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Crocheted Surface shown here is entirely hyperbolic, made of negative Gauss Curvature patches.

The Manifold smoothly grows unbounded exponentially into $\mathbb R^3$ ... as any one watching it gradually evolve by Crochet needles making new rings of area added on to the outer boundary... perhaps would agree.

David Hilbert's theorem asserts that there is no complete regular surface $ S $ of constant negative gaussian curvature $K$ immersed in $ \mathbb {R} ^{3}$ and that to me means that a growing patch must eventually encounter cuspidal edges with infinite curvature defining a Ridge.

So, one could be logically lead a view that corals crocheted in a similar form from yarn can never have constant negative K but should necessarily compromise to variable $K$... else there would be a contradiction.

Is this conclusion correct?

Partially at least it seems me to reinforce an unstated implication why a parametrization for such type of coral surfaces could as yet not be found.

Coral Reef Geometry TED is another relevant reference.

Narasimham
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  • I think the curvature is constant on a finite region, but the region cannot be extended because it reaches a ridge. Look at the tractricoid: Near the ridge, the normal curvature in one direction approaches $0$, while the normal curvature in the other direction approaches $\infty$, so their product $K$ is constant. (Of course it's undefined at the edge.) This restriction of constant $K$ is exactly what forces the ridge to appear. With variable $K<0$, we have for example $z=xy$ or $x^2+y^2-z^2=1$, which extend infinitely with no ridges. – mr_e_man Sep 26 '19 at 19:47
  • The discussion is only about a constant $K=-1$ attainability for the crochet, assuming for sake of this argument a sufficiently fine wool used. – Narasimham Sep 27 '19 at 05:17

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