Clebsch Cubic

Differential Equations, Mechanics, and Computation
clebsch cubic
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A classical result from algebraic geometry says that on every smooth cubic surface there are 27 lines (possibly complex). Clebsch Cubic is a surface with 3-fold symmetry that has all 27 real lines. Its function is:

81*(x^3 + y^3 + z^3) - 189*(x^2*y + x^2*z + y^2*x + y^2*z + z^2*x + z^2*y) + 54*(x*y*z) + 126*(x*y + x*z + y*z) - 9*(x^2 + y^2 + z^2) - 9*(x + y + z) + 1

Clebsch Cubic surface
Another view of the Clebsch cubic, showing its 3-fold symmetry.

Mathematica file

clebsch_cubic.gcf

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