a far more complete gallery (in some respects at least) can be found at

Steiner Surface
Given by an implicit degree four equation in three variables. This is the Steiner surface of type 4.
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Decomposed with respect to random projections. The red line is a singular embedded curve.
Nordstrand's Weird Surface
An interesting surface, with 120° rotational symmetry about the axis passing through the small cone. A surface generated by a single degree four equation in x,y,z.
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Banchoff 3-3-3
The Banchoff class of systems are defined by a single function in three variables, where the function is generated by taking the sum of three Chebyschev polynomials (each in x, y, or z) of specified degree, and setting it equal to 0.
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Raw unrefined decomposition
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Critical curve of Banchoff
The classical object, and the first decomposed by Bertini_real.
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One of the first genus>0 objects studied. We also play lots of video games on these!
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A system with two holes and two singular curves, one equation in three variables. The red line is one of the singular lines -- it passes for an interval outside the surface itself. One singularity is a self-crossing, the other is a cusp.
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