It was said such real objects of famous topological surfaces were plotted by the cooperation between mathematician Richard Palais and artist Luc Benard, and published as the cover page of Science magazine (issue on Sept 22 in 2006):
The cover page of Science, Sept 22, 2006
How to create such surfaces with the same color, rendering, lighting and opacity effects via Mathematica?
(answers with only similar rendering results for these surfaces one by one are also acceptable if the same rendering effects as the samples are too difficult to realize)
Another similar question with anwser can be seen from this link: another topological surface example with answer
Surface information are : Klein bottle; symmetric 4-noid; breather surface; Boy surface; Sievert-Enneper surface.
It was said these surfaces in the figure were created via 3D-XplorMath software.
EDITS:
unfortunately it is voted as on hold
; but I personally believe Mathematica's rendering should be powerful enough to realize such effects( at least more powerful than the open source 3D-XplorMath The author on MathOverflow); Here I add two of the surfaces obtained and rendered which are already good enough;
There are documents on these surfaces from this link: documents on surfaces
There are five different surfaces; I have found four of them; but only three as below look beautiful; so I will not post others here;
Clear["Global`*"]; (* Sievert-Enneper surface *)
ϕ := -u/Sqrt[c + 1] + ArcTan[Tan[u] Sqrt[c + 1]]
a := 2/(c + 1 - c Sin[v]^2 Cos[u])
r := a Sqrt[(c + 1) (1 + c Sin[u]^2)] Sin[v]/Sqrt[c]
Clear[x, y, z, p]; p =
ParametricPlot3D[{x = r Cos[ϕ] - 2, y = r Sin[ϕ],
z = ((Log[Tan[v/2]] + a (c + 1) Cos[v])/Sqrt[
c])} /. {c -> .4}, {u, -Pi/2, Pi/2}, {v, 0, Pi},
PlotPoints -> {30, 30},
PlotStyle ->
Directive[Opacity[0.65], LightPurple, Specularity[White, 20]],
Mesh -> None, RegionFunction -> Function[{x, y, z}, Abs[z] < 2.2],
Axes -> False, Boxed -> False]
Graphics3D
rendering is Gouraud shading. $\endgroup$