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How might one manage to find the surface area and volume of the following constructs?

Rotini: A geometric figure I've found to be similar is the helicoid. However, the helicoid (having been formed from a 2D plane) has an edge width equal to epsilon, whereas the edges of a fully 3D rotini-like construct would be roughly catenarian in shape. 

Fusillibucati/Cavatappi: The only difference between the two is length. Cavalieri's principle could possibly be at play here, but the topography could potentially suggest otherwise. 

Campanelle: No hints for this one, good luck. 

Casarecce/Gemelli: The only difference between them is the topographic helical pitch. Cavalieri's principle is more than likely at play here, but you might have to use it a bit differently. 

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Is this research driven by the need to find the specific amount of sauce to coat the pasta ? :)
I suggest next, you look at density/viscosity of different sauces to achieve the same goal. :D

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1 hour ago, MigL said:

Is this research driven by the need to find the specific amount of sauce to coat the pasta ? :)
I suggest next, you look at density/viscosity of different sauces to achieve the same goal. :D

It was more a thought experiment than anything, but that is a good idea!  : )

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Had pasta ( rotini ) in a tomato/meat sauce with Parm and chili flakes sprinkled on top, for supper.
It certainly was a good idea; for real, not a thought experiment.

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58 minutes ago, MigL said:

Had pasta ( rotini ) in a tomato/meat sauce with Parm and chili flakes sprinkled on top, for supper.
It certainly was a good idea; for real, not a thought experiment.

Sounds delightful! I have two recommendations; you could make a chili-lime seasoning with the chili flakes, or instead you could replace chili with cayenne and add some garlic salt.

 

I wouldn't recommend both simultaneously, however... Lime and garlic salt? 🤢

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Lime and garlic salt works really well, if you have something to bind it together. Like in a sauce, rather than just seasoning. Often the tomato delivers enough acidity though a sprinkle of light vinegar or citrus can brighten it up. Or you can make it denser with aceto balsamico.

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5 hours ago, CharonY said:

Lime and garlic salt works really well, if you have something to bind it together. Like in a sauce, rather than just seasoning. Often the tomato delivers enough acidity though a sprinkle of light vinegar or citrus can brighten it up. Or you can make it denser with aceto balsamico.

That, is actually brilliant. Maybe tomato-hachiya sauce with a little corn vinegar.

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