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Biodiversity: The Computational Pumpkin

Posted 3 years ago
3 Replies
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3 Replies
Posted 3 years ago

As a sort of addendum to Jofre's lovely excursion into pumpkin-adjacent miscellanea, here's a parametric equation I made up a number of years ago to simulate a pumpkin:

With[{p = 1, r = 1/20, b = 2/3}, 
     ParametricPlot3D[Append[((p + r) {Cos[u], Sin[u]} -
                              r Exp[-v^2] Through[{Cos, Sin}[(p + r) u/r]])
                             b^2 Cos[v], b Sin[v]]/(b^2 Cos[v]^2 + Sin[v]^2),
                     {u, 0, 2 ?}, {v, -?/2, ?/2}, Exclusions -> None, 
                     Lighting -> "Neutral", Mesh -> None, PlotPoints -> 75, 
                     PlotStyle -> Directive[Orange, Specularity[1, 20]]]]

pumpkin-like surface

(You can add a stem yourself, if you wish.)

The observant will notice that the surface was built-up from a hippopede and an epicycloid.

For the chemically minded, there is a family of molecules whimsically named the cucurbiturils, in part because the shape of these molecules reminded their discoverers of a pumpkin. As the default rendering done by Molecule[]/MoleculePlot3D[] from the molecule's SMILES string leaves something to be desired, I'll instead import the model featured on the Molecule of the Month webpage:



This particular member is called cucurbit[6]uril, as there are six glycoluril units involved in the assembly. Higher and lower oligomers of it are known.

(As an aside, the molecule featured in the MoleculeViewer home page is a lower oligomer, cucurbit[5]uril.)


Wonderful presentation Jofre! Also thanks for the info :D

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