To celebrate Halloween, last year we discussed what you can do with 1,818 pounds of pumpkin. It was a popular blog post, and it put an awful lot of smiles on peoples’ faces. An entire lamina (filled shape) of smiles, in fact.

smiley equation

We at Wolfram|Alpha were recently inspired to make a lamina out of everything possible. That said, Halloween is—to me, at least—about making jack-o’-lanterns. Before we can delve into jack-o’-lantern laminae, first you need to find a quality pumpkin. One that, if it were a curved parametric surface made by extending the epicycloid into 3D via spherical coordinates, would look like this:

pumpkin surface

A typical jack-o’-lantern usually has triangles for eyes and a nose and angled teeth (because angles are scary). For the jack-o’-lantern lamina, we give closed forms for the area, centroid, and area moment of inertia tensor, which is sure to earn us some street cred.

filled jack-o-lantern face

Of course, some people prefer to avoid that kind of imagery altogether, opting instead for something a bit sillier. One of Wolfram’s most talented mathematicians/artists put together a pirate jack-o’-lantern lamina that we simply had to share with you.

pirate face equation

If you want to check out how to make a jack-o’-lantern of your own using other Wolfram technology, I strongly recommend Yu-Sung Chang’s blog post “Industrial Pumpkin Carving with Mathematica.” It’s just terrific, and you should read it.

We here at Wolfram|Alpha hope you have a happy Halloween! And stay tuned to our blog, because we have a whole series of new shapes, curves, and laminae to show off coming soon.

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