chess board equations

I got the idea from V.I. Arnold’s Real Algebraic Geometry, where he multiplies lines together (e.g. x-4=0 and y-4=0 \To (x-4)(y-4)=0 ) then increases the RHS above 0 to create disconnected shapes. Can make pretty much any shape that way. So immediately I thought of chess boards.
Well, this was my first effort in the area:
sunflower
Then chess boards.
\prod\limits_{n=1}^9 (x-n)\prod\limits_{n=1}^9 (y-n)=0 gives the “traditional” square grid; increasing the RHS from 0 makes the squares gradually circular, eventually the central ones disappear. (I used Mac Grapher for these pictures; it has problems drawing it accurately, even on highest settings.)
pcb1000
pcb5000
pcb10000
pcg20000
10000 (although I haven’t tried to play on it yet!) seems a good balance between craziness and playability:
justboard
Or in coloured form:
justboard
Well that looks awful; I turned the original into b&w which messed up the smoothing. Will do a better version. But you get the idea.


Another try with some border lines added; I thought it might slow down the (possibly excessive) circlification in the centre:
boardv2
I would have made them closer together, but Grapher can’t draw that at all.


Next: Why not just use circles. Or those squarish circles x^c+y^c=1 with c>2?

references

V.I. Arnold – Real Algebraic Geometry, p.41

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