27 Lines on a Cubic Surface

Here is an interactive visualization of the 27 lines on a Clebsch diagonal cubic surface rendered:

Click button to open the graphic in a separate full window. The figure can be rotated by clicking and dragging with a mouse or swiping on a touch device, as well as zoomed in or out with a mouse wheel or pinching on a touch device.

As stated in the post, the isosurface is defined by

$\begin{array}{l} 81 (x^{3} + y^{3} + z^{3}) - 189 (x^{2} y + x^{2} z + x y^{2} + x z^{2} + y^{2} z + y z^{2}) \\ + 54 x y z + 126 (x y + x z + y z) - 9 (x^{2} + y^{2} + z^{2}) - 9 (x + y + z) + 1 = 0 \end{array}$