We get negative Gauss curvature K surfaces of quasi polar symmetry surface form generated from:

$$ Re (x+ i y)^n = a^n $$ (n integer) with n humps above plane $ z =0$.

($ n =2,3,4 $ hyperbolic paraboloids, monkey saddles, four humped frill/ pleated shells respectively).

"Quasi" due to $n>2$ introducing circumferential pleats or frills.

What polynomial or differential relation function yields constant negative surfaces of $K = -1/a^2 $?

How to set up its differential equation?

Instead of $ z^n $ if a simpler $ \cos z $ is taken, no clue or handle how to modify the function in order to get a constant negative$ K. ( \cos x \cosh y = $ const)

Z= cos_x cosh_y

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    $\begingroup$ Embeddings of portions of the hyperbolic plane into Euclidean 3-space has been classically studied. mathoverflow.net/questions/533/… gives some references. $\endgroup$ – Willie Wong Oct 26 '15 at 20:14
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    $\begingroup$ There is no unique mapping it appears. About the Coral ruffles (in comments of Oct 15 above) ,I sent a message to Prof. Daina Taimina sometime back, says there is no known parameterization. $\endgroup$ – Narasimham Oct 26 '15 at 21:12

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