# Newton's cubic curve [closed]

This article from The Guardian discusses Isaac Newton's investigation of the properties of the curve $$x^3 – abx + a^3 – cy^2 = 0$$, where $$a, b$$ and $$c$$ are constants. The image above is the curve of this equation when $$a = 1, c = 4$$ and $$b$$ ranges from –8 to 8.

The image seemsextremely aesthetic to me , almost artistic work. I'd have image on the wall like a poster, but with

ContourPlot[
Evaluate@Table[x^3 + b x == 4 y^2, {b, -8, 8, 1}], {x, -5, 5}, {y, -5, 5},
Frame -> None, ImageSize -> 600,  ContourStyle -> Black, PlotPoints -> 20]


I can not reproduce that image. The result is a bit different. Can someone please explain what is wrong with my code?

## closed as off-topic by MarcoB, Henrik Schumacher, Bob Hanlon, AccidentalFourierTransform, JohuOct 10 '18 at 6:31

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• Well, you seem to have done your substitutions incorrectly. The correct equation you should plot is x^3 - b x - 4 y^2 + 1 == 0. You are missing a constant, and the b x term seems to have the wrong sign. If you change the equation to the one I showed, the plots is much more similar to the one in the link. You might also want to restrict the range of x to e.g. {x, -5, 3.5}. – MarcoB Oct 9 '18 at 16:26
• I'm sorry, I do not know how I managed to make such a ridiculous mistake. – Darko G Oct 9 '18 at 17:07

You are plotting wrong equation, See this.

a = 1; c = 4;
ContourPlot[
Evaluate@Table[x^3 - a*b *x + a^3 - c*y^2 == 0, {b, -8, 8, 0.5}], {x, -3.,
3.}, {y, -3.5, 3.5}, Frame -> None, ImageSize -> 600,
ContourStyle -> Directive[Thickness[0.0015],PlotPoints -> 20]


I have changed the step size of $$b$$ and plot range of axis to mimic with the figure in that link.

• You can maintain sharpness while increasing the number of steps ({b, -8, 8, 1/2}) by using ContourStyle -> Directive[Thin, Black] – Bob Hanlon Oct 9 '18 at 17:14
• @BobHanlon, Modified, Thank you – math Oct 9 '18 at 18:25