$.
$(-2,-4,3,1)$ at place $-2$ and position $(0, -14)$.
$(-2,-4,5,3,1)$ at place $3$ and position $(-35, 7)$.
$(-2,-4,-6,5,3,1)$ at place $-3$ and position $(35, -21)$.
$(-2,-4,7,-6,5,3,1)$ at place $3$ and position $(-21, 21)$.
$(-2,-4,7,-6,5,-8,3,1)$ at place $-6$ and position $(28, 0)$.
$(-2,9,-4,7,-6,5,-8,3,1)$ at place $2$ and position $(5, 11)$.
$(-2,9,-4,7,-6,5,-8,3,-10,1)$ at place $-9$ and position $(21,9.8)$.
Here's a larger example based on curve $4 x - 4 x^3 + 27 y - 3 x^2 y + 6 x y^2 - 3 y^3=0$. The value $7$ is outside of the frame of this snapshot at position $(7,4)$. Blue and Red indicate positive and negative.
More of these curves are at Infinite Magic Elliptic Curves.
- Are all such magic cubic curves symmetric?
- Are all such magic cubic curves connected curves?
- Is there a good way to generate these curves?
- Is the number of distinct placement series finite or infinite? If finite, how many?
- The second curve above places values -34 to 34 so that the spacing between them is fairly uniform. Are there curves with greater uniformity over a range of values? Here's a uniform representation for $-8$ to $8$ based on curve $\sqrt{3} y \left(x^2-3 y^2+48\right)+x \left(5 x^2-3 y^2-60\right)$. In terms of the placement series, this is actually the same as the first one.
UPDATE: William R. Somsky sent me a disconnected asymmetric curve.
